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Glossary G-1

A

Addition Property of Equality

The addition property of equality states: “If a 5 b, then a 1 c 5 b 1 c.”

Example

If x 5 2, then x 1 5 5 2 1 5, or x 1 5 5 7 is an example of the Addition Property of Equality.

Addition Rule for Probability

The Addition Rule for Probability states: “The probability that Event A occurs or Event B occurs is the probability that Event A occurs plus the probability that Event B occurs minus the probability that both A and B occur.”

P(A or B) 5 P(A) 1 P(B) 5 P(A and B)

Example

You flip a coin two times. Calculate the probability of flipping a heads on the first flip or flipping a heads on the second flip.

Let A represent the event of flipping a heads on the first flip. Let B represent the event of flipping a heads on the second flip.

P(A or B) 5 P(A) 1 P(B) 2 P(A and B)

P(A or B) 5 1 __ 2 1 1 __

2 2 1 __

4

P(A or B) 5 3 __ 4

So, the probability of flipping a heads on the first flip or flipping a heads on the second flip is 3 __

4 .

adjacent angles

Adjacent angles are angles that share a common side and a common vertex, and lie on opposite sides of their common side.

Example

Angle BAC and angle CAD are adjacent angles. Angle FEG and angle GEH are adjacent angles.

A D

B

C

E

F G H

adjacent arcs

Adjacent arcs are two arcs of the same circle sharing a common endpoint.

Example

Arcs ZA and AB are adjacent arcs.

O

Z

B

A

adjacent side

The adjacent side of a triangle is the side adjacent to the reference angle that is not the hypotenuse.

Example

reference angleadjacent side

opposite side

altitude

An altitude is a line segment drawn from a vertex of a triangle perpendicular to the line containing the opposite side.

Example

Segment EG is an altitude of triangle FED.

D

E

F G

3 in.

8 in.

Glossary

Glossary

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Glossary

G-2 Glossary

angle

An angle is a figure that is formed by two rays that extend from a common point called the vertex.

Example

Angles A and B are shown.

A B

angle bisector

An angle bisector is a ray that divides an angle into two angles of equal measure.

Example

Ray AT is the angle bisector of angle MAH.

M

T

H

A

angular velocity

Angular velocity is a type of circular velocity described as an amount of angle movement in radians over a specified amount of time. Angular velocity can be expressed as � 5 � __

t , where � 5 angular velocity,

� 5 angular measurement in radians, and t 5 time.

annulus

An annulus is the region bounded by two concentric circles.

Example

The annulus is the shaded region shown.

r

R

arc

An arc is the curve between two points on a circle. An arc is named using its two endpoints.

Example

The symbol used to describe arc BC is � BC .

AB

C

arc length

An arc length is a portion of the circumference of a circle. The length of an arc of a circle can be calculated by multiplying the circumference of the circle by the ratio of the measure of the arc to 360°.

arc length 5 2�r ? x� _____ 360�

Example

In circle A, the radius ___

AB is 3 centimeters and the measure of arc BC is 83 degrees.

( 2�r ) ( m � BC _____ 360�

) � 2�(3) ( 83� _____ 360�

) � 4.35

So, the length of arc BC is approximately 4.35 centimeters.

AB

C

3 cm

83°

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Glossary

Glossary G-3

axis of symmetry

An axis of symmetry is a line that passes through a figure and divides the figure into two symmetrical parts that are mirror images of each other.

Example

Line k is the axis of symmetry of the parabola.

k

B

base angles of a trapezoid

The base angles of a trapezoid are either pair of angles that share a base as a common side.

Example

Angle T and angle R are one pair of base angles of trapezoid PART. Angle P and angle A are another pair of base angles.

T R

base angles

base

base

leg legbase angles

P A

bases of a trapezoid

The parallel sides of a trapezoid are the bases of the trapezoid.

Example

Line segment TR and line segment PA are the bases of trapezoid PART.

T Rbase

base

leg leg

P A

biconditional statement

A biconditional statement is a statement written in the form “if and only if p, then q.” It is a combination of both a conditional statement and the converse of that conditional statement. A biconditional statement is true only when the conditional statement and the converse of the statement are both true.

Example

Consider the property of an isosceles trapezoid: “The diagonals of an isosceles trapezoid are congruent.” The property states that if a trapezoid is isosceles, then the diagonals are congruent. The converse of this statement is true: “If the diagonals of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.” So, this property can be written as a biconditional statement: “A trapezoid is isosceles if and only if its diagonals are congruent.”

C

categorical data (qualitative data)

Categorical data are data that each fit into exactly one of several different groups, or categories. Categorical data are also called “qualitative data.”

Example

Animals: lions, tigers, bears, etc. U.S. Cities: Los Angeles, Atlanta, New York City, Dodge City, etc.

The set of animals and the set of U.S. cities are two examples of categorical data sets.

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Glossary

G-4 Glossary

Cavalieri’s principle

Cavalieri’s principle states that if all one-dimensional slices of two-dimensional figures have the same lengths, then the two-dimensional figures have the same area. The principle also states that given two solid figures included between parallel planes, if every plane cross section parallel to the given planes has the same area in both solids, then the volumes of the solids are equal.

center of a circle

The center of a circle is a fixed point in the plane that is at an equal distance from every point on the circle.

Example

Point H is the center of the circle.

H

central angle

A central angle of a circle is an angle whose sides are radii. The measure of a central angle is equal to the measure of its intercepted arc.

Example

In circle O, /AOC is a central angle and � AC is its intercepted arc. If m/AOC 5 45º, then m � AC 5 45º.

A

O

C

45°

centroid

The centroid of a triangle is the point at which the medians of the triangle intersect.

Example

Point X is the centroid of triangle ABC.

A

BC

X

chord

A chord is a line segment whose endpoints are points on a circle. A chord is formed by the intersection of the circle and a secant line.

Example

Segment CD is a chord of circle O.

OC

D

circular permutation

A circular permutation is a permutation in which there is no starting point and no ending point. The circular permutation of n objects is (n 2 1)!.

Example

A club consists of four officers: a president (P), a vice-president (VP), a secretary (S), and a treasurer (T). There are (4 2 1)!, or 6 ways for the officers to sit around a round table.

circumcenter

The circumcenter of a triangle is the point at which the perpendicular bisectors intersect.

Example

Point X is the circumcenter of triangle ABC.

X

A

C B

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Glossary G-5

Glossary

complement of an event

The complement of an event is an event that contains all the outcomes in the sample space that are not outcomes in the event. In mathematical notation, if E is an event, then the complement of E is often denoted as

__ E or Ec.

Example

A number cube contains the numbers 1 though 6. Let E represent the event of rolling an even number. The complement of Event E is rolling an odd number.

complementary angles

Two angles are complementary if the sum of their measures is 90º.

Example

Angle 1 and angle 2 are complementary angles. m�1 � m�2 � 90�

1

2

composite figure

A composite figure is formed by combining different shapes.

Example

The composite figure shown is formed by a square and a semicircle.

compound event

A compound event combines two or more events, using the word “and” or the word “or.”

Example

You roll a number cube twice. Rolling a six on the first roll and rolling an odd number on the second roll are compound events.

circumscribed polygon

A circumscribed polygon is a polygon drawn outside a circle such that each side of the polygon is tangent to the circle.

Example

Triangle ABC is a circumscribed triangle.

A

B

C

P

collinear points

Collinear points are points that are located on the same line.

Example

Points A, B, and C are collinear.

A CB

combination

A combination is an unordered collection of items. One notation for the combinations of r elements taken from a collection of n elements is:

nCr 5 C(n, r) 5 Cnr

Example

The two-letter combinations of the letters A, B, and C are: AB, AC, BC.

compass

A compass is a tool used to create arcs and circles.

Example

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G-6 Glossary

Glossary

concurrent

Concurrent lines, rays, or line segments are three or more lines, rays, or line segments intersecting at a single point.

Example

Lines �, m, and n are concurrent lines.

X

�

m

n

conditional probability

A conditional probability is the probability of event B, given that event A has already occurred. The notation for conditional probability is P(B|A), which reads, “the probability of event B, given event A.”

Example

The probability of rolling a 4 or less on the second roll of a number cube, given that a 5 is rolled first, is an example of a conditional probability.

conditional statement

A conditional statement is a statement that can be written in the form “If p, then q.”

Example

The statement “If I close my eyes, then I will fall asleep” is a conditional statement.

congruent line segments

Congruent line segments are two or more line segments that have equal measures.

Example

Line segment AB is congruent to line segment CD.

A DB C

conjecture

A conjecture is a hypothesis that something is true. The hypothesis can later be proved or disproved.

construct

A constructed geometric figure is created using only a compass and a straightedge.

concavity

The concavity of a parabola describes the orientation of the curvature of the parabola.

Example

y

concave up

x

y

concave right

x

y

concave down

x

y

concave left

x

concentric circles

Concentric circles are circles in the same plane that have a common center.

Example

The circles shown are concentric because they are in the same plane and have a common center H.

H

conclusion

Conditional statements are made up of two parts. The conclusion is the result that follows from the given information.

Example

In the conditional statement “If two positive numbers are added, then the sum is positive,” the conclusion is “the sum is positive.”

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Glossary G-7

Glossary

coplanar lines

Coplanar lines are lines that lie in the same plane.

Example

Line A and line B are coplanar lines. Line C and line D are not coplanar lines.

A B

C D

corresponding parts of congruent

triangles are congruent (CPCTC)

CPCTC states that if two triangles are congruent, then each part of one triangle is congruent to the corresponding part of the other triangle.

Example

In the triangles shown, �XYZ � �LMN. Because corresponding parts of congruent triangles are congruent (CPCTC), the following corresponding parts are congruent.

/X � /L /Y � /M /Z � /N ___

XY � ___

LM ___

YZ � ____

MN ___

XZ � ___

LN

X

Y

Z

M

L N

construction proof

A construction proof is a proof that results from creating a figure with specific properties using only a compass and straightedge.

Example

A construction proof is shown of the conditional statement: If

___ AB �

___ CD , then

___ AC �

___ BD .

A B C D

A B

C D

A B

B(AC)

(BD)

C

D C

C B

A

B D

C

contrapositive

To state the contrapositive of a conditional statement, negate both the hypothesis and the conclusion and then interchange them.

Conditional Statement: If p, then q. Contrapositive: If not q, then not p.

Example

Conditional Statement: If a triangle is equilateral, then it is isosceles.

Contrapositive: If a triangle is not isosceles, then it is not equilateral.

converse

To state the converse of a conditional statement, interchange the hypothesis and the conclusion.

Conditional Statement: If p, then q.Converse: If q, then p.

Example

Conditional Statement: If a � 0 or b � 0, then ab � 0.Converse: If ab � 0, then a � 0 or b � 0.

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G-8 Glossary

Glossary

cosecant (csc)

The cosecant (csc) of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side opposite the angle.

Example

In triangle ABC, the cosecant of angle A is:

csc A � length of hypotenuse

_________________________ length of side opposite /A

� AB ___ BC

The expression “csc A” means “the cosecant of angle A.”

A C

B

cosine (cos)

The cosine (cos) of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Example

In triangle ABC, the cosine of angle A is:

cos A 5 length of side adjacent to �A

___________________________ length of hypotenuse

5 AC ___ AB

The expression “cos A” means “the cosine of angle A.”

A C

B

cotangent (cot)

The cotangent (cot) of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.

Example

In triangle ABC, the cotangent of angle A is:

cot A � length of side adjacent to /A

___________________________ length of side opposite /A

� AC ___ BC

The expression “cot A” means “the cotangent of angle A.”

A C

B

counterexample

A counterexample is a single example that shows that a statement is not true.

Example

Your friend claims that you add fractions by adding the numerators and then adding the denominators. A counterexample is 1 __

2 1 1 __

2 . The sum of these two

fractions is 1. Your friend’s method results in 1 1 1 ______ 2 1 2

, or 1 __

2 . Your friend’s method is incorrect.

Counting Principle

The Counting Principle states that if action A can occur in m ways and for each of these m ways action B can occur in n ways, then actions A and B can occur in m ? n ways.

Example

In the school cafeteria, there are 3 different main entrées and 4 different sides. So, there are 3 ? 4, or 12 different lunches that can be created.

D

deduction

Deduction is reasoning that involves using a general rule to make a conclusion.

Example

Sandy learned the rule that the sum of the measures of the three interior angles of a triangle is 180 degrees. When presented with a triangle, she concludes that the sum of the measures of the three interior angles is 180 degrees. Sandy reached the conclusion using deduction.

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Glossary G-9

Glossary

diameter of a sphere

The diameter of a sphere is a line segment with each endpoint on the sphere that passes through the center of the sphere.

Example

great circle

hemisphere

diameter

radius

center

direct proof

A direct proof begins with the given information and works to the desired conclusion directly through the use of givens, definitions, properties, postulates, and theorems.

directrix of a parabola

The directrix of a parabola is a line such that all points on the parabola are equidistant from the focus and the directrix.

Example

The focus of the parabola shown is the point (0, 2). The directrix of the parabola shown is the line y 5 22. All points on the parabola are equidistant from the focus and the directrix.

4

6

8

–6

–4

–8

2 4 6 8 –6 –8 –4

y

x–2

d1 = d2

d1

d2

(0, 2) (x, y)

y = –2

degree measure of an arc

The degree measure of a minor arc is equal to the degree measure of its central angle. The degree measure of a major arc is determined by subtracting the degree measure of the minor arc from 360°.

Example

The measure of minor arc AB is 30°. The measure of major arc BZA is 360° 2 30° 5 330°.

O

Z

B

A

dependent events

Dependent events are events for which the occurrence of one event has an impact on the occurrence of subsequent events.

Example

A jar contains 1 blue marble, 1 green marble, and 2 yellow marbles. You randomly choose a yellow marble without replacing the marble in the jar, and then randomly choose a yellow marble again. The events of randomly choosing a yellow marble first and randomly choosing a yellow marble second are dependent events because the 1st yellow marble was not replaced in the jar.

diameter

The diameter of a circle is a line segment with each endpoint on the circle that passes through the center of the circle.

Example

In circle O, ___

AB is a diameter.

A

B

O

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G-10 Glossary

Glossary

E

element

A member of a set is called an element of that set.

Example

Set B contains the elements a, b, and c.

B 5 {a, b, c}

endpoint of a ray

An endpoint of a ray is a point at which a ray begins.

Example

Point C is the endpoint of ray CD.

CD

endpoints of a line segment

An endpoint of a line segment is a point at which a segment begins or ends.

Examples

Points A and B are endpoints of segment AB.

A B

disc

A disc is the set of all points on a circle and in the interior of a circle.

disjoint sets

Two or more sets are disjoint sets if they do not have any common elements.

Example

Let N represent the set of 9th grade students. Let T represent the set of 10th grade students. The sets N and T are disjoint sets because the two sets do not have any common elements. Any student can be in one grade only.

Distance Formula

The Distance Formula can be used to calculate the distance between two points.

The distance between points (x1, y1) and (x2, y2) is

d 5 √___________________

(x2 2 x1)2 1 (y2 2 y1)

2 .

Example

To calculate the distance between the points (21, 4) and (2, 25), substitute the coordinates into the Distance Formula.

d 5 √___________________

(x2 2 x1)2 1 (y2 2 y1)

2

d 5 √___________________

(2 1 1)2 1 (25 2 4)2

d 5 √__________

32 1 (29)2

d 5 √_______

9 1 81

d 5 √___

90

d < 9.49

So, the distance between the points (21, 4) and(2, 25) is approximately 9.49 units.

draw

To draw is to create a geometric figure using tools such as a ruler, straightedge, compass, or protractor. A drawing is more accurate than a sketch.

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Glossary G-11

Glossary

experimental probability

Experimental probability is the ratio of the number of times an event occurs to the total number of trials performed.

Example

You flip a coin 100 times. Heads comes up 53 times. The experimental probability of getting heads is 53 ____

100 .

exterior angle of a polygon

An exterior angle of a polygon is an angle that is adjacent to an interior angle of a polygon.

Examples

Angle JHI is an exterior angle of quadrilateral FGHI.

Angle EDA is an exterior angle of quadrilateral ABCD.

G

H

F

J

I

A

B

D

E

C

external secant segment

An external secant segment is the portion of each secant segment that lies outside of the circle. It begins at the point at which the two secants intersect and ends at the point where the secant enters the circle.

Example

Segment HC and segment PC are external secant segments.

GH

N

B P

C

F

factorial

The factorial of n, written as n!, is the product of all non-negative integers less than or equal to n.

Example

3! 5 3 3 2 3 1 5 6

Euclidean geometry

Euclidean geometry is a complete system of geometry developed from the work of the Greek mathematician Euclid. He used a small number of undefined terms and postulates to systematically prove many theorems.

Euclid’s first five postulates are:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.5. If two lines are drawn that intersect a third line in

such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (This postulate is equivalent to what is known as the parallel postulate.)

Example

Euclidean geometry

Non-Euclidean geometry

event

An event is an outcome or a set of outcomes in a sample space.

Example

A number cube contains the numbers 1 through 6. Rolling a 6 is one event. Rolling an even number is another event.

expected value

The expected value is the average value when the number of trials in a probability experiment is large.

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G-12 Glossary

Glossary

focus of a parabola

The focus of a parabola is a point such that all points on the parabola are equidistant from the focus and the directrix.

Example

The focus of the parabola shown is the point (0, 2). The directrix of the parabola shown is the line y 5 22. All points on the parabola are equidistant from the focus and the directrix.

4

6

8

–6

–4

–8

2 4 6 8 –6 –8 –4

y

x–2

d1 = d2

d1

d2

(0, 2) (x, y)

y = –2

flow chart proof

A flow chart proof is a proof in which the steps and corresponding reasons are written in boxes. Arrows connect the boxes and indicate how each step and reason is generated from one or more other steps and reasons.

Example

A flow chart proof is shown for the conditional statement: If ___

AB � ___

CD , then ___

AC � ___

BD .

Given: ___

AB � ___

CD

Prove: ___

AC � ___

BD

AB CD Given

m AB m CD Definition of congruent segments

m AB m BC m CD m BC Addition Property of Equality

m AC m BD Substitution Property

m BC m BC Identity Property

Segment Additionm AB m BC m AC

m BC m CD m BD Segment Addition

AC BD Definition of congruent segments

� � �

�

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Glossary G-13

Glossary

geometric mean

The geometric mean of two positive numbers a and b is the positive number x such that a __ x 5 x __

b .

Example

The geometric mean of 3 and 12 is 6.

3 __x 5 x___

12

x2 5 36x 5 6

geometric probability

Geometric probability is probability that involves a geometric measure, such as length, area, volume, and so on.

Example

A dartboard has the size and shape shown. The gray shaded area represents a scoring section of the dartboard. Calculate the probability that a dart that lands on a random part of the target will land in a gray scoring section.

20 in.

20 in.

8 in.

Calculate the area of the dartboard: 20(20) 5 400 in.2

There are 4 gray scoring squares with 8-in. sides and a gray scoring square with 20 2 8 2 8 5 4-in. sides. Calculate the area of the gray scoring sections: 4(8)(8) 1 4(4) 5 272 in.2

Calculate the probability that a dart will hit a gray

scoring section: 272 ____ 400

5 0.68 5 68%.

frequency table

A frequency table shows the frequency of an item, number, or event appearing in a sample space.

Example

The frequency table shows the number of times a sum of two number cubes occurred.

Sum of Two

Number CubesFrequency

2 1

3 2

4 3

5 4

6 5

7 6

8 5

9 4

10 3

11 2

12 1

G

general form of a parabola

The general form of a parabola centered at the origin is an equation of the form Ax2 1 Dy 5 0 or By2 1 Cx 5 0.

Example

The equation for the parabola shown can be written in general form as x2 2 2y 5 0.

4

2

6

8

–6

–4

–2

–8

2 4 6 8 –6 –8 –4

y

x–2 O

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G-14 Glossary

Glossary

I

image

An image is a new figure formed by a transformation.

Example

The figure on the right is the image that has been created by translating the original figure 3 units to the right horizontally.

y

1

2

3

4

5

6

7

1 2 3 4 5 76

incenter

The incenter of a triangle is the point at which the angle bisectors of the triangle intersect.

Example

Point X is the incenter of triangle ABC.

X

A

C B

included angle

An included angle is an angle formed by two consecutive sides of a figure.

Example

In triangle ABC, angle A is the included angle formed by consecutive sides

___ AB and

___ AC .

C

A

B

great circle of a sphere

The great circle of a sphere is a cross section of a sphere when a plane passes through the center of the sphere.

Example

A

great circle

H

hemisphere

A hemisphere is half of a sphere bounded by a great circle.

Example

A hemisphere is shown.

hemisphere

hypothesis

A hypothesis is the “if” part of an “if-then” statement.

Example

In the statement, “If the last digit of a number is a 5, then the number is divisible by 5,” the hypothesis is “If the last digit of a number is a 5.”

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Glossary G-15

Glossary

indirect proof or proof by contradiction

An indirect proof, or proof by contradiction, uses the contrapositive. By proving that the contrapositive is true, you prove that the statement is true.

Example

Given: Triangle DEF

Prove: A triangle cannot have more than one obtuse angle.

Given �DEF, assume that �DEF has two obtuse angles. So, assume m�D � 91� and m�E � 91�. By the Triangle Sum Theorem, m�D � m�E � m�F � 180�. By substitution, 91� � 91� � m�F � 180�, and by subtraction, m�F 5 22�. But, it is not possible for a triangle to have a negative angle, so this is a contradiction. This proves that a triangle cannot have more than one obtuse angle.

induction

induction is reasoning that involves using specific examples to make a conclusion.

Example

Sandy draws several triangles, measures the interior angles, and calculates the sum of the measures of the three interior angles. She concludes that the sum of the measures of the three interior angles of a triangle is 180º. Sandy reached the conclusion using induction.

inscribed angle

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

Example

Angle BAC is an inscribed angle. The vertex of angle BAC is on the circle and the sides of angle BAC contain the chords

___ AB and

___ AC .

A

B

C

included side

An included side is a line segment between two consecutive angles of a figure.

Example

In triangle ABC, ___

AB is the included side formed by consecutive angles A and B.

C

A

B

independent events

Independent events are events for which the occurrence of one event has no impact on the occurrence of the other event.

Example

You randomly choose a yellow marble, replace the marble in the jar, and then randomly choose a yellow marble again. The events of randomly choosing a yellow marble first and randomly choosing a yellow marble second are independent events because the 1st yellow marble was replaced in the jar.

indirect measurement

Indirect measurement is a technique that uses proportions to determine a measurement when direct measurement is not possible.

Example

You can use a proportion to solve for the height x of the flagpole.

5.5 ft

x

19 ft 11 ft

x ___ 5.5

� 19 � 11 ________ 11

x ___ 5.5

� 30 ___ 11

11x � 165

x � 15

The flagpole is 15 feet tall.

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G-16 Glossary

Glossary

intersecting sets

Two or more sets are intersecting sets if they have common elements.

Example

Let V represent the set of students who are on the girls’ volleyball team. Let M represent the set of students who are in the math club. Julia is on the volleyball team and belongs to the math club. The sets V and M are intersecting sets because the two sets have at least one common element, Julia.

inverse

To state the inverse of a conditional statement, negate both the hypothesis and the conclusion.

Conditional Statement: If p, then q. Inverse: If not p, then not q.

Example

Conditional Statement: If a triangle is equilateral, then it is isosceles.

Inverse: If a triangle is not equilateral, then it is not isosceles.

inverse cosine

The inverse cosine, or arc cosine, of x is the measure of an acute angle whose cosine is x.

Example

In right triangle ABC, if cos A � x, then cos–1 x � m�A.

A C

B

inverse sine

The inverse sine, or arc sine, of x is the measure of an acute angle whose sine is x.

Example

In right triangle ABC, if sin A � x, then sin–1 x � m�A.

A C

B

inscribed polygon

An inscribed polygon is a polygon drawn inside a circle such that each vertex of the polygon is on the circle.

Example

Quadrilateral KLMN is inscribed in circle J.

L

M

N

J

K

intercepted arc

An intercepted arc is formed by the intersections of the sides of an inscribed angle with a circle.

Example

___

PR is an intercepted arc of inscribed angle PSR.

Q

P R

S

interior angle of a polygon

An interior angle of a polygon is an angle which is formed by consecutive sides of the polygon or shape.

Example

The interior angles of �ABC are �ABC, �BCA, and �CAB.

A

B

C

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Glossary G-17

Glossary

L

Law of Cosines

The Law of Cosines, ora2 5 c2 1 b2 2 2bc ? cos Ab2 5 a2 1 c2 2 2ac ? cos Bc2 5 a2 1 b2 2 2ab ? cos C

can be used to determine the unknown lengths of sides or the unknown measures of angles in any triangle.

B

A Cb

ac

Example

In triangle ABC, the measure of angle A is 65º, the length of side b is 4.4301 feet, and the length of side c is 7.6063 feet. Use the Law of Cosines to calculate the length of side a.

a2 5 4.43012 1 7.60632 2 2(4.4301)(7.6063) cos 65º

The length of side a is 7 feet.

Law of Sines

The Law of Sines, or sin A _____ a 5 sin B _____ b

5 Sin C _____ c , can be used to determine the unknown side lengths or the unknown angle measures in any triangle.

Example

B

A Cb

ac

In triangle ABC, the measure of angle A is 65º, the measure of angle B is 80º, and the length of side a is 7 feet. Use the Law of Sines to calculate the length of side b.

7 _______ sin 65º

5 b _______ sin 80º

The length of side b is 7.6063 feet.

inverse tangent

The inverse tangent (or arc tangent) of x is the measure of an acute angle whose tangent is x.

Example

In right triangle ABC, if tan A � x, then tan–1 x � m�A.

A C

B

isometric paper

Isometric paper is often used by artists and engineers to create three-dimensional views of objects in two dimensions.

Example

The rectangular prism is shown on isometric paper.

isosceles trapezoid

An isosceles trapezoid is a trapezoid whose nonparallel sides are congruent.

Example

In trapezoid JKLM, side ___

KL is parallel to side ___

JM , and the length of side

___ JK is equal to the length of side

___ LM ,

so trapezoid JKLM is an isosceles trapezoid.

K L

J M

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G-18 Glossary

Glossary

linear velocity

Linear velocity is a type of circular velocity described as an amount of distance over a specified amount of time. Linear velocity can be expressed as v 5 s __

t , where

v 5 velocity, s 5 arc length, and t 5 time.

locus of points

A locus of points is a set of points that satisfy one or more conditions.

Example

A circle is defined as a locus of points that are a fixed distance, called the radius, from a given point, called the center.

y

x

radius

Center

M

major arc

Two points on a circle determine a major arc and a minor arc. The arc with the greater measure is the major arc. The other arc is the minor arc.

Example

Circle Q is divided by points A and B into two arcs, arc ACB and arc AB. Arc ACB has the greater measure, so it is the major arc. Arc AB has the lesser measure, so it is the minor arc.

C

A

B

Q

legs of a trapezoid

The non-parallel sides of a trapezoid are the legs of the trapezoid.

Example

legs

line

A line is made up of an infinite number of points that extend infinitely in two opposite directions. A line is straight and has only one dimension.

Example

The line below can be called line k or line AB.

A

B k

line segment

A line segment is a portion of a line that includes two points and all of the collinear points between the two points.

Example

The line segment shown is named ___

AB or ___

BA .

A B

linear pair

A linear pair of angles are two adjacent angles that have noncommon sides that form a line.

Example

The diagram shown has four pairs of angles that form a linear pair.

Angles 1 and 2 form a linear pair. Angles 2 and 3 form a linear pair. Angles 3 and 4 form a linear pair. Angles 4 and 1 form a linear pair.

m

n

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Glossary

midsegment of a triangle

A midsegment of a triangle is a line segment formed by connecting the midpoints of two sides of a triangle.

Example

Segment AB is a midsegment.

A B

minor arc

Two points on a circle determine a minor arc and a major arc. The arc with the lesser measure is the minor arc. The other arc is the major arc.

Example

Circle Q is divided by points A and B into two arcs, arc ACB and arc AB. Arc AB has the lesser measure, so it is the minor arc. Arc ACB has the greater measure, so it is the major arc.

C

A

B

Q

N

non-uniform probability model

When all probabilities in a probability model are not equivalent to each other, it is called a non-uniform probability model.

Example

Spinning the spinner shown represents a non-uniform probability model because the probability of landing on a shaded space is not equal to the probability of landing on a non-shaded space.

median

The median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.

Example

The 3 medians are drawn on the triangle shown.

midpoint

The midpoint of a line segment is the point that divides the line segment into two congruent segments.

Example

Because point B is the midpoint of ___

AC , ___

AB ˘ ___

BC .

A B C

Midpoint formula

The Midpoint Formula can be used to calculate the midpoint between two points. The midpoint between

(x1, y1) and (x2, y2) is ( x1 1 x2 _______ 2 ,

y1 1 y2 _______ 2 ) .

Example

To calculate the midpoint between the points (21, 4) and (2, 25), substitute the coordinates into the Midpoint Formula.

( x1 1 x2 _______ 2 ,

y1 1 y2 _______ 2 ) 5 ( 21 1 2 _______

2 , 4 2 5 ______

2 )

5 ( 1 __ 2 , 21 ___

2 )

So, the midpoint between the points (21, 4) and

(2, 25) is ( 1 __ 2 , 2

1 __ 2 ) .

midsegment of a trapezoid

The midsegment of a trapezoid is a line segment formed by connecting the midpoints of the legs of the trapezoid.

Example

Segment XY is the midsegment of trapezoid ABCD.

A

B C

D

X Y

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G-20 Glossary

Glossary

opposite side

The opposite side of a triangle is the side opposite the reference angle.

Example

reference angleadjacent side

opposite side

organized list

An organized list is a visual model for determining the sample space of events.

Example

The sample space for flipping a coin 3 times can be represented as an organized list.

HHH THHHHT THTHTH TTHHTT TTT

orthocenter

The orthocenter of a triangle is the point at which the altitudes of the triangle intersect.

Example

Point X is the orthocenter of triangle ABC.

X

A

C B

outcome

An outcome is the result of a single trial of an experiment.

Example

Flipping a coin has two outcomes: heads or tails.

O

oblique cylinder

When a circle is translated through space in a direction that is not perpendicular to the plane containing the circle, the solid formed is an oblique cylinder.

Example

The prism shown is an oblique cylinder.

oblique rectangular prism

When a rectangle is translated through space in a direction that is not perpendicular to the plane containing the rectangle, the solid formed is an oblique rectangular prism.

Example

The prism shown is an oblique rectangular prism.

oblique triangular prism

When a triangle is translated through space in a direction that is not perpendicular to the plane containing the triangle, the solid formed is an oblique triangular prism.

Example

The prism shown is an oblique triangular prism.

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Glossary G-21

Glossary

permutation

A permutation is an ordered arrangement of items without repetition.

Example

The permutations of the letters A, B, and C are:

ABC ACB

BAC BCA

CAB CBA

perpendicular bisector

A perpendicular bisector is a line, line segment, or ray that intersects the midpoint of a line segment at a 90-degree angle.

Example

Line k is the perpendicular bisector of ___

AB . It is perpendicular to

___ AB , and intersects

___ AB at midpoint M

so that AM � MB.

k

M BA

plane

A plane is a flat surface with infinite length and width, but no depth. A plane extends infinitely in all directions.

Example

Plane A is shown.

A

point

A point has no dimension, but can be visualized as a specific position in space, and is usually represented by a small dot.

Example

point A is shown.

A

P

parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix.

Example

The focus of the parabola shown is the point (0, 2). The directrix of the parabola shown is the line y 5 22. All points on the parabola are equidistant from the focus and the directrix.

4

6

8

–6

–4

–8

2 4 6 8 –6 –8 –4

y

x–2

d1 = d2

d1

d2

(0, 2) (x, y)

y = –2

paragraph proof

A paragraph proof is a proof that is written in paragraph form. Each sentence includes mathematical statements that are organized in logical steps with reasons.

Example

The proof shown is a paragraph proof that vertical angles 1 and 3 are congruent.

Angle 1 and angle 3 are vertical angles. By the definition of linear pair, angle 1 and angle 2 form a linear pair. Angle 2 and angle 3 also form a linear pair. By the Linear Pair Postulate, angle 1 and angle 2 are supplementary. Angle 2 and angle 3 are also supplementary. Angle 1 is congruent to angle 3 by the Congruent Supplements Theorem.

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G-22 Glossary

Glossary

point of concurrency

A point of concurrency is the point at which three or more lines intersect.

Example

Point X is the point of concurrency for lines �, m, and n.

X

�

m

n

point of tangency

A tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency.

Example

Line RQ is tangent to circle P. Point Q is the point of tangency.

PQ

R

point-slope form

The point-slope form of a linear equation that passes through the point (x1, y1) and has slope m is y � y1 � m(x � x1).

Example

A line passing through the point (1, 2) with a slope of 1 __

2 can be written in point-slope form as

y � 2 � 1 __ 2 (x � 1).

postulate

A postulate is a statement that is accepted to be true without proof.

Example

The following statement is a postulate: A straight line may be drawn between any two points.

pre-image

A pre-image is the figure that is being transformed.

Example

The figure on the right is the image that has been formed by translating the pre-image 3 units to the right horizontally.

y

1

2

3

4

5

6

7

1 2 3 4 5 76

probability

The probability of an event is the ratio of the number of desired outcomes to the total number of possible

outcomes, P(A) 5 desired outcomes __________________ possible outcomes

.

Example

When flipping a coin, there are 2 possible outcomes: heads or tails. The probability of flipping a heads is 1 __

2 .

probability model

A probability model lists the possible outcomes and the probability for each outcome. In a probability model, the sum of the probabilities must equal 1.

Example

The table shows a probability model for flipping a fair coin once.

Outcomes Heads (H) Tails (T)

Probability 1 __ 2

1 __ 2

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Glossary G-23

Glossary

rationalizing the denominator

Rationalizing the denominator is the process of eliminating a radical from the denominator of an expression. To rationalize the denominator, multiply by a form of one so that the radicand of the radical in the denominator is a perfect square.

Example

Rationalize the denominator of the expression 5 ___ √

__ 3 .

5 ___ √

__ 3 5 5 ___

√__

3 ? √

__ 3 ___

√__

3

5 5 √__

3 ____ √

__ 9

5 5 √__

3 ____ 3

ray

A ray is a portion of a line that begins with a single point and extends infinitely in one direction.

Example

The ray shown is ray AB.

A

B

reference angle

A reference angle is the angle of the right triangle being considered. The opposite side and adjacent side are named based on the reference angle.

Example

reference angleadjacent side

opposite side

Reflexive Property

The reflexive property states that a � a.

Example

The statement 2 � 2 is an example of the reflexive property.

propositional form

When a conditional statement is written using the propositional variables p and q, the statement is said to be written in propositional form.

Example

Propositional form: “If p, then q.”p → q

propositional variables

When a conditional statement is written in propositional form as “If p, then q,” the variables p and q are called propositional variables.

R

radian

One radian is defined as the measure of a central angle whose arc length is the same as the radius of the circle.

radius

The radius of a circle is a line segment with one endpoint on the circle and one endpoint at the center.

Example

In circle O, ___

OA is a radius.

O

A

radius of a sphere

The radius of a sphere is a line segment with one endpoint on the sphere and one endpoint at the center.

Example

great circle

hemisphere

diameter

radius

center

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G-24 Glossary

Glossary

right rectangular prism

A rectangle translated through space in a direction perpendicular to the plane containing the rectangle forms a right rectangular prism.

Example

right triangular prism

A triangle translated through space in a direction perpendicular to the plane containing the triangle forms a right triangular prism.

Example

rigid motion

A rigid motion is a transformation of points in space. Translations, reflections, and rotations are examples of rigid motion.

relative frequency

A relative frequency is the ratio or percent of occurrences within a category to the total of the category.

Example

John surveys 100 students in his school about their favorite school subject. Of the 100 students, 37 chose math as their favorite subject. The relative frequency of students show selected math as their favorite subject

is 37 ____ 100

, or 37%.

remote interior angles of a triangle

The remote interior angles of a triangle are the two angles that are not adjacent to the specified exterior angles.

Example

The remote interior angles with respect to exterior angles 4 are angles 1 and 2.

1

2

34

right cylinder

A disc translated through space in a direction perpendicular to the plane containing the disc forms a right cylinder.

Example

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Glossary G-25

Glossary

secant of a circle

A secant of a circle is a line that intersects the circle at two points.

Example

The line intersecting the circle through points A and B is a secant.

BA

secant segment

A secant segment is formed when two secants intersect outside of a circle. A secant segment begins at the point at which the two secants intersect, continues into the circle, and ends at the point at which the secant exits the circle.

Example

Segment GC and segment NC are secant segments.

GH

N

B P

C

sector of a circle

A sector of a circle is a region of the circle bounded by two radii and the included arc.

Example

In circle Y, arc XZ, radius XY, and radius YZ form a sector.

Z

X

Y

Rule of Compound Probability

involving “and”

The Rule of Compound Probability involving “and” states: “If Event A and Event B are independent, then the probability that Event A happens and Event B happens is the product of the probability that Event A happens and the probability that Event B happens, given that Event A has happened.”

P(A and B) 5 P(A) ? P(B)

Example

You flip a coin two times. Calculate the probability of flipping a heads on the first flip and flipping a heads on the second flip.

Let A represent the event of flipping a heads on the first flip. Let B represent the event of flipping a heads on the second flip.

P(A and B) 5 P(A) ? P(B)

P(A and B) 5 1 __ 2 ? 1 __

2

P(A or B) 5 1 __ 4

So, the probability of flipping a heads on the first flip and flipping a heads on the second flip is 1 __

4 .

S

sample space

A list of all possible outcomes of an experiment is called a sample space.

Example

Flipping a coin two times consists of four outcomes: HH, HT, TH, and TT.

secant (sec)

The secant (sec) of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side adjacent to the angle.

Example

In triangle ABC, the secant of angle A is:

sec A � length of hypotenuse

___________________________ length of side adjacent to /A

� AB ___ AC

The expression “sec A” means “the secant of angle A.”

A C

B

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G-26 Glossary

Glossary

semicircle

A semicircle is an arc whose endpoints form the endpoints of a diameter of the circle.

Example

Arc XYZ and arc ZWX are semicircles of circle P.

PX

Y

Z

W

set

A set is a collection of items. If x is a member of set B, then x is an element of set B.

Example

Let E represent the set of even whole numbers.E 5 {2, 4, 6, 8, . . .}

similar triangles

Similar triangles are triangles that have all pairs of corresponding angles congruent and all corresponding sides are proportional.

Example

Triangle ABC is similar to triangle DEF.

A

B

CD

E

F

simulation

A simulation is an experiment that models a real-life situation.

Example

You can simulate the selection of raffle numbers by using the random number generator on a graphing calculator.

segment bisector

A segment bisector is a line, line segment, or ray that intersects a line segment so that the line segment is divided into two segments of equal length.

Example

Line k is a segment bisector of segment AC. The lengths of segments AB and BC are equal.

A B C

k

segment of a circle

A segment of a circle is a region bounded by a chord and the included arc.

Example

In circle A, chord ___

BC and arc BC are the boundaries of a segment of the circle.

A

B

C

segments of a chord

Segments of a chord are the segments formed on a chord if two chords of a circle intersect.

Example

The segments of chord ___

HD are ___

EH and ___

ED . The segments of chord

___ RC are

___ ER and

___ EC .

O

E

HR

C

D

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Glossary G-27

Glossary

sphere

A sphere is the set of all points in space that are a given distance from a fixed point called the center of the sphere.

Example

A sphere is shown.

great circle

hemisphere

diameter

radius

center

standard form of a parabola

The standard form of a parabola centered at the origin is an equation of the form x2 5 4py or y2 5 4px, where p represents the distance from the vertex to the focus.

Example

The equation for the parabola shown can be written in standard form as x2 = 2y.

4

2

6

8

–6

–4

–2

–8

2 4 6 8 –6 –8 –4

y

x–2 O

straightedge

A straightedge is a ruler with no numbers.

Substitution Property of Equality

The Substitution Property of Equality states: “If a and b are real numbers and a 5 b, then a can be substituted for b.”

Example

If AB 5 12 ft and CD 5 12 ft, then AB 5 CD.

sine (sin)

The sine (sin) of an acute angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Example

In triangle ABC, the sine of angle A is:

sin A 5 length of side opposite /A

_________________________ length of hypotenuse

5 BC ___ AB

The expression “sin A” means “the sine of angle A.”

A C

B

sketch

To sketch is to create a geometric figure without using tools such as a ruler, straightedge, compass, or protractor. A drawing is more accurate than a sketch.

skew lines

Skew lines are two lines that do not intersect and are not parallel. Skew lines do not lie in the same plane.

Example

Line m and line p are skew lines.

p

m

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G-28 Glossary

Glossary

tangent of a circle

A tangent of a circle is a line that intersects the circle at exactly one point, called the point of tangency.

Example

Line RQ is tangent to circle P.

P

R

Q

tangent segment

A tangent segment is a line segment formed by connecting a point outside of the circle to a point of tangency.

Example

Line segment AB and line segment AC are tangent segments.

B

A

C

E

m

n

theorem

A theorem is a statement that has been proven to be true.

Example

The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and hypotenuse of length c, then a2 � b2 � c2.

theoretical probability

Theoretical probability is the mathematical calculation that an event will happen in theory.

Example

The theoretical probability of rolling a 1 on a number cube is 1 __

6 .

Subtraction Property of Equality

The Subtraction Property of Equality states: “If a � b, then a 2 c � b 2 c.”

Example

If x 1 5 5 7, then x 1 5 2 5 5 7 2 5, or x 5 2 is an example of the subtraction property of equality.

supplementary angles

Two angles are supplementary if the sum of their measures is 180º.

Example

Angle 1 and angle 2 are supplementary angles.

If m�1 � 75°, then m�2 � 180° 2 75° 5 105°.

1 2

T

tangent (tan)

The tangent (tan) of an acute angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Example

In triangle ABC, the tangent of angle A is:

tan A 5 length of side opposite /A

___________________________ length of side adjacent to /A

5 BC ___ AC

The expression “tan A” means “the tangent of angle A.”

A C

B

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Glossary

translation

A translation is a transformation in which a figure is shifted so that each point of the figure moves the same distance in the same direction. The shift can be in a horizontal direction, a vertical direction, or both.

Example

The top trapezoid is a vertical translation of the bottom trapezoid by 5 units.

x–4–5–6–7 –3 –2 –1 1

–3

y

–4

–2

–1

1

2

3

4

tree diagram

A tree diagram is a diagram that illustrates sequentially the possible outcomes of a given situation.

Example

Boy

Boy Girl

Boy Girl Boy Girl

transformation

A transformation is an operation that maps, or moves, a figure, called the preimage, to form a new figure called the image. Three types of transformations are reflections, rotations, and translations.

Example

reflection over a line

rotation about a point

translation

Transitive Property of Equality

The Transitive Property of Equality states: “If a 5 b and b 5 c, then a 5 c.”

Example

If x � y and y � 2, then x � 2 is an example of the Transitive Property of Equality.

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G-30 Glossary

Glossary

truth value

The truth value of a conditional statement is whether the statement is true or false. If a conditional statement could be true, then the truth value of the statement is considered true. The truth value of a conditional statement is either true or false, but not both.

Example

The truth value of the conditional statement “If a quadrilateral is a rectangle, then it is a square” is false.

two-column proof

A two-column proof is a proof consisting of two columns. In the left column are mathematical statements that are organized in logical steps. In the right column are the reasons for each mathematical statement.

Example

The proof shown is a two-column proof.

Statements Reasons

1. �1 and �3 are vertical angles.

1. Given

2. �1 and �2 form a linear pair. �2 and �3 form a linear pair.

2. Definition of linear pair

3. �1 and �2 are supplementary. �2 and �3 are supplementary.

3. Linear Pair Postulate

4. �1 � �3 4. Congruent Supplements Theorem

truth table

A truth table is a table that summarizes all possible truth values for a conditional statement p → q. The first two columns of a truth table represent all possible truth values for the propositional variables p and q. The last column represents the truth value of the conditional statement p → q.

Example

The truth value of the conditional statement p → q is determined by the truth value of p and the truth value of q.

If p is true and q is true, then p → q is true.If p is true and q is false, then p → q is false.If p is false and q is true, then p → q is true.If p is false and q is false, then p → q is true.

p q p → qT T TT F FF T TF F T

two-way frequency table

A two-way frequency table, also called a contingency table, shows the number of data points and their frequencies for two variables. One variable is divided into rows, and the other is divided into columns.

Example

The two-way frequency table shows the hand(s) favored by people who do and do not participate in individual or team sports.

Sports Participation

Fa

vo

red

Ha

nd

Individual Team Does Not Play Total

Left 3 13 8 24

Right 6 23 4 33

Mixed 1 3 2 6

Total 10 39 14 63

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Glossary G-31

Glossary

two-way relative frequency table

A two-way relative frequency table displays the relative frequencies for two categories of data.

Example

The two-way relative frequency table shows the hand(s) favored by people who do and do not participate in individual or team sports.

Individual Team Does Not Play Total

Left 3 ___ 63

� 4.8% 13 ___ 63

< 20.6% 8 ___ 63

< 12.7% 24 ___ 63

< 38.1%

Right 6 ___ 63

< 9.5% 23 ___ 63

< 36.5% 4 ___ 63

< 6.3% 33 ___ 63

< 52.4%

Mixed 1 ___ 63

< 1.6% 3 ___ 63

< 4.8% 2 ___ 63

< 3.2% 6 ___ 63

< 9.5%

Total 10 ___ 63

< 15.9% 39 ___ 63

< 61.9% 14 ___ 63

< 22.2% 63 ___ 63

5 100%

two-way table

A two-way table shows the relationship between two data sets, one data set is organized in rows and the other data set is organized in columns.

Example

The two-way table shows all the possible sums that result from rolling two number cubes once.

2nd Number Cube

1st

Nu

mb

er

Cu

be

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

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G-32 Glossary

Glossary

vertical angles

Vertical angles are two nonadjacent angles that are formed by two intersecting lines.

Examples

Angles 1 and 3 are vertical angles.

Angles 2 and 4 are vertical angles.

12

34

U

uniform probability model

A uniform probability model occurs when all the probabilities in a probability model are equally likely to occur.

Example

Rolling a number cube represents a uniform probability model because the probability of rolling each number is equal.

V

vertex angle of an isosceles triangle

The vertex angle of an isosceles triangle is the angle formed by the two congruent legs.

Example

vertex angle

vertex of a parabola

The vertex of a parabola, which lies on the axis of symmetry, is the highest or lowest point on the parabola.

Example

The vertex of the parabola is the point (1, 24), the minimum point on the parabola.

x28 26 24 22 O 2 4 6 8

6

y

8

4

2

4

6

8

(1, 24)

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Index I-1

central angle, 844, 850, 851, 853inscribed angle, 853–858measuring, 864–876radian measure, 932–933

complementary, 153–155copying/duplicating, 54–56cosecant of, 689cosine of, 696cotangent of, 679–681defined, 52included, 458inverse cosine of, 702inverse sine of, 690inverse tangent of, 681–683linear pairs, 158–159of perpendicular lines, 66reference, 659–664right, 600of rotation, 518secant of, 700sine of, 687supplementary, 152, 154–155symbol (∠ ), 52tangent of, 674–679translating on coordinate plane, 52–54of triangles

of congruent triangles, 538exterior, 393–399interior, remote, 394–395interior, side length and, 389–393remote, 394–395similar triangles, 442, 446,

452–459, 461spherical triangles, 625See also specific types of triangles

vertex, 626vertical, 160–161

Angle Addition Postulate, 168, 189Angle-Angle-Angle (AAA), 582Angle-Angle-Side (AAS) Congruence

Theorem, 568–577, 584, 586congruence statement for, 575–576congruent triangles on coordinate

plane, 571–573constructing congruent triangles,

568–570defined, 568proof of, 574

Angle-Angle (AA) Similarity Theorem, 461defined, 453in indirect height measurement, 499in indirect width measurement, 500–502similar triangles, 452–454

Angle Bisector/Proportional Side Theorem, 464–468

applying, 466–468defined, 464proving, 465

Angle bisectors, 57–59, 96–100Angle of rotation, 518Angle postulates

Corresponding Angle Converse Postulate, 202–203

Corresponding Angle Postulate, 192–194

Angle relationships, 152–163adjacent angles, 156–157complementary angles, 153–155linear pairs, 158–159supplementary angles, 152, 154–155vertical angles, 160–161

Angle-Side-Angle (ASA) Congruence Theorem, 562–566, 584, 585

congruence statement for, 575–576congruent triangles on coordinate

plane, 564–566constructing congruent triangles,

562–563defined, 563proof of, 566

Angle theoremsAlternate Exterior Angle Converse

Theorem, 202, 205Alternate Exterior Angle Theorem, 196Alternate Interior Angle Converse

Theorem, 202, 204Alternate Interior Angle Theorem,

194–195, 447Same-Side Exterior Angle Converse

Theorem, 203, 207Same-Side Exterior Angle

Theorem, 198Same-Side Interior Angle Converse

Theorem, 202, 206Same-Side Interior Angle Theorem, 197

Angular velocity, 949–950Annulus, 340Arc Addition Postulate, 852Arc cosine, 702–704Arc length, 923–931, 946

defined, 926formula for, 926and radius, 927–928

Arcsadjacent, 852Arc Addition Postulate, 852arc length, 923–931, 946and chords, 884–886in copying line segments, 28defined, 28, 844intercepted, 852, 853, 924major, 844, 850, 851, 931minor, 844, 850, 851, 924–925, 931Parallel Lines–Congruent Arcs

Theorem, 859radian measure, 932–933

Index

Index

AAcute scalene triangle, 386Acute triangles

altitudes of, 106angle bisectors of, 96on coordinate plane, 380identifying, 385medians of, 101perpendicular bisectors of, 91points of concurrency for, 111scalene, 386

Addition Property of Equality, 170Addition Rule for Probability, 1093Adjacent angles, 156–157Adjacent arcs, 852Adjacent side

defined, 659of 45°–45°–90° triangles, 659–664of 30°–60°–90° triangles, 664–667

Algebrafor equation of a circle

to determine center and radius, 977–987

in standard form vs. in general form, 977–979

using Pythagorean Theorem, 974–976

with points of concurrency, 112–117proving Hypotenuse-Leg Congruence

Theorem with, 603proving Side-Angle-Side Theorem

with, 554Algebraic reasoning

angles of right triangles, 678–679proving Pythagorean Theorem

with, 492Alternate Exterior Angle Converse

Theorem, 202, 205Alternate Exterior Angle Theorem, 196Alternate Interior Angle Converse

Theorem, 202, 204Alternate Interior Angle Theorem,

194–195, 447Altitude, 106–110

defined, 106drawn to hypotenuse of right

triangles, 482–488geometric mean, 485–488Right Triangle Altitude/Hypotenuse

Theorem, 485Right Triangle Altitude/Leg

Theorem, 485Right Triangle/Altitude Similarity

Theorem, 482–484Angle

adjacent, 156–157bisecting, 57–59of circles

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and diameters, 878–882inscribed angles formed by, 854segments of, 887–888

circular velocities, 949–950circumference of, 946circumscribed figures

polygons, 916quadrilaterals, 920–921squares, 952triangles, 916

concentric, 936–937congruent, 841on coordinate plane, 966–971in copying line segments, 27–30defined, 852diameter of, 946

and chords, 842, 878–882and radius, 841

discs, 301drawn with a compass, 27equation of

to determine center and radius, 977–987

in standard form vs. in general form, 977–979

using Pythagorean Theorem, 974–976

inscribed angle of, 853–858inscribed figures

polygons, 912–915quadrilaterals, 917–918squares, 951triangles, 912–915

measuring angles ofdetermining measures, 875–876inside the circle, 864–865outside the circle, 866–870vertices on the circle, 871–874

parallel lines intersecting, 859points on, 990–997radian measure, 932–933radius of, 840–841rotated through space, 301secant of, 894–897

defined, 842and tangent, 843

sectors of, 936–939defined, 937determining area of, 937–939,

947–948segments of

area of, 940–942defined, 940

similar, 846–847in solving problems, 946–958tangent of, 890–893

defined, 843and secant, 843

Circular velocities, 949–950Circumcenter

algebra used to locate, 113, 115–116constructing, 91–95defined, 95

Circumference, 946Circumscribed figures

polygons, 916quadrilaterals, 920–921

defined, 57with patty (tracing) paper, 57

a line segment, 45–49by construction, 46–49defined, 45with patty (tracing) paper, 45–46

Bisectorsangle, 57–59, 96–100perpendicular, 76–79, 91–95, 878segment, 45–49

CCavalieri’s principle, 319–324

for area, 320–321for volume, 322–324

Center of a circlealgebraic determination of, 977–987defined, 840

Central angle (circles), 850, 851, 853defined, 844determining, 850radians, 932–933

Centroidalgebra used to locate, 113, 114constructing, 101–105defined, 105of right triangles, 329–333

Chords (circles), 878–888and arcs, 884–886congruent, 880defined, 841diameter as, 842and diameters, 878–882inscribed angles formed by, 854segments of, 887–888

CirclesArc Addition Postulate, 852arc length, 923–931, 946

defined, 926formula for, 926and radius, 927–928

arcs of, 28adjacent, 852Arc Addition Postulate, 852arc length, 923–931, 946and chords, 884–886in copying line segments, 28defined, 28, 844intercepted, 852, 853, 924major, 844, 850, 851, 931minor, 844, 850, 851, 924–925, 931Parallel Lines–Congruent Arcs

Theorem, 859radian measure, 932–933

center ofalgebraic determination of,

977–987defined, 840

central angle of, 850, 851, 853defined, 844determining, 850radian measure, 932–933

chords, 878–888and arcs, 884–886congruent, 880defined, 841diameter as, 842

Arc sine, 690–691Arc tangent, 681–683Area

Cavalieri’s principle for, 320–321circles

sectors of, 937–939, 947–948segments of, 940–942

of composite figureson coordinate plane, 283–284hexagons, 285

of cross sectionsof cones, 340in hemispheres, 341

in geometric probability, 1214of hexagons, 285inscribed polygons, 953–958inside inscribed squares, 951outside of inscribed squares, 952of parallelograms

and area of trapezoids, 274–275Cavalieri’s principle for, 324on coordinate plane, 256–266doubling, 269

of polygons, 813of quadrilaterals

Distance Formula for, 229, 232transformations for, 231–232

of squareson coordinate plane, 228–230Distance Formula for, 229

of trapezoidsand area of a parallelogram,

274–275on coordinate plane, 272–273rectangle method for, 276–278

of triangles, 718–719, 725–727on coordinate plane, 236–237,

239–241, 243–247, 249–250doubling, 253rectangle method for, 265–266transformations for, 238–241, 250

of two-dimensional figures, approximating, 320–321

Axis of symmetry (parabolas), 1004–1006, 1012–1015

BBases

area of, 334of cones, 339of parallelograms, 257, 259, 260of solid figures, 334of a trapezoid, 273, 274of a triangle, 237, 243

Bases of the trapezoid, 273, 274Biconditional statements

Congruent Chord–Congruent Arc Converse Theorem, 865

Congruent Chord–Congruent Arc Theorem, 885

defined, 778Equidistant Chord Converse

Theorem, 882Equidistant Chord Theorem, 882

Bisectingan angle, 57–59

by construction, 58–59

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defined, 192from postulates, 165theorems from, 165, 194writing, 192–193

Construct (geometric figures)circles, 27–28congruent triangles

Angle-Angle-Side Congruence Theorem, 568–570

Angle-Side-Angle Congruence Theorem, 562–563

Hypotenuse-Leg Congruence Theorem, 602–603

Side-Angle-Side Congruence Theorem, 552–553

Side-Side-Side Congruence Theorem, 544–545

defined, 8equilateral triangle, 8245°–45°–90° triangles, 417isosceles right triangle, 417isosceles triangle, 83kites, 772parallelograms, 758rectangle, 85rectangles, 748rhombus, 762similar triangles, 444, 452–459squares, 8, 84, 74330°–60°–90° triangles, 425trapezoids, 775

Construction proof, 179Constructions

bisecting angles, 57–59bisecting line segments, 45–49centroid, 101–105circumcenter, 91–95copy/duplicate

angles, 55–56a line segment, 27–34

incenter, 96–100orthocenter, 106–111parallel lines, 80–81perpendicular lines, 76–79

through a point not on a line, 78–79

through a point on a line, 76–77, 79

points of concurrency, 88–89Contingency tables, 1146

See also Two-way (contingency) frequency tables

Contradiction, proof by, 638See also Indirect proof (proof by

contradiction)Contrapositive

of conditional statements, 634–637

in indirect proof, 638–639Converse, 202Converse of the Pythagorean Theorem,

493–494Converse of Triangle Proportionality

Theorem, 474Conversion ratios, 658–664

for 45°–45°–90° triangles, 658–664for 30°–60°–90° triangles, 664–667

as rotation of triangles, 301from stacking two-dimensional

figures, 312, 313tranformations for, 314volume of, 314, 324, 332–335

Congruencesymbol (>), 12understanding, 536–538

Congruence statementfor Angle-Angle-Side Congruence

Theorem, 575–576for Angle-Side-Angle Congruence

Theorem, 575–576for Side-Angle-Side Congruence

Theorem, 557–560for Side-Side-Side Congruence

Theorem, 558–560Congruent angles, 59, 442, 453–459Congruent Chord–Congruent Arc

Converse Theorem, 865Congruent Chord–Congruent Arc

Theorem, 885Congruent Complement Theorem,

184–186Congruent line segments, 12–13, 29Congruent quadrilaterals, 232Congruent Supplement Theorem,

181–184Congruent triangles, 239, 536–541

and Angle-Angle-Angle as not a congruence theorem, 582

congruence statements for, 539–540, 575–560

Congruence Theorems in determining, 584–588

constructingAngle-Angle-Side Congruence

Theorem, 568–570Angle-Side-Angle Congruence

Theorem, 562–563Side-Angle-Side Congruence

Theorem, 552–553Side-Side-Side Congruence

Theorem, 544–545on coordinate plane

Angle-Angle-Side Congruence Theorem, 571–573

Angle-Side-Angle Congruence Theorem, 564–566

Side-Angle-Side Congruence Theorem, 554–556

Side-Side-Side Congruence Theorem, 546–549

corresponding angles of, 538corresponding parts of, 618–624corresponding parts of congruent

triangles are congruent concept, 618–624

corresponding sides of, 536–537points on perpendicular bisector

of line segment equidistant to endpoints of line segment, 580–581

and Side-Side-Angle as not a congruence theorem, 583

Conjecturesconverse, 204–207

rhombus, 919–921squares, 952triangles, 916

Collinear pointsdefined, 5in similar triangles, 439

Combinations, 1184–1187defined, 1184for probability of multiple trials of two

independent events, 1197–1204Compass, 8, 27Complement angle relationships (right

triangles), 708–715Complementary angles, 153–155Complement of an event, 1042Completing the square, 977Composite figures

area of, 274–275, 283–285on coordinate plane, 282–284defined, 282hexagons, 284–285perimeter of, 282, 285trapezoids, 274–275volume of, 347–350

Compound eventsdefined, 1074involving “and,” 1074–1085involving “or,” 1088–1099

Compound probabilities, on two-way tables, 1138–1153

frequency tables, 1141–1144two-way (contingency) frequency

tables, 1145–1147two-way relative frequency tables,

1147–1153Concavity (parabolas), 1004, 1006,

1008, 1012–1015Concentric circles, 936–937Conclusions

of conditional statements, 144defined, 144false, recognizing, 143through induction or deduction,

138–143Concurrent, 90

See also Points of concurrencyConditional probability, 1156–1168

defined, 1158dependent and independent events,

1164–1165formula for

building, 1161–1163using, 1166–1168

on two-way tables, 1158–1162Conditional statements, 144–149

converse of, 202defined, 144inverse and contrapositive of, 634–637rewriting, 148–149truth tables for, 146–147truth value of, 144–147See also Biconditional statements

Conesbuilding, 329–332cross-section shapes for, 356diameter of, 302height of, 302, 339

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as longest chord, 842and radius, 841

of concentric circles, 936–937of cones, 302of spheres, 338

Diameter–Chord Theorem, 879Dilations

proving similar triangles, 457, 459of rectangles, 443of similar triangles, 438–442, 444–445

Direct proof, 638Directrix of a parabola, 1000, 1010,

1012–1015Discs

of cylinders, 326defined, 301

Disjoint sets, 1061, 1093Distance

Angle Bisector/Proportional Side Theorem for, 466–468

on coordinate plane, 18–20, 24Distance Formula, 21–23, 71on a graph, 20to horizon, 874between lines and points not on lines,

71–73between points, 18–23from three or more points. See Points

of concurrencyusing Pythagorean Theorem, 21velocity-time graph for, 279–280

Distance Formula, 21–23, 71Distributive Property, 340Dot paper, 304Draw (geometric figures), 8Duplicating

an angle, 55–56a line segment

with an exact copy, 31–33with circles, 27–31

EElement (of a set), 1061

combinations of, 1184–1187repeated, permutations with,

1177–1181Elliptic geometry, 165Endpoint(s)

of angles, 52, 54of a line segment, 11, 26of a ray, 10

EqualityAddition Property of, 170Subtraction Property of, 171

Equal symbol (=), 12Equidistant Chord Converse

Theorem, 882Equidistant Chord Theorem, 881Equilateral triangles

altitudes of, 109angle bisectors of, 99constructing, 82on coordinate plane, 381defined, 13exterior angles of polygons, 806medians of, 104perpendicular bisectors of, 94

Cosine ratios, 695–706inverse cosine, 702–704secant ratio, 700–701

Cotangent (cot), 679Cotangent ratio, 679–681Counterexamples, 143Counting Principle, 1066–1069Cross sections

area offor cylinders, 339–340for hemispheres, 341–342

determining shapes of, 352–358cones, 356cubes, 354–355cylinders, 352hexagons, 357pentagons, 357pyramids, 355spheres, 353

Cubes, cross-section shapes for, 354–355

Cylindersannulus of, 340building, 326–328cross-section shapes for, 352height of, 300, 344, 345oblique, 309, 323radius of, 344, 345right, 309, 311, 323as rotation of rectangles, 300tranformations for, 314from two-dimensional figures

by stacking, 310by translation, 308–309

volume of, 313, 314, 326–328, 334–335, 344–345

DData, median of, 327Deduction

defined, 137identifying, 138–142

Degree measuresconverting to radian measures, 932defined, 850of intercepted arcs, 924of minor arcs, 924–925

Dependent events, 1062–1065, 1099compound probability of

with “and,” 1081with “or,” 1094–1098

conditional probability of, 1164–1168on two-way tables, 1139–1140

Diagonalsof kites, 772, 818of parallelograms, 758, 818of quadrilaterals, 818of rectangles, 748, 750of rhombi, 762, 818of squares, 746–747of three-dimensional solids, 360–366two-dimensional, 360

Diagonal translation, of three-dimensional figures, 305, 307

Diameterof circles, 946

and chords, 878–882

Coordinate planearea on

composite figures, 274–275, 283–285

parallelograms, 256–266squares, 228–230trapezoids, 272–273triangles, 236–237, 239–241,

243–247, 249–250circles and polygons on, 966–971classifying quadrilaterals on,

820–825congruent triangles on

Angle-Angle-Side Congruence Theorem, 571–573

Angle-Side-Angle Congruence Theorem, 564–566

Side-Angle-Side Congruence Theorem, 554–556

Side-Side-Side Congruence Theorem, 546–549

dilations on, 444–445Distance Formula on, 21–23, 71distance on, 18–20, 24line segments on

midpoint of, 36–38, 41, 43–44translating, 24–26

parallel lines on, 65perimeter on

composite figures, 282parallelograms, 258triangles, 236–237, 239–242,

247–248, 252reflecting geometric figures on,

527–532rotating geometric figures on,

518–525translating angles on, 52–54translating geometric figures on,

515–517Coplanar lines, 9Copying

an angle, 55–56a line segment, 27–33

with an exact copy, 31–33using circles, 27–30

Corresponding Angle Converse Postulate, 202–203

Corresponding Angle Postulate, 192–194

Corresponding parts of congruent triangles are congruent (CPCTC), 618–624

applications of, 622–624Isosceles Triangle Base Angle

Converse Theorem proved by, 621

Isosceles Triangle Base Angle Theorem proved by, 620

Cosecant (csc), 689Cosecant ratio, 689Cosine (cos)

defined, 696Law of Cosines

appropriate use of, 728defined, 724deriving, 722–725

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Hinge Converse Theorem, 642–645Hinge Theorem, 640–641Horizontal lines, 69–70

identifying, 69–70writing equations for, 70

Horizontal translation, 25, 26of angles, 53of quadrilaterals, 232of three-dimensional figures, 305

Hyperbolic geometry, 165Hypotenuse

of 45°–45°–90° triangles, 413, 659–664

of right triangles, altitudes drawn to, 482–488

of 30°–60°–90° triangles, 422, 664–667

Hypotenuse-Angle (HA) Congruence Theorem, 607–608

Hypotenuse-Leg (HL) Congruence Theorem, 600–604

Hypothesesof conditional statement, 144conjectures as, 192defined, 144rewriting, 148–149

IImage

of angles, 54defined, 24–26of line segments, 25–26pre-image same as, 54

Incenteralgebra used to locate, 113constructing, 96–100defined, 100

Included angle, 458Included side, 458Independent events, 1062–1065, 1099

compound probability ofwith “and,” 1081with “or,” 1088–1093

conditional probability of, 1164–1168multiple trials of two, 1195–1207

using combinations, 1197–1204using formula for, 1205–1207

Rule of Compound Probability involving “and,” 1078

two trials of two, 1192–1194on two-way tables, 1139, 1140

Indirect measurement, 496–502defined, 496of height, 496–499of width, 500–502

Indirect proof (proof by contradiction), 638–639

Hinge Converse Theorem, 642–645Hinge Theorem, 640–641Tangent to a Circle Theorem, 872

Indivisibles, method of, 321Induction

defined, 137identifying, 138–142

Inscribed angles (circles), 844, 853–858

Inscribed Angle Theorem, 855–858

Alternate Interior Angle Theorem, 195Congruent Complement Theorem, 185Congruent Supplement Theorem,

181–183defined, 175Right Angle Congruence Theorem, 180Same-Side Exterior Angle Theorem, 198Same-Side Interior Angle Converse

Theorem, 206Triangle Proportionality Theorem,

469–472Vertical Angle Theorem, 187

Focus of a paraboladefined, 1000distance from vertex to, 1009–1015on a graph, 1012–1015

45°–45°–90° triangles, 412–417, 658–66445°–45°–90° Triangle Theorem, 413–416Fractions

involving factorials, 1173rationalizing the denominator of, 673

Frequency tablesdefined, 1141two-way (contingency), 1145two-way relative frequency, 1147–1153

GGeneral form

of a circle, 977–979of a parabola, 1002

Geometric figurescreating, 8–9reflecting

on coordinate plane, 527–532without graphing, 532–533

rotatingon coordinate plane, 518–525without graphing, 525–526

translatingon coordinate plane, 515–517without graphing, 518

See also specific topics; specific types of figures

Geometric mean, 485–488Geometric probability, 1210–1214Great circle of a sphere, 338

HHeight

of cones, 302, 339of cylinders, 300, 344, 345of hemispheres, 341, 342indirect measurement of, 496–499of parallelograms, 257, 259–261of prisms, 335of solid figures, 334of trapezoids, 274of triangles, 237, 243, 245, 252

Hemispheres, 341defined, 338height of, 341, 342

Hexagons, 284–285area of, 285cross-section shapes for, 357exterior angles of, 804, 806, 810interior angles of, 797–800perimeter of, 285

Error, in indirect measurement, 497Euclid, 164Euclidean geometry

defined, 164non-Euclidean geometry vs., 164–165

Events (probability), 1040, 1061–1066complements of, 1042compound

defined, 1074involving “and,” 1074–1085involving “or,” 1088–1099with replacements, 1102–1105, 1107on two-way tables, 1138–1153without replacements, 1105–1107

defined, 1040dependent, 1062–1065, 1099

compound probability of, 1081, 1094–1098

conditional probability of, 1164–1168

on two-way tables, 1139–1140expected value of, 1215–1221independent, 1062–1065, 1078, 1099

compound probability of, 1081, 1088–1093

conditional probability of, 1164–1168

Rule of Compound Probability involving “and,” 1078

on two-way tables, 1139, 1140simulating, 1114–1122

Expected valuedefined, 1216probability of receiving, 1215–1221

Experimental probability, 1120, 1122Exterior Angle Inequality Theorem,

397–399Exterior angles

of circlesExterior Angles of a Circle

Theorem, 868–870vertices of, 871–874

of polygons, 802–812defined, 802equilateral triangles, 806hexagons, 804, 806, 810measures of, 807–809nonogons, 808–809pentagons, 803–804, 806, 812quadrilaterals, 803squares, 806, 811sum of, 802–806

of triangles, 393–399Exterior Angle Inequality Theorem,

397–399Exterior Angle Theorem, 396

Exterior Angles of a Circle Theorem, 868–870

Exterior Angle Theorem, 396External secant segment, 894–897

FFactorials, 1172–1173Flow chart proof, 175–177

Alternate Exterior Theorem, 196Alternate Interior Angle Converse

Theorem, 204

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copying/duplicating, 27–34with an exact copy, 31–33using circles, 27–31

defined, 11end-points of, 11measures of, 11Midpoint Formula, 39–43midpoint of, 36–44

by bisecting, 45–49on coordinate plane, 36–38, 41,

43–44Midpoint Formula, 39–43

naming, 11, 14points on perpendicular bisector

of equidistant to endpoints of segment, 580–581

symbol (s), 11tangent, 891–893translating, 24–26

Locus of points, 1000

MMajor arc (circles), 851

defined, 844degree measure of, 850length of, 931naming, 844

Measurementdegrees of error in, 497indirect, 496–502

Mediandefined, 327of a triangle, 101

Method of indivisibles, 321Midpoint Formula, 39–43Midpoints

and characteristics of polygons, 968–970

of line segmentby bisecting, 45–49on coordinate plane, 36–38, 41,

43–44Midpoint Formula, 39–43

Midsegments (of trapezoids), 781–784Minor arc (circles), 850, 851

defined, 844degree measure of, 850, 924–925length of, 931

NNon-Euclidean geometry, 153Nonogons, exterior angles of, 808–809Non-square rectangles, properties of, 229Non-uniform probability model, 1043–1044

OOblique cylinders, 309, 323Oblique rectangular prisms, 308, 322Oblique triangular prism, 306Obtuse scalene triangles, 545Obtuse triangles

altitudes of, 107angle bisectors of, 97on coordinate plane, 381medians of, 102perpendicular bisectors of, 92points of concurrency for, 111

properties of, 772–774proving, 773–774solve problems using, 785

LLaw of Cosines

appropriate use of, 728defined, 724deriving, 722–725

Law of Sinesappropriate use of, 728defined, 721deriving, 720–721

Leg-Angle (LA) Congruence Theorem, 609–610, 622

Leg-Leg (LL) Congruence Theorem, 605–606

Legs of the trapezoid, 273Linear Pair Postulate, 166, 774, 802–804Linear pair(s)

of angles, 158–159defined, 159

Linear velocity, 949–950Line(s), 4–5

concurrent, 90coplanar, 9defined, 4dilating, 439distance between points not on line

and, 71–73horizontal, 69–70intersection of plane and, 7naming, 14parallel, 62–65

constructing, 80–81converse conjectures, 204–207equations of, 63, 64identifying, 64, 67intersecting circles, 859Parallel Lines–Congruent Arcs

Theorem, 859Perpendicular/Parallel Line

Theorem, 743–747slopes of, 62–65, 67

perpendicular, 66–68conditional statements about, 637constructing, 76–79equations of, 68identifying, 67Perpendicular/Parallel Line

Theorem, 743–747slope of, 67–68through a point not on a line,

77–79through a point on a line, 76–77

skew, 9symbol (↔), 4through points, 4, 5unique, 4vertical, 69–70

Line segment(s)bisecting, 45–49

by construction, 46–49defined, 45with patty (tracing) paper, 45–46

concurrent, 90congruent, 12–13, 29

Inscribed figuresparallelograms, 969polygons, 912–915, 953–958quadrilaterals, 917–918squares, 951, 968triangles, 912–915

Inscribed Right Triangle–Diameter Converse Theorem, 915

Inscribed Right Triangle–Diameter Theorem, 913–914

Integers, conditional statements about, 635

Intercepted arcs (circles), 853defined, 852degree measures of, 924

Interior anglesof circles, vertices of, 864–865of polygons, 790–800

defined, 790measures of, 794, 795sum of measures of, 790–800Triangle Sum Theorem, 791

of trianglesremote, 394–395and side length, 389–393, 412

Interior Angles of a Circle Theorem, 865Intersecting sets, 1061Inverse, of conditional statements,

634–637Inverse cosine (arc cosine), 702–704Inverse sine (arc sine), 690–691Inverse tangent (arc tangent), 681–683Irregularly shaped figures

approximating area of, 320–321volume of, 346–350See also Composite figures

Isometric paper (dot paper), 304Isometric projection, 303Isosceles right triangle, 417Isosceles trapezoids, 776–780, 970Isosceles Triangle Altitude to Congruent

Sides Theorem, 629Isosceles Triangle Angle Bisector to

Congruent Sides Theorem, 629Isosceles Triangle Base Angle Converse

Theorem, 621, 622Isosceles Triangle Base Angle Theorem,

620, 623Isosceles Triangle Base Theorem, 626Isosceles Triangle Perpendicular

Bisector Theorem, 628Isosceles triangles

constructing, 8, 83on coordinate plane, 381defined, 13identifying, 384similar, 454vertex angle of, 626

Isosceles Triangle Vertex Angle Theorem, 627

KKites

characteristics of, 814–816constructing, 772defined, 772diagonals of, 772, 818

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shapes of intersections of solids and, 352–358

Point of rotation, 518Point of tangency, 843, 890Point(s), 4

on circles, 990–997collinear, 5defined, 4distance between, 18–23distance between lines and,

71–73lines passing through, 4, 5locus of, 1000reflecting, 527–530See also Points of concurrency

Point-slope form, 63Points of concurrency, 87–120

for acute, obtuse, and right triangles, 111

algebra used to locate, 112–117centroid, 101–105circumcenter, 91–95constructing, 88–89defined, 90incenter, 96–100orthocenter, 106–111

Polygonsarea, 813

of hexagons, 285in inscribed polygons, 953–958

circumscribed, 916conditional statements about, 636on coordinate plane, 967–971exterior angles of, 802–812

defined, 802equilateral triangles, 806hexagons, 804, 806, 810measures of, 807–809nonogons, 808–809pentagons, 803–804, 806, 812squares, 806, 811sum of, 802–806

four-sided. See Quadrilateralshexagons, 284–285, 797–800

area of, 285cross-section shapes for, 357exterior angles of, 804,

806, 810interior angles of, 797–800perimeter of, 285

identifying, 817inscribed, 912–915, 953–958interior angles of, 790–800

defined, 790measures of, 794, 795sum of measures of, 790–800Triangle Sum Theorem, 791

nonogons, 808–809octagons, 800pentagons, 796

cross-section shapes for, 357exterior angles of, 803–804,

806, 812interior angles of, 796

reflecting, 531rotating, 523, 525undecagons, 796

Parallelogramsarea of, 813

and area of trapezoids, 274–275Cavalieri’s principle for, 324on coordinate plane, 256–266doubling, 269rectangle method for, 256–258,

263–266characteristics of, 814–816constructing, 758defined, 758diagonals of, 758, 818height of, 257, 259–261inscribed, 969perimeter of, 257, 258properties of, 758–761

proving, 758–761solve problems using, 765–767

rectangles vs., 256rhombus, 762–765rotating, 525–526

Patterns, identifying through reasoning, 142

Penrose Triangle, 597Pentagons

cross-section shapes for, 357exterior angles of, 803–804, 806, 812interior angles of, 796

Perimeterof composite figures

on coordinate plane, 282hexagons, 285

constructing a rectangle given, 85constructing a square given, 84of hexagons, 285of parallelograms, 257, 258of trapezoids, 273of triangles

on coordinate plane, 236–237, 239–242, 247–248, 252

transformations for, 238, 240, 241using Triangle Midsegment

Theorem, 479Permutations, 1174–1176

circular, 1182–1183and combinations, 1184–1187defined, 1174with repeated elements, 1177–1181

Perpendicular bisectorsof chords, 878defined, 76–79of triangles, 91–95

Perpendicular lines, 66–68conditional statements about, 637constructing, 76–79equations of, 68identifying, 67slope of, 67–68through a point not on a line, 77–79through a point on a line, 76–77

Perpendicular/Parallel Line Theorem, 743–747

Plane(s), 6–7defined, 6intersection of, 6–7intersection of line and, 7naming, 6

Octagons, interior angles of, 800Opposite side

defined, 659of 45°–45°–90° triangles, 659–664of 30°–60°–90° triangles, 664–667

Organized lists, 1051, 1053Orthocenter

algebra used to locate, 113, 116–117

constructing, 106–111defined, 110

Outcomes (probability), 1040defined, 1040in independent and dependent

events, 1062–1065in probability models, 1040–1046

PParabolas

applications of, 1016–1018axis of symmetry, 1004–1006,

1012–1015concavity of, 1004, 1006, 1008,

1012–1015defined, 1000directrix of, 1000, 1010, 1012–1015equations of, 1001–1003focus of, 1012–1015

defined, 1000distance from vertex to, 1009–1015on a graph, 1012–1015

general form of, 1002graphing, 1012–1015key characteristics of, 1004–1008as sets of points, 1000solving problems with, 1022–1032standard form of, 1002, 1007vertex of, 1004

coordinates of, 1006distance from vertex to focus,

1009–1015on graphs, 1012–1015

Paragraph proof, 178defined, 178of Triangle Proportionality

Theorem, 469Parallel lines, 62–65

constructing, 80–81converse conjectures

Alternate Exterior Angle Converse Conjecture, 205

Alternate Interior Angle Converse Conjecture, 204

Same-Side Exterior Angle Converse Conjecture, 207

Same-Side Interior Angle Converse Conjecture, 206

equations of, 63, 64identifying, 64, 67intersecting circles, 859Perpendicular/Parallel Line Theorem,

743–747slopes of, 62–65, 67

Parallel Lines–Congruent Arcs Theorem, 859

Parallelogram/Congruent-Parallel Side Theorem, 761

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Right Angle Congruence Theorem, 179–180

Side-Angle-Side Congruence Theorem, 557

supplementary and complementary angles, 152–155

two-column. See Two-column prooftypes of reasoning, 137Vertical Angle Theorem, 186–188

Properties of real numbers, 170–174Addition Property of Equality, 170Reflexive Property, 172Substitution Property, 173Subtraction Property of Equality, 171Transitive Property, 174

Proportionalityin similar triangles, 452, 453, 455–459

proving Pythagorean Theorem with, 490–491

Right Triangle/Altitude Similarity Theorem, 484, 490–491

theorems, 464–479Angle Bisector/Proportional Side

Theorem, 464–468Converse of Triangle

Proportionality Theorem, 474Proportional Segments

Theorem, 475Triangle Midsegment Theorem,

476–479Triangle Proportionality Theorem,

469–473Proportional Segments Theorem, 475Proportions, in indirect measurement,

496–502Propositional variables, 144Protractor, 8Pyramids

cross-section shapes for, 355rectangular, 312from stacking two-dimensional

figures, 312, 313tranformations for, 314triangular, 312volume of, 314, 335

Pythagorean Theoremand complement angle relationships,

710–711Converse of, 493–494distance using, 21, 874for equation of a circle, 974–976to identify right triangles, 383for points on a circle, 990–994proving

with algebraic reasoning, 492Converse of, 493–494with Right Triangle/Altitude

Similarity Theorem, 490–491with similar triangles, 490–491

proving 45°–45°–90° Triangle Theorem with, 413, 414

proving 30°–60°–90° Triangle Theorem with, 422

for side length of triangles, 341for three-dimensional diagonals,

360, 362for triangles on coordinate plane, 967

circular, 1182–1183and combinations, 1184–1187defined, 1174with repeated elements, 1177–1181

sample spaces, 1040calculating, 1069–1070compound, 1048–1060defined, 1040determining, 1043factorials, 1172–1173organized lists, 1051, 1053with replacements, 1102–1105, 1107strings, 1170–1171tree diagrams, 1048–1052,

1054–1057without replacements, 1105–1107

sets, 1061–1062simulation

defined, 1120using random number generator,

1114–1122theoretical, 1120, 1122two trials of two independent events,

1192–1194Probability models, 1040–1046

defined, 1040non-uniform, 1043–1044uniform, 1042

Proof, 135–150Alternate Exterior Angle Theorem, 196Alternate Interior Angle Theorem,

194–195angle relationships, 156–163coming to conclusions, 138–142conditional statements, 144–149Congruent Complement Theorem,

184–186Congruent Supplement Theorem,

181–184construction, 179by contradiction, 638with Corresponding Angle Converse

Postulate, 202–203with Corresponding Angle Postulate,

192–194deduction, 137–142defined, 175direct, 638flow chart. See Flow chart proofindirect, 638–639induction, 137–142paragraph. See Paragraph proofof parallel line converse conjectures,

204–207Perpendicular/Parallel Line Theorem,

743–747postulates and theorems, 164–168properties of quadrilaterals

kites, 773–774parallelograms, 758–761rectangles, 749–750rhombus, 762–764squares, 744–747trapezoids, 776–779

and properties of real numbers, 170–174

recognizing false conclusions, 143

Postulates, 164–168conjectures from, 165defined, 164of Euclid, 164See also individual postulates

Pre-imageof angles, 54defined, 24–26image same as, 54of line segments, 25–26

Prismsheight of, 335rectangular, 306–307

diagonals of, 363–364oblique, 308, 322right, 308, 310, 322

right, 311tranformations for, 314triangular, 304–305

oblique, 306right, 306from stacking two-dimensional

figures, 311volume of, 313, 314, 322, 335

ProbabilityAddition Rule for Probability, 1093combinations, 1184–1187compound

with “and,” 1072–1086calculating, 1102–1112for data displayed in two-way

tables, 1138–1153with “or,” 1088–1099with replacements, 1102–1105,

1107on two-way tables, 1138–1153without replacements, 1105–1107

conditional, 1156–1168building formula for, 1161–1163defined, 1158dependability of, 1164–1165on two-way tables, 1158–1162using formula for, 1166–1168

Counting Principle, 1066–1069defined, 1040events, 1040, 1061–1066

complements of, 1042defined, 1040dependent, 1062–1065, 1081,

1094–1099, 1139–1140, 1164–1168

expected value of, 1215–1221independent, 1062–1065, 1078,

1081, 1088–1093, 1099, 1139, 1140, 1164–1168

simulating, 1114–1122expected value, 1215–1221experimental, 1120, 1122geometric, 1210–1214models, probability, 1040–1046Monty Hall problem, 1071multiple trials of two independent

events, 1195–1207using combinations, 1197–1204using formula, 1205–1207

outcome, 1040permutations, 1174–1176

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naming, 14symbol (→), 10

Real numbers, properties of, 170–174Reasoning

deduction, 137–142identifying types of, 138–142induction, 137–142

Rectangle method (for area)parallelograms, 256, 263–265trapezoids, 276–278triangles, 265–266

Rectanglesarea of

and area of a parallelogram, 256, 263–265

and area of a triangle, 265–266on coordinate plane, 228–230Distance Formula for, 229, 232transformations for, 231–232

characteristics of, 814–816congruent, 232constructing, 85, 748defined, 229, 748diagonals of, 748, 750dilation of, 443non-square, 229parallelograms vs., 256Perpendicular/Parallel Line Theorem,

743–747properties of, 748–750

proving, 749–750solve problems using, 751–753

rotated through space, 300, 327–328squares vs., 229, 230

Rectangular prismsdiagonals of, 363–364oblique, 308, 322right, 308, 310, 322from translating two-dimensional

figures, 306–307Rectangular pyramids, 312Rectangular solids, diagonals of,

360–366Reference angle

defined, 659of 45°–45°–90° triangles, 659–664of 30°–60°–90° triangles, 664–667

Reflectioncongruent triangles

Angle-Angle-Side Congruence Theorem, 571–573

Angle-Side-Angle Congruence Theorem, 564

on coordinate plane, 527–532defined, 527shape and size preserved in, 232of triangles, 240, 241, 540

for congruence, 548–549proving similarity, 448for similarity, 457

without graphing, 532–533Reflexive Property, 172, 612Regular tetrahedron, 1207Relative frequency

defined, 1147two-way relative frequency tables,

1147–1153

circumscribed, 919–921constructing, 762on coordinate plane, 822–824defined, 762diagonals of, 762, 818formed from isosceles

trapezoids, 970properties of, 762–765, 768–769

squaresarea of, 228–230, 813characteristics of, 814–816circumscribed, 952constructing, 8, 84, 743on coordinate plane, 820–822, 825diagonals of, 746–747exterior angles of polygons,

806, 811inscribed, 951, 968Perpendicular/Parallel Line

Theorem, 743–747properties of, 229, 742–747,

754–755rectangles vs., 229, 230

Trapezoid Midsegment Theorem, 783–784

trapezoidsarea of, 272–278base angles of, 775bases of, 273, 274characteristics of, 814–816constructing, 775, 780defined, 272, 775isosceles, 776–780, 970legs of, 273, 775midsegments of, 781–784perimeter of, 273properties of, 775–779, 786–787reflecting, 527–533rotating, 518–526translating, 515–518on velocity-time graphs, 279–280

RRadians, 932–933Radius(—i)

and arc length, 927–928of circles, 840–841

algebraic determination of, 977–987

as congruent line segments, 29defined, 840and diameter, 841length of, 28

of cylinders, 344, 345of spheres, 301, 338

Random number generator, 1114–1122Rationalizing the denominator, 673Ratio(s)

in probability, 1040of similar rectangles, 443of similar triangles, 442, 455, 456slope, 534

Ray(s)of angles, 52concurrent, 90defined, 10endpoint of, 10

QQuadrilateral–Opposite Angles Theorem,

917–918Quadrilaterals

area ofDistance Formula for, 229, 232parallelograms, 256–266, 269,

274–275, 324, 813polygons, 813rectangles, 228–232, 263–266squares, 228–230, 813transformations for, 231–232trapezoids, 272–278

characteristics of, 814, 815circumscribed, 920–921classifying on coordinate plane,

820–825conditional statements about, 634congruent, 232defined, 742diagonals of, 818exterior angles of, 803identifying, 766inscribed, 917–918kites

characteristics of, 814–816constructing, 772defined, 772diagonals of, 772, 774, 818properties of, 772–774, 785

Parallelogram/Congruent-Parallel Side Theorem, 761

parallelogramsarea of, 256–266, 269, 274–275,

324, 813characteristics of, 814–816constructing, 758defined, 758diagonals of, 758, 818height of, 257, 259–261inscribed, 969perimeter of, 257, 258properties of, 758–761, 765–767rectangles vs., 256rhombus, 762–765rotating, 525–526

Perpendicular/Parallel Line Theorem, 743–747

properties of, 742, 814–818rectangles

area of, 228–232, 256, 263–266characteristics of, 814–816congruent, 232constructing, 85, 748defined, 229, 748diagonals of, 748, 750dilation of, 443non-square, 229parallelograms vs., 256Perpendicular/Parallel Line

Theorem, 743–747properties of, 748–753rotated through space, 300,

327–328squares vs., 229, 230

rhombicharacteristics of, 814–816

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Scalene trianglesacute, 386on coordinate plane, 381identifying, 382obtuse, 545

Secant (sec), 894–897defined, 700, 842and tangent, 843

Secant ratio, 700–701Secant segments

defined, 894external, 894–897length of, 967

Secant Segment Theorem, 895Secant Tangent Theorem, 897Sector of a circle, 936–939

defined, 937determining area of, 937–939, 947–948number of, 937

Segment Addition Postulate, 167Segment bisector

constructing, 45–49defined, 45

Segment–Chord Theorem, 888Segments

of a chord, 887–888of a circle

area of, 940–942defined, 940

Semicircle, 844Sequences, identifying, 142Sets, 1061–1062

defined, 1061disjoint, 1061, 1093intersecting, 1061

Side-Angle-Side (SAS) Congruence Theorem, 552–560, 584–586, 624

congruence statements for, 557–560congruent triangles on coordinate

plane, 554–556constructing congruent triangles,

552–553defined, 552proof of, 557

Side-Angle-Side (SAS) Similarity Theorem, 458–459, 461

Side-Side-Angle (SSA), 582Side-Side-Side (SSS) Congruence

Theorem, 543–549, 585congruence statement for, 558–560congruent triangles on coordinate

plane, 546–549constructing congruent triangles,

544–545proof of, 549

Side-Side-Side (SSS) Similarity Theorem, 455–457, 461

Similar circles, 846–847Similar triangles

constructing, 444–445with Angle-Angle Similarity

Theorem, 452–454with Side-Angle-Side Similarity

Theorem, 458–459with Side-Side-Side Similarity

Theorem, 455–457defined, 446

Right Triangle Altitude/Hypotenuse Theorem, 485

Right Triangle Altitude/Leg Theorem, 485

Right Triangle/Altitude Similarity Theorem, 482–484

sine ratios, 685–694cosecant ratio, 689inverse sine, 690–691

tangent ratios, 670–679, 683cotangent ratio, 679–681inverse tangent, 681–683

Right triangular prism, 306Rigid motion

defined, 25to determine points on a circle,

995–997to prove similar circles, 846–847in proving points on perpendicular

bisector of equidistant to endpoints of segment, 581

See also Rotation; TranslationRotation

congruent trianglesAngle-Side-Angle Congruence

Theorem, 564Side-Angle-Side Congruence

Theorem, 554–556on coordinate plane, 518–525defined, 518proving similar triangles, 448, 457, 459shape and size preserved in, 232of triangles, 539of two-dimensional figures through

space, 300–302to form cones, 329to form cylinders, 327

Rule of Compound Probability involving “and,” 1078

SSame-Side Exterior Angle Converse

Theorem, 203, 207Same-Side Exterior Angle Theorem, 198Same-Side Interior Angle Converse

Theorem, 202, 206Same-Side Interior Angle Theorem, 197Sample spaces (probability), 1040

calculating, 1069–1070compound, 1048–1060defined, 1040determining, 1043factorials, 1172–1173organized lists, 1051, 1053with replacements, 1102–1105, 1107strings, 1170–1171tree diagrams, 1048–1052,

1054–1057without replacements, 1105–1107

Scale factorwith dilations

rectangles, 445similar triangles, 445

with similar trianglesconstructing similar triangles,

455, 457proving similarity, 448

Remote interior angles, 394–395Rhombus(—i)

characteristics of, 814–816circumscribed, 919–921constructing, 762on coordinate plane, 822–824defined, 762diagonals of, 762, 818formed from isosceles trapezoids, 970properties of, 762–765

proving, 762–764solve problems using, 768–769

Right Angle Congruence Theorem, 179–180

Right angles, congruence of, 600Right cylinders

from stacking two-dimensional figures, 311

from translating two-dimensional figures, 309

volume of, 323Right prisms, 311Right rectangular prisms

from stacking two-dimensional figures, 310

from translating two-dimensional figures, 308

volume of, 322Right Triangle Altitude/Hypotenuse

Theorem, 485Right Triangle Altitude/Leg Theorem, 485Right Triangle/Altitude Similarity

Theoremdefined, 484proving, 482–484proving Pythagorean Theorem with,

490–491Right triangles

altitudes of, 108, 485–488angle bisectors of, 98complement angle relationships in,

708–715congruence theorems, 599–616

applying, 611–615Hypotenuse-Angle Congruence

Theorem, 607–608Hypotenuse-Leg Congruence

Theorem, 600–604Leg-Angle Congruence Theorem,

609–610Leg-Leg Congruence Theorem,

605–606conversion ratios, 658–667

for 45°–45°–90° triangles, 658–664

for 30°–60°–90° triangles, 664–667on coordinate plane, 380cosine ratios, 695–706

inverse cosine, 702–704secant ratio, 700–701

identifying, 383, 536isosceles, 417medians of, 103perpendicular bisectors of, 93points of concurrency for, 111similar

geometric mean, 485–488

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as rotation of rectangles, 300by stacking two-dimensional

figures, 310tranformations for, 314by translation of two-dimensional

figures, 308–309volume of, 313, 314, 326–328,

334–335, 344–345diagonals of, 360–366prisms

height of, 335rectangular, 306–308, 310, 322,

363–364right, 311right rectangular, 308, 322tranformations for, 314triangular, 304–306, 311volume of, 313, 314, 322, 335

pyramidscross-section shapes for, 355rectangular, 312from stacking two-dimensional

figures, 312, 313tranformations for, 314triangular, 312volume of, 314, 335

shapes of intersections of planes and, 352–358

spherescross-section shapes for, 353defined, 338diameter of, 338great circle of, 338radius of, 338as rotation of circles, 301volume of, 336–342

from two-dimensional figuresrotated, 300–302stacked, 310–315translated, 304–309

volume ofCavalieri’s principle for, 322–324cones, 314, 324, 332–335cylinders, 313, 314, 326–328,

334–335, 344–345prisms, 313, 314, 322, 335pyramids, 314, 335spheres, 336–342

Transformationsfor area

of quadrilaterals, 231–232of triangles, 238–241, 250

for cones, 314for cylinders, 314defined, 25dilations

proving similar triangles, 457, 459of rectangles, 443similar triangles, 438–442, 444–445

identifying, 440for perimeter

of trapezoids, 273of triangles, 238, 240, 241

for prisms, 314proving similar triangles, 448–449,

457, 459for pyramids, 314

Stacking, 310–315cones from, 312, 313cylinders from, 310, 311, 326prisms from, 310, 311pyramids from, 312, 313

Standard formequation of a circle, 977–979of a parabola, 1002, 1007

Straightedge, 8Strings, 1170–1171, 1184–1187Substitution Property, 173Subtraction Property of Equality, 171Supplementary angles, 152, 154–155Symmetry

in determining points on a circle, 992, 995–997

of parabolasaxis of, 1004–1006on coordinate plane, 1003lines of, 1004

TTangent (tan), 890–893

defined, 674, 843and secant, 843

Tangent ratios, 670–679, 683cotangent ratio, 679–681inverse tangent, 681–683

Tangent segments, 891–893Tangent Segment Theorem, 892Tangent to a Circle Theorem, 871–873Terms, defined by undefined terms,

10–15Tetrahedron, regular, 1207Theorems, 164

from conjectures, 165defined, 164as proved conjectures, 194proving similar triangles, 446–447See also individual theorems

Theoretical probability, 1120, 112230°–60°–90° triangles, 420–427,

664–66730°–60°–90° Triangle Theorem, 422–424Three-dimensional solids

Cavalieri’s principle for volume of, 322–324

conesbuilding, 329–332cross-section shapes for, 356diameter of, 302height of, 302, 339as rotation of triangles, 301from stacking two-dimensional

figures, 312, 313tranformations for, 314volume of, 314, 324, 332–335

cubes, cross-section shapes for, 354–355

cylindersannulus of, 340building, 326–328cross-section shapes for, 352height of, 300, 344, 345oblique, 309, 323radius of, 344, 345right, 309, 323

dilations, 438–442, 444–445geometric theorems proving, 446–447indirect measurement using, 496–502proving Pythagorean Theorem with,

490–491right

geometric mean, 485–488Right Triangle Altitude/Hypotenuse

Theorem, 485Right Triangle Altitude/Leg

Theorem, 485Right Triangle/Altitude Similarity

Theorem, 482–484sides and angles not ensuring

similarity, 461transformations proving, 448–449

Simulationdefined, 1120using random number generator,

1114–1122Sine (sin)

defined, 687Law of Sines

appropriate use of, 728defined, 721deriving, 720–721

Sine ratios, 685–694cosecant ratio, 689inverse sine, 690–691

Sketch (of geometric figures), 8Skew lines, 9Slope

cotangent ratio, 679–681of horizontal lines, 69–70inverse tangent, 681–683of parallel lines, 62–65, 67of perpendicular lines, 67of rotated lines, 534tangent ratio, 670–679, 683of vertical lines, 69–70

Slope ratio, 534Spheres

cross-section shapes for, 353defined, 338diameter of, 338great circle of, 338radius of, 338as rotation of circles, 301volume of, 336–342

Spherical triangles, 625Squares

area of, 813on coordinate plane, 228–230Distance Formula for, 229

characteristics of, 814–816circumscribed, 952constructing, 8, 84, 743on coordinate plane, 820–822, 825diagonals of, 746–747exterior angles of polygons, 806, 811inscribed, 951, 968Perpendicular/Parallel Line Theorem,

743–747properties of, 229, 742–747

proving, 744–747solve problems using, 754–755

rectangles vs., 229, 230

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perpendicular bisectors of, 91points of concurrency for, 111

altitudes of, 106–110analyzing, 389–393angle bisectors of, 96–100area of

on coordinate plane, 236–237, 239–241, 243–247, 249–250

doubling, 253rectangle method for, 265–266transformations for, 238–241, 250

centroid of, 105circumcenter of, 95circumscribed, 916classifying, 380–386congruent, 239, 536–541

and Angle-Angle-Angle as not a congruence theorem, 582

Angle-Angle-Side Congruence Theorem, 568–573

Angle-Side-Angle Congruence Theorem, 562–566

congruence statements for, 539–540Congruence Theorems in

determining, 584–588corresponding angles of, 538corresponding parts of, 618–624corresponding parts of congruent

triangles are congruent, 618–624corresponding sides of, 536–537points on perpendicular bisector

of line segment equidistant to endpoints of line segment, 580–581

Side-Angle-Side Congruence Theorem, 552–556

and Side-Side-Angle as not a congruence theorem, 583

Side-Side-Side Congruence Theorem, 544–549

constructing, 82–83on coordinate plane, 380defined, 13equilateral

altitudes of, 109angle bisectors of, 99constructing, 82on coordinate plane, 381defined, 13exterior angles of polygons, 806medians of, 104perpendicular bisectors of, 94

exterior angles, 393–399Exterior Angle Inequality

Theorem, 397Exterior Angle Theorem, 396

exterior angles of, 393–39945°–45°–90°, 412–417, 658–664height of, 237, 243, 245, 252incenter of, 100inscribed in circles, 912–915interior angles

remote, 394–395and side length, 389–393, 412

isoscelesconstructing, 8, 83on coordinate plane, 381

on coordinate plane, 515–517without graphing, 518

horizontal, 25, 26, 53of angles, 53of quadrilaterals, 232of three-dimensional figures, 305

of line segments, 24–26of parallel lines, 65proving similar triangles, 457, 459of quadrilaterals, 232shape and size preserved in, 232of trapezoids, 275of triangles, 238, 240, 241, 250,

536–537of two-dimensional figures through

space, 304–309vertical, 25, 26, 53

of angles, 53of three-dimensional figures, 305of triangles, 238, 241

Trapezoid Midsegment Theorem, 783–784

Trapezoidsarea of

and area of a parallelogram, 274–275

on coordinate plane, 272–273rectangle method for, 276–278

base angles of, 775bases of, 273, 274characteristics of, 814–816constructing, 775, 780defined, 272, 775isosceles, 776–780

constructing, 780defined, 776proving properties of, 776–779rhombus formed from, 970

legs of, 273, 775midsegments of, 781–784

on coordinate plane, 781defined, 782Trapezoid Midsegment

Theorem, 783perimeter of, 273properties of, 775–779

proving, 776–779solve problems using, 786–787

reflecting, 527–533rotating, 518–526translating, 515–518on velocity-time graphs, 279–280

Tree diagrams, 1048–1052, 1054–1057Triangle Inequality Theorem, 406–409Triangle Midsegment Theorem, 476–479Triangle Proportionality Theorem,

469–473Converse of, 474defined, 469proving, 469–473

Trianglesacute, 380

altitudes of, 106angle bisectors of, 96on coordinate plane, 380identifying, 385medians of, 101

Transformations (Cont.)reflection

congruent triangles, 564, 571–573on coordinate plane, 527–532defined, 527shape and size preserved in, 232of trapezoids, 527–533of triangles, 240, 241, 448, 457,

540, 548–549without graphing, 532–533

rigid motiondefined, 25to determine points on a circle,

995–997to prove similar circles, 846–847in proving points on perpendicular

bisector of equidistant to endpoints of segment, 581

rotationcongruent triangles, 554–556, 564on coordinate plane, 518–525defined, 518proving similar triangles, 448,

457, 459shape and size preserved in, 232of trapezoids, 518–526of triangles, 539of two-dimensional figures through

space, 300–302, 327, 329translation

of angles, 52–54on coordinate plane, 25–26,

515–517by copying/duplicating line

segments, 27–33defined, 25diagonal, 305to form three-dimensional figures,

304–309horizontal, 25, 26, 53, 232, 305of line segments, 24–26of parallel lines, 65proving similar triangles, 457, 459of quadrilaterals, 232shape and size preserved in, 232of trapezoids, 275, 515–518of triangles, 238, 240, 241, 250,

536–537of two-dimensional figures through

space, 304–309vertical, 25, 26, 53, 238, 241, 305without graphing, 518

of trapezoidsreflecting, 527–533rotating, 518–526translating, 515–518

Transitive Property, 174Translation

of angles, 52–54on coordinate plane, 25–26by copying/duplicating line segments,

27–33defined, 25diagonal, 305to form three-dimensional figures,

304–309of geometric figures

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Index I-13

Index

Truth tables, 146–147Truth values

of conditional statements, 144–147and their contrapositives, 637and their inverses, 637

defined, 144on truth tables, 146–147

Two-column proof, 178Alternate Exterior Angle Converse

Theorem, 205Alternate Interior Angle Theorem, 195with Angle Addition Postulate, 189Angle-Angle-Side Congruence

Theorem, 574Angle Bisector/Proportional Side

Theorem, 465Angle-Side-Angle Congruence

Theorem, 566Congruent Chord–Congruent Arc

Converse Theorem, 865Congruent Chord–Congruent Arc

Theorem, 885Congruent Complement Theorem, 186Congruent Supplement Theorem, 184with CPCTC, 618–621defined, 178Diameter–Chord Theorem, 879Equidistant Chord Converse

Theorem, 882Equidistant Chord Theorem, 881Exterior Angles of a Circle Theorem,

868–870Hypotenuse-Leg Congruence

Theorem, 601indirect, 638, 639, 641, 643Inscribed Angle Theorem, 855–857Inscribed Right Triangle–Diameter

Converse Theorem, 915Inscribed Right Triangle–Diameter

Theorem, 914Interior Angles of a Circle Theorem, 865of isosceles triangle theorems,

626–630Perpendicular/Parallel Line Theorem,

744–746points on perpendicular bisector

of line segment equidistant to endpoints of line segment, 580–581

properties of quadrilateralsisosceles trapezoids, 776, 777kites, 773–774parallelograms, 758, 759rectangles, 749

Quadrilateral–Opposite Angles Theorem, 918

of right triangle congruence theorems, 613, 614

Same-Side Exterior Angle Converse Theorem, 207

Same-Side Interior Angle Theorem, 197

Secant Segment Theorem, 895Secant Tangent Theorem, 897Segment–Chord Theorem, 888Side-Side-Side Congruence

Theorem, 549

side lengths, 406–409of congruent triangles, 536–537,

546, 547geometric mean for, 485–488and interior angles, 389–393, 412of similar triangles, 452–453,

455–459, 461Triangle Inequality Theorem, 409

similarconstructing, 444–445, 452–459defined, 446dilations, 438–442, 444–445geometric theorems proving,

446–447indirect measurement using,

496–502proving Pythagorean Theorem

with, 490–491right, 482–488sides and angles not ensuring

similarity, 461transformations proving, 448–449

spherical, 62530°–60°–90°, 420–427, 664–667translation of, 238, 240, 241, 250,

536–537Triangle Inequality Theorem, 406–409Triangle Sum Theorem, 388, 394,

403, 447, 493–494, 791vertices’ coordinates, 478

Triangle Sum Theorem, 394, 403, 791Converse of the Pythagorean

Theorem proved with, 493–494defined, 388in proving similar triangles, 447

Triangular prisms, 304–305oblique, 306right, 306from stacking two-dimensional

figures, 311Triangular pyramids, 312Trigonometry

area of triangleapplying, 725–727deriving, 718–719

complement angle relationships, 708–715

conversion ratios, 658–664for 45°–45°–90° triangles, 658–664for 30°–60°–90° triangles, 664–667

cosine ratios, 695–706inverse cosine, 702–704secant ratio, 700–701

Law of Cosinesappropriate use of, 728defined, 724deriving, 722–725

Law of Sinesappropriate use of, 728defined, 721deriving, 720–721

sine ratios, 685–694cosecant ratio, 689inverse sine, 690–691

tangent ratios, 670–679, 683cotangent ratio, 679–681inverse tangent, 681–683

defined, 13identifying, 384Isosceles Triangle Altitude to

Congruent Sides Theorem, 629Isosceles Triangle Angle Bisector

to Congruent Sides Theorem, 629Isosceles Triangle Base Angle

Converse Theorem, 622Isosceles Triangle Base Angle

Theorem, 623Isosceles Triangle Base

Theorem, 626Isosceles Triangle Perpendicular

Bisector Theorem, 628Isosceles Triangle Vertex Angle

Theorem, 627similar, 454vertex angle of, 626

medians of, 101–105obtuse, 381

altitudes of, 107angle bisectors of, 97on coordinate plane, 381medians of, 102perpendicular bisectors of, 92points of concurrency for, 111

orthocenter of, 110perimeter of

on coordinate plane, 236–237, 239–242, 247–248, 252

transformations for, 238, 240, 241using Triangle Midsegment

Theorem, 479perpendicular bisectors of, 91–95proportionality theorems, 464–479

Angle Bisector/Proportional Side Theorem, 464–468

Converse of Triangle Proportionality Theorem, 474

Proportional Segments Theorem, 475Triangle Midsegment Theorem,

476–479Triangle Proportionality Theorem,

469–473right

altitudes of, 108, 485–488angle bisectors of, 98complement angle relationships in,

708–715congruence theorems, 599–616conversion ratios, 658–667on coordinate plane, 380cosine ratios, 695–706identifying, 383, 536isosceles, 417medians of, 103perpendicular bisectors of, 93points of concurrency for, 111similar, 482–488sine ratios, 685–694tangent ratios, 670–679, 683

rotated through space, 302, 329scalene

acute, 386on coordinate plane, 381identifying, 382obtuse, 545

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I-14 Index

Index

Vertical angles, 160–161Vertical Angle Theorem

defined, 186proof of, 186–188in proving similar triangles, 447

Vertical lines, 69–70identifying, 69–70writing equations for, 70

Vertical translation, 25, 26of angles, 53of three-dimensional figures, 305of triangles, 238, 241

Volumeapproximating, 346Cavalieri’s principle for, 322–324of composite figures, 347–350of cones, 314, 332–333of cylinders, 313, 314, 326–328,

334–335, 344–345of irregularly shaped figures,

346–350of prisms, 313, 314, 335of pyramids, 314, 335solving problems involving,

344–350of spheres, 336–342

WWidth, indirect measurement of,

500–502

VVelocity

angular, 949in circular motion, 949–950linear, 949

Velocity-time graph, 279–280Venn diagrams, 815Vertex angle (isosceles triangles), 626Vertex(—ices)

of angles of circles, 864central angles, 864inscribed angles, 864located inside the circle, 864–865located on the circle, 871–874located outside the circle, 866–870

of inscribed polygons, 912of a parabola, 1009–1015

coordinates of, 1006, 1008defined, 1004distance to focus from, 1009–1015on a graph, 1012–1015

of parallelograms, 525, 532–533of polygons, 233of quadrilaterals, 232

classification based on, 822–825determining, 820–822

of trapezoids, 518of triangles

coordinates of, 478similar triangles, 483

Two-column proof (Cont.)Tangent Segment Theorem, 892Trapezoid Midsegment Theorem, 784Triangle Midsegment Theorem, 476Triangle Proportionality Theorem, 473Vertical Angle Theorem, 188

Two-dimensional figuresarea of, 320–321diagonals of, 360rotating through space, 299–302stacking, 310–315translating, 304–309

Two-way (contingency) frequency tables, 1145–1147

Two-way relative frequency tables, 1147–1153

Two-way tables, 1156–1157compound probabilities on, 1138–1153

frequency tables, 1141–1144two-way (contingency) frequency

tables, 1145–1147two-way relative frequency tables,

1147–1153conditional probability on, 1158–1162defined, 1138

UUndecagons, interior angles of, 796Undefined terms, defining new terms

with, 10–15Uniform probability model, 1042