A Review of the Entire Course...DEFINITION OF MIDPOINT Two Equal Parts AM = MB DEFINITION OF SEGMENT BISECTOR Forms a Midpoint C is the midpoint of AB DEFINITION OF MEDIAN Vertex To

Integrated Geometry: Review of Entire Course 1

Integrated Geometry:

A Review of the Entire

Course

Niagara Wheatfield High School


Locus:

The five basic locus conditions: Fixed distance from a point Fixed distance from a line

Circle Two Parallel Lines

Equidistant from two points Equidistant from two parallel lines

Perpendicular One Parallel Line

Bisector

Equidistant from two intersecting lines

Angle Bisectors

Compound Locus (two locus conditions)

Draw both locus conditions separately, the solution are the points where the dotted lines intersect. Example: Point P is on line l. How many points are 3 units from line l and five units from point P?

Four points satisfy both conditions.

Equation of a circle: 222 )()( rkyhx

Change signs Square root

Center: (h,k) radius: r

Example:

16)2()4( 22 yx

Center (4, -2) r = 4


Transformations: Isometry: Any transformation that preserves distance Direct Isometry: Preserves distance and orientation Opposite Isometry: Preserves distance but not orientation

Line Reflection Point Reflection Translations Rotations Dilations

FLIP

FLIP SLIDE TURN

CHANGE SIZE

Isometry

Isometry

Isometry

Isometry NOT AN

ISOMETRY

Opposite Direct Direct Direct Not an isometry but orientation is

direct

Notation:

Notation:

),( yxrorigin

Negate Both x and y

Notation:

),(

),(,

byax

yxT ba

ADD or

SUBTRACT

Notation:

Turn LEFT (positive)

Turn RIGHT

(negative)

Notation:

Multiply

Glide Reflection: Line reflection combined with a translation PARALLEL

to the line of reflection

SLIDE and FLIP

Isometry

Opposite

Composition of Transformations:

Work backwards, read from right to left: Example:

)3,7(4,2Tr axisy

Translate first, then reflect.

(7,-3) (5, 1) (-5,1) If a question says which of the following is NOT an

isometry: look for the one with a Dilation.

If a question asks which of the following does not

preserve orientation or is an indirect or opposite isometry look for the answer with some sort of line

reflection.


Logic:

Negate: ~ add the word not to the statement (or take the word not out)

Disjunction:

“OR”, example: It is raining or it is sunny

Only False is both statements are False

Conjunction:

“AND” , example: 2 is even and 3 is odd

Only True if both statements are True

Conditional

“IF” “THEN”, example: If it snows then it is cold.

Only False if T F

Inverse: NEGATE BOTH

o Example: If it rains, then I bring an umbrella. Inverse: If it does not rain then I did not bring an umbrella.

Converse: SWITCH

o Example: Converse: If I bring an umbrella then it rains

Contra positive: SWITCH and NEGATE

o Example: If I do not bring an umbrella then it is not raining.

LOGICALLY EQUIVALENT: have the same truth value.

Biconditional

“IF AND ONLY IF”, example: Two lines are parallel if and only if they never

intersect.

True if both statements are True, True if both statements are False.

Angles:

Supplementary Angles: 2 angles whose sum in 180

Complementary Angles: 2 angles whose sum is 90

Acute Angle: an angle less than 90

Obtuse Angle: an angle greater than 90

Linear Pair: 2 angles that form a line


Definition/Axiom/Theorem

(Reason)

Key Words

(Picture)

Example

(Statement) DEFINITION OF

RIGHT ANGLE

90

m 1 = 90

DEFINITION OF AN

ANGLE BISECTOR

Two

Equal

Angles

m 1 = m 2

DEFINITION OF

COMPLEMENTARY

ANGLES

Angles

Equal

90

m 1 + m 2 = 90

DEFINITION OF

SUPPLEMENTARY

ANGLES

Angles

Equal

180

m 1 + m 2 = 180

DEFINITION OF

PERPENDICULAR

LINES

Form

Right

Angles ABC is a right angle

DEFINITION OF

ALTITUDE

Perpendicular

Lines BD AC

DEFINITION OF

MIDPOINT

Two

Equal

Parts AM = MB

DEFINITION OF

SEGMENT BISECTOR

Forms a

Midpoint C is the midpoint of AB

DEFINITION OF

MEDIAN

Vertex

To Midpoint D is the midpoint of AC

ADDITION AXIOM

Equals added to equals are equal

a = b and c = d

then

a + c = b + d

SUBTRACTION AXIOM

Equals subtracted from equals

are equal.

a + c = b + d and a = d

then

c = b

PARTITION AXIOM

The whole is equal

To the sum of its parts

AC = AB + BC

REFLEXIVE AXIOM

Anything is equal to itself AB = AB

SUBSTITUTION AXIOM

Equals may be substituted for

equals

a = b and b = c

then

a = c

SUPPLEMENT AXIOM

Two angles on a straight line are

supplementary

m 1 and m 2 are supplementary


Definition/Axiom/Theorem

(Reason)

Key Words

(Picture)

Example

(Statement)

ALL RIGHT ANGLES ARE EQUAL

m 1 = m 2

VERTICAL ANGLES ARE EQUAL

m 1 = m 2

SSS

SAS

ASA

AAS

HL (HY-LEG)

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

A D or DEAB

ALTERNATE INTERIOR ANGLES ON

PARALLEL LINES ARE EQUAL

m 1 = m 2

CORRESPONDING ANGLES ON PARALLEL

LINES ARE EQUAL

m 1 = m 2

AA

Angle Angle

Δ ABC ~ Δ DEF

DEFINITION OF SIMILAR POLYGONS

Corresponding Sides

form a proportion EF

BC

DE

AB

MEANS EXTREMES PROPERTY

Cross Multiply (AB)(EF) = (DE)(BC)


Triangles:

Sum of the interior angles of a triangle: 1800

Types of Triangles:

Isosceles Triangle: 2 = angles across from 2 = sides

Equilateral Triangle: 3 = sides, 3 = angles (all 60 )

Scalene Triangle: all 3 sides are NOT =

Obtuse Triangle: a triangle with ONE obtuse angle and 2 acute angles

Acute Triangle: ALL THREE angles are acute. (< 90 )

Right Triangle: Has one right angle

Pythagorean Theorem

222 cba , where c has to be the longest side (the hypotenuse)

only works for a right triangle

The Longest Side is across from the largest angle.

The smallest side is across from the smallest angle

Triangle Inequality: the sum of any two sides of a triangle needs to be greater than the third

side.

RULE: AB+BC> AC

BC+AC> AB AB+AC>BC

The exterior angle of a triangle is always greater than either of the two non-adjacent interior

angles.

Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two

non-adjacent interior angles.

o RULE: a = b + c

Similar Triangles: Set up a proportion and cross multiply. o All angles are equal but sides are proportional

A

B

C

a b

c


Parallel Lines:

Alternate interior angles: equal: 4 & 6 and 5 & 3 Corresponding angles: equal: 2 & 5, 4 & 7, 1 & 6, and 3 & 8 Alternate Exterior angles: equal: 2 & 8 and 1 & 7 Same side interior: supplementary: 4 & 5, and 3 & 6 Same side Exterior angles: supplementary: 2 & 7 and 1 & 8

TO USE FOR PROOFS

Alternate interior angles on parallel lines are equal

o Highlight the parallel lines and the transversal and look for the “Z”

Corresponding angles on parallel lines are equal

Highlight the parallel lines and the transversal and look for the “F”

Quadrilaterals and other Polygons:

Sum of interior angles = (number of sides-2) 180 = (n-2)180

One interior angle of a regular polygon = n

n 180)2( (n = # of sides)

Sum of exterior angles of a polygon = 360 ( no matter how many sides)

One exterior angle of a polygon=n

360

Properties of a parallelogram: use these for algebra problems and proofs

Opposite sides parallel

o To prove using coordinate: find slopes of opposite sides and show that the slopes are equal

Opposite sides are equal

Opposite angles are equal

Consecutive angles are supplementary

Diagonals bisect each other

One set of opposite sides are equal and parallel Properties of a Rectangle: use these for algebra problems and proofs

Opposite sides parallel o To prove using coordinate: find slopes of opposite sides and show that the

slopes are equal





4 right angles o to prove using coordinate: show that the slope of two adjacent sides are

negative reciprocals.

Congruent diagonals

Properties of a Rhombus: use these for algebra problems and proofs

Opposite sides parallel

o To prove using coordinate: find slopes of opposite sides and show that the slopes are equal



1

5 6

A B

t

D C 8

3

2

4

7




Diagonals are perpendicular

4 equal sides o use distance formula 4 times to show all sides are =

Diagonals bisect opposite angles

2 consecutive sides are equal

Properties of a Square: use these for algebra problems and proofs

Opposite sides parallel o To prove using coordinate: find slopes of opposite sides and show that the

slopes are equal





Diagonals are perpendicular

4 equal sides o use distance formula 4 times to show all sides are =

Diagonals bisect opposite angles

4 right angles

o find slope of 2 consecutive sides and show that they are negative reciprocals and therefore perpendicular

Congruent diagonals

Properties of a Trapezoid: use these for algebra problems and proofs o Only one pair of opposite sides parallel

Isosceles trapezoid: 2 = legs, = base angles Congruent diagonals


Circles:



Coordinate Geometry:

PARALLEL LINES have EQUAL SLOPES

Slope formula: 12

12

xx

yym

PERPENDICULAR LINES have NEGATIVE RECIPROCAL SLOPES

Examples of negative reciprocals: 3

2and

2

3

Two lines with a slope of 0 and undefined are considered negative reciprocals and are therefore perpendicular

General equation of a line: y = mx+b

m is the slope and b is the y-intercept

If they give you a point that is on the line, plug in the coordinate (x,y) into y = mx+b

and plug in the slope. Then solve for b.

if they give you two points first find the slope using the slope formula, then pick one

of the points (x,y) and plug it into y = mx+b and plug in the slope. Then solve for b .

Midpoint formula:

Use to show that two segments bisect each other by showing that they have the same midpoint.

Midpoint= 2

,2

2121 yyxx

Distance Formula

Used to find the length of a segment

Used to prove the four sides of a square or rhombus are equal

Distance = 2

12

2

12 )()( yyxx


Formula

Reason for using

Slope

12

12

xx

yym

Proving Segments Parallel Equal Slopes

Proving Segments Perpendicular

Negative Reciprocal SLOPES

(change sign and flip)

Distance

2

12

2

12 )()( yyxxd

Proving Segments have equal length

EQUAL Distances

Midpoint

2,

2int 2121 yyxx

Midpo

Proving Segments bisect each other

Equal Midpoints


Shape Formula(s) and

# of times used

Properties being proved

Parallelogram

4 Slopes

Opposite Sides Parallel

Rectangle

4 Slopes

Opposite Sides Parallel

One right angle

Rhombus

4 distances

All sides equal in length

Square

4 distances 2 Slopes

All sides equal in length One right angle

Trapezoid

4 Slopes

One pair of sides parallel

One pair of sides not

parallel

Isosceles Trapezoid

4 Slopes

2 Distances

One pair of sides parallel

One pair of sides not

parallel

Legs equal in distance


Solids:

Area

Rectangle: A= lw

Triangle: A= ½ (bh)

Square: A= s2

Trapezoid: A= ½ (base1 +base2)h

Volume

Rectangular Prism: V= lwh

Cube: V= s3

Triangular Prism: V= Bh = (½ bh)(Height of Prism)

Cylinder: V= Bh= r2h

Cone: V= (1/3)Bh= (1/3) r2h

Pyramid: V= (1/3)Bh= (1/3)(side of base)(side of base)(altitude)

Sphere: (4/3) r3

Surface Area

Prism: Find the area of all sides and add them together

Cylinder: SA= 2 r2+ 2 rh

Cone: SA= r2 + rl (l is the slant height)

Sphere: SA= 4 r2

Lateral Area

Always refer to reference sheet on exam.

Right Circular Cone: rlL , where l is the slant height

Right Circular Cylinder: rhL 2


Three Dimensional Geometry

2 planes intersect to form a line

There are four ways to form a plane o Intersecting lines, parallel lines, three non-collinear points, a line and a point

not on the line

Coplanar: points or lines that are on the same plane

Collinear: points that are on the same line

There is one plane that can be made that is perpendicular to a point on a line

A plane that intersects two parallel planes forms two parallel lines.

Two lines that are perpendicular to the same plane and parallel and coplanar

There are an infinite number of planes that are perpendicular to a plane and go through a point not on the plane

There is only one line that can be perpendicular to a plane that goes through a point not on the plane.


Constructions:

Perpendicular Bisector Perpendicular at a point on a line

Perpendicular to a line through an

external point

Bisect an Angle

Parallel Lines through a point

Constructing A Median in a Triangle


Miscellaneous Topics:

Name Picture Description

Incenter

Located at intersection of the angle bisectors.

Circumcenter

Located at intersection of the perpendicular

bisectors of the sides.

For an obtuse triangle the circumcenter is located

outside the triangle

Centroid

Located at intersection of medians.

Always located inside the triangle

The centroid is two-thirds the way along each median

The centroid divides each median into two

segments whose lengths are in the ratio 2: 1, the longest segment is near the vertex

Orthocenter

Located at intersection of the altitudes of the

triangle.

For an obtuse triangle the orthocenter is located

outside the triangle

Mid-Segment of a Triangle

When you connect the midpoints of two sides of a triangle it is called the mid-segment. Mid-segments have the following properties:

1. The mid-segment of a triangle joins the midpoints of two sides of a triangle such that it is

parallel to the third side of the triangle.

2. The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its

length is half the length of the third side of the triangle.

Mid-Segment of a Trapezoid When you connect the midpoints of the two legs of a trapezoid it is called the mid-segment The mid-segment of a Trapezoid has the following properties:

1. The mid-segment is parallel to both bases.

2. The mid-segment has length equal to the average of the length of the bases.

Given: D is the midpoint of AC. E is the midpoint of BC. Mid-Segment DE Therefore: DE || AB and DE = ½ AB

http://www.mathopenref.com/triangleincenter.html

http://www.mathopenref.com/trianglecircumcenter.html

http://www.mathopenref.com/triangleorthocenter.html


Tips in Proofs:

3 types of triangle proofs: Congruent Triangle Only, CPCTC and Indirect

Congruent Triangle only: Prove statement looks like: ABC DEF

o Last reason will be: SAS, ASA, AAS, SSS or HL o NO ASS OR SSA

o HL only works for right triangles

CPCTC: When proving parts, like angles or segments. Prove statement may look

like: AB CD or ABC DEF o CPCTC COMES AFTER PROVING TWO TRIANGLES ARE

CONGRUENT o Use CPCTC when proving lines are parallel o

The keyword from the statement helps you find the reason for the next statement

BUILD PROOFS o Use when they give you pieces and you have to put them together to make

bigger segments o Parts Wholes

Given

Reflexive Axiom (only if needed)

Addition Axiom

Partition Axiom

Substitution Axiom

CHOP PROOFS o Use when they give you longer pieces and you have to chop some of it off to

get the segment that you want o Wholes Parts

Given

Partition Axiom

Substitution Axiom

Reflexive or Given Subtraction Axiom

Documents

A Review of the Entire Course...DEFINITION OF MIDPOINT Two Equal Parts AM = MB DEFINITION OF SEGMENT BISECTOR Forms a Midpoint C is the midpoint of AB DEFINITION OF MEDIAN Vertex To