LinesLesson 1

Parallel and Perpendicular Lines

Parallel Perpendicular

Symbols

Define it in your own words

Draw it

Describe a real-world example of

it

Transversals and Angles

Interior angles:

Exterior angles:

Alternate Interior angles:

Alternate Exterior angles:

Corresponding angles:

Transversals and Angles

Example 1Classify each pair of angles in the figure as alternate interior, alternate exterior, or corresponding.

corresponding angles

alternate exterior angles

Got it? 1Classify the relationship between

alternate interior angles

Missing Angle MeasuresIf = 50˚, find , , and

m2 = 130˚ because and are supplementary.

m = 50˚ because and 3 are vertical angles. m4 = 130˚

because and are supplementary.

Example 2A furniture designer built the bookcase shown. Line a is parallel to line b. If m2 = 105˚, find m6 and m. Justify your answer. Since 2 and 6 are supplementary, the m6 = 75˚.

Since 6 and 3 are interior angles, so the m3 is 75˚.

Got it? 2Find the measure of angle 4. 105º; 2 and 4 are corresponding angles, so their measures are equal.

Example 3In the figure, line m is parallel to line n, and line q is perpendicular to line p. The measure of 1 is 40˚. What is the measure of 7.

Since 1 and 6 are alternate exterior angles, m6 = 40˚.

Since 6, 7, and 8 form a straight line, the sum is 180˚.

40 + 90 + m7 = 180 So m7 is 50˚.

Geometric ProofLesson 2

Deductive vs. Inductive ReasoningEvery time Bill watches his favorite team on TV, the

team loses. So, he decides to not watch the

team play on TV. In order to play sports, you need to have a B average.

Simon has a B average, so he concludes that he can play

sports. All triangles have 3 sides and 3 angles. Mariah has a figure with 3 sides and 3

angles so it must be a triangle.

After performing a science experiment, LaDell

concluded that only 80% of tomato seeds would grow

into plants.

DeductiveReasoning

InductiveReasoning

The Proof ProcessSTEP 1: List the given information, or what you know. Draw a diagram if needed.

STEP 2: State what is to be proven.

STEP 3: Create a deductive argument by forming a logical chain of statements linking the given information.

STEP 4: Justify each statement with definitions, properties, and theorems

STEP 5: State what it is you have proven.

VocabularyA proof is a logical argument where each statement is justified by a reason.

A paragraph proof or informal proof involves writing a paragraph.

A two-column proof or formal proof contains statements and reason organized in two columns.

Once a statement has been proven, it is a theorem.

Example 1 – Paragraph ProofThe diamondback rattlesnake has a diamond pattern on its back. An enlargement of the skin is shown. If m1 = m4, write a paragraph proof to show that m = m3.

Given: m1 = m4Prove: m = m3Proof: m1 = m2 because they are vertical angles. Since m1 = m4, and m = m4. The measure of angle 3 and 4 are the same since they are vertical angles. Therefore, m = m3.

Got it? 1Refer to the diagram shown. AR = CR and DR = BR. Write a paragraph proof to show that AR + DR = CR + BR.

Given: AR = ___________ and DR = ____________.

Prove: _________________ = CR + BR.Proof: You know that AR = CR and DR = BR.

AR + DR = CR + BR by the _____________ Property of

Equality. So, AR + DR = CR + BR by ___________________.

CR BR

AR + DR

Addition

substitution

Example 2Write a two-column proof to show that if two angles are vertical angles, then they have the same measure. Given: lines m and n intersect; 1 and 3 are vertical. Prove: m1 = m3 Statements Reasons

Given

Definition of linear pair

Definition of supplemental anglesSubstitutionSubtraction Property of Equality

a. Lines m and n intersect; 1 and 3 are vertical.

b. 1 and 2 are a linear pair and 3 and 2 are a linear pair.

c. m 1 and m2 = 180˚ m3 and m2 = 180˚d. m 1 and m2 = m3 and m2 e. m1 = m3

Got it? 2The statements for a two-column proof toshow that if mY = mZ, then x =100 are given below. Complete the proof by providing the reasons.

Statements Reasons

Given

Substitution

Subtraction Property of Equality

Addition Property of Equality

a. m Y = mZ, m Y = 2x – 90 mZ = x + 10

b. 2x – 90 = x + 10

c. x – 90 = 10

d. x = 100

Angles of TrianglesLesson 3

Real-World Link1. What is true about the measures of 1 and 2? Explain. They are equal because they are alternate interior angles.

2. What is true about the measures of 3 and 4?They are equal because they are alternate exterior angles.

3. What kind of angle is formed by 1, 5, and 3? Write an equation representing the relationship between the 3 angles.

Straight angle = 1 + 5, + 3 = 180˚

4. Draw a conclusion about ΔABC. The sum of the angles in ΔABC is 180˚.

Angle Sum of a TriangleWords: The sum of the measures of the interior angles of a triangle is 180˚.

Symbols: x + y + z = 180˚.

Model:

Example 1Find the value of x in the Antigua and Barbuda flag.

x + 55 + 90 = 180x + 145 = 180

x = 35

The value of x is 35.

Got it? 1In ΔXYZ, if mX = 72˚ and mY = 74˚, what is mZ?

72 + 74 + Z = 180146 + Z = 180

Z = 34

The measure of angle Z is 34 degrees.

Example 2The measures of the angles of ΔABC are in the ratio 1:4:5. What are the measures of the angles?

Let x represent angle A, 4x angle B, and 5x angle C

x + 4x + 5x = 18010x = 180

x = 18

Angle A = 18˚Angle B = 18(4) = 72˚Angle C = 18(5) = 90˚

Got it? 2The measures of the angles of ΔLMN are in the ratio 2:4:6. What are the measures of the angles?

Let x represent angle L, 4x angle M, and 5x angle N

2x + 4x + 6x = 18012x = 180

x = 15

Angle L = 15(2) = 30˚Angle M = 15(4) = 60˚Angle N = 15(6) = 90˚

Exterior Angles of a TriangleWords: The measure of an exterior angle is equal to the sum of the measures of its two remote interior angles.

Symbols: mA + mB = m1

Model:

Interior and Exterior AnglesEach exterior angle of the triangle has two remote interior angles that are not adjacent to the exterior angle.

interior

1

2

3exterior

4

5

64 is an exterior angle.It’s two remote angles

are 2 and 3.

m4 = m2 + m3

Example 3Suppose m4 = 135˚. Find the measure of 2.

First Way:Angle 4 is the exterior

angle with angle 2 and angle K as the

remote interior.

2 + K = 42 + 90 = 135

2 = 45˚Second Way:

4 and 1 are supplementary, so they equal 180˚.

4 + 1 = 180135 + 1 = 180

1 = 45

1 + 2 + K = 18045 + 2 + 90 = 180

2 = 45˚

Got it? 3Suppose m 5 = 147˚. Find m 1.

m1 = 57˚

Polygons and AnglesLesson 4

Real-World LinkA polygon is a closed figure with three of more line segments. List the states that are in a shape of a polygon.

New Mexico

Utah

Wyoming

Colorado

North Dakota

Interior Angle Sum of a Polygon

Words: The sum of the measures of the interior angles of a polygon is (n – 2)180, where n is the number of sides.

Symbols: S = (n – 2)180

Regular Polygons – an equilateral (all sides are the same) and a equiangular (all angles are the same)

Interior Angle Sum of a Polygon

Example 1Find the sum of the measures of the interior angles of a decagon.

S = (n -2)180S = (10 – 2)180

S = (8)180S = 1,440

The sum of the interior angles of a 10-sided polygon is 1,440˚.

Got it? 1Find the sum of the measures of the interior angles of each polygon.

a. hexagon720˚

b. octagon1,080˚

c. 15-gon2, 340˚

Example 2Each chamber of a bee honeycomb is a regular hexagon. Find the measure of an interior angle of a regular hexagon.STEP 1:Find the sum of the measures of angle.

S = (n – 2)180S = (6 – 2)180

S = (4)180S = 720˚

STEP 2: Divide 720 by 6, since there are six angles in a hexagon.

720˚÷ 6 = 120Each angle in a hexagon is 120˚

Got it? 2Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary.

a. octagon135˚

b. heptagon128.6˚

c. 20-gon162˚

Exterior Angles of a PolygonWords: The sum of the measures of the exterior angles, one at each vertex, is 360˚.

Symbols: m1 + m 2 + m 3 + m 4 + m 5 = 360˚

Model: Examples:

Example 3Find the measure of an exterior angle in a regular hexagon.

A hexagon has a 6 exterior angles.

6x = 360x = 60

Each exterior angle is 60˚.

Got it? 3Find the measure of an exterior angle in a regular polygon.a. triangle

120 ˚

b. quadrilateral90 ˚

c. octagon45 ˚

The Pythagorean Theorem

Lesson 5

Pythagorean TheoremWords: In a right triangle, the sum of the squares of the legs equal the square of the hypotenuse.

Symbols: a2 + b2 = c2

Model:

a

b

c

Example 1Find the missing length. Round to the nearest tenth.

12 in

9 in

c

a2 + b2 = c2

92 + 122 = c2

81 + 144 = c2

225 = c2

= cc = 15 and -15

The equation has two solutions, -15 and 15.

However, the length of the side must be positive.

The hypotenuse is 15 inches long.

Example 2Find the missing length. Round to the nearest tenth.

b

8 cm

24 cm

a2 + b2 = c2

82 + b2 = 242

64 + b2 = 57664 – 64 + b2 = 576 - 64

b2 = 512b =

b 22.6 or -22.6

The length of leg b is 22.6 cm long.

Got it? 1 and 2Find the missing length. Round to the nearest tenth if necessary. a. b.

The length of the hypotenuse is 30

yards long.

The length of leg a is 10.5 cm long.

Converse of Pythagorean Theorem

STATEMENT: If a triangle is a right triangle, then a2 + b2 = c2.

CONVERSE:If a2 + b2 = c2, then a triangle is a right triangle.

The converse of the Pythagorean Theorem is also true.

Example 3The measures of three sides of a triangle are 5 inches,12 inches and 13 inches. Determine whether the triangle is a right triangle.

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169169 = 169

The triangle is a right triangle.

Got it? 3Determine if these side lengths makes a right triangle.

a. 36 in, 48 in, 60 in b. 4 ft, 7ft, 5ft

yes no

Use the Pythagorean Theorem

Lesson 6

Example 1Write an equation that can be used to find the length of the ladder. Then solve. Round to the nearest tenth.

a2 + b2 = c2

8.752 + 182 = x2

76.5625 + 324 = x2400.5625 = x2

= x20.0 x

The ladder is about 20 feet.

Example 2Write an equation that can be used to find the length of the ladder. Then solve. Round to the nearest tenth.

a2 + b2 = c2

102 + b2 = 122

100 + b2 = 144b2 = 44

b = b 6.6

The height of the plane is about 6.6 miles.

Got it? 1 & 2Mr. Parsons wants to build a new banister for the staircase shown. If the rise of the stairs of a building is 5 feet and the run is 12 feet, what will be the length of the new banister?

The length of the new banister is about 13

feet.

Example 3A 12-foot flagpole is placed in the center of a square area. To stabilize the pole, a wire will stretch from the top of the pole to each corner of the square. The flagpole is 7 feet from each corner of the square. what is the length of each wire. Round to the nearest tenth.

a2 + b2 = c2

72 + 122 = c2

49 + 144 = c2

193 = c2

= c2

13.9 c

The length of the wire is about 13.9 feet.

Got it? 3The top part of a circus tent is in the shape of a cone. The tent has a radius of 50 feet. The distance from the top of the tent to the edge is 61 feet. How tall is the top part of the tent? Round to the nearest whole number.

a2 + b2 = c2

a2 + 502 = 612

a2 + 2,500 = 3,721a2 = 1,221

a = a 34.9

The leg has a length of 35 feet.

Distance on the Coordinate Plane

Lesson 7

Example 1Graph the ordered pairs (3, 0) and (7, 5). Then find the distance c between the two points. Round to the nearest tenth.

a2 + b2 = c2

52 + 42 = c2

25 + 16 = c2

41 = c2

= c2

6.4 c

The points are about 6.4 units apart.

Got it? 1Graph the ordered pairs (1, 3) and (-2, 4). Then find the distance c between the two points. Round to the nearest tenth. 3.2 units

The Distance FormulaSymbols: The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by the formula

d =

Model:

Example 2On the map, each unit represents 45 miles. West Point, New York is located at (1.5, 2) and Annapolis, Maryland, is located at (-1.5, -1.5). What is the approximate distance between West Point and Annapolis?

METHOD 1:Use the Pythagorean

Theorema2 + b2 = c2

32 + 3.52 = c2

21.25 = c2 = c

4.6 c

Since the map units equals 45 miles, the distance between the cities is 4.6(45) or about

207 miles.

Example 2On the map, each unit represents 45 miles. West Point, New York is located at (1.5, 2) and Annapolis, Maryland, is located at (-1.5, -1.5). What is the approximate distance between West Point and Annapolis?METHOD 2:

Use the Distance Formulac =

c =

c =

c =

c = 4.6

Since the map units equals 45 miles, the

distance between the cities is 4.6(45) or about

207 miles.

Got it? 2Cromwell Field is located at (2.5, 3.5) and Deadwoods Field is at (1.5, 4.5) on a map. If each map unit is 0.1 mile, about how far apart are the fields?

d = d = d =

d 1.4

1.4(0.1) = 0.14

The field are about 0.14 miles apart.

Example 3Use the Distance Formula to find the distance between X(5, -4) and Y(-3, -2). Round to the nearest tenth if necessary.

d = d = d = d =

d 8.2

This distance between the points is about 8.2 units.