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### Lesson 1-5

Angle Relationships

Lesson Outline

- Five-Minute Check
- Then & Now and Objectives
- Vocabulary
- Key Concept
- Examples
- Lesson Checkpoints
- Summary and Homework

Then and Now

You measured and classified angles.

- Identify and use special pairs of angles
- Identify perpendicular lines

Objectives

- Identify and use special pairs of angles
- Identify perpendicular lines

Vocabulary

- Adjacent angles – two coplanar angles that have a common vertex, a common side, but no common interior points
- Linear pair – a pair of adjacent angles whose noncommon sides are opposite rays (always supplementary)
- Vertical angles – two non adjacent angles formed by two intersecting lines
Vertical angles are congruent (measures are equal)!!

- Complementary Angles – two angles whose measures sum to 90°
- Supplementary Angles – two angles whose measures sum to 180°
- Perpendicular – two lines or rays are perpendicular if the angle (s) formed measure 90°

Example 1A

A. ROADWAYS Name an angle pair that satisfies the condition two angles that form a linear pair.

A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays.

Sample Answers: PIQ and QIS, PIT and TIS, QIU and UIT

Example 1B

B. ROADWAYS Name an angle pair that satisfies the condition two acute vertical angles.

Sample Answers: PIU and RIS, PIQ and TIS, QIR and TIU

Key Concept

- m = m
- Add to 90
- Add to 180

Example 2

ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.

Let the measure of one angle be x.

Solve

Given

6x – 6 = 180 Simplify.

6x = 186 Add 6 to each side.

x = 31 Divide each side by 6.

Example 2 cont

ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.

Use the value of x to find each angle measure.

mA = x mB = 5x – 6

= 31 = 5(31) – 6 or 149

CheckAdd the angle measures to verify that the angles are supplementary.

mA + mB = 180

31 + 149 = 180

180 = 180

Answer:mA = 31, mB = 149

Key Concept

- perpendicular means right angle

ALGEBRA Find x and y so thatKO and HM are perpendicular.

Example 3If KO HM, then mKJH = 90. To find x, use KJI and IJH

Example 3 cont

mKJH = mKJI + mIJH Sum of parts = whole

90 = (3x + 6) + 9x Substitution

90 = 12x + 6 Combine like terms.

84 = 12x Subtract 6 from each side.

7 = x Divide each side by 12.

To find y, use mMJO.

mMJO = 3y + 6 Given

90 = 3y + 6 Substitution

84 = 3y Subtract 6 from each side.

28 = y Divide each side by 3.

Answer: x = 7 and y = 28

Key Concept

- Be careful what you assume!
- 89° and 91° may be hard to tell apart from 90°

Example 4A

Answer: Yes; VY and TX are perpendicular

A. Determine whether the following statement can be justified from the figure below. Explain.

mVYT = 90

The diagram is marked to show that VY TX. From the definition of perpendicular, perpendicular lines intersect to form congruent adjacent angles.

Example 4B

B. Determine whether the following statement can be justified from the figure below. Explain.

TYW andTYU are supplementary.

Answer: Yes, they form a linear pair of angles.

Example 4C

C. Determine whether the following statement can be justified from the figure below. Explain.

VYW andTYS are adjacent angles.

Answer: No, they do not share a common side.

Summary & Homework

- Summary:
- There are many special pairs of angles such as adjacent angles, vertical angles, complementary angles, supplementary angles, and linear pairs.

- Homework:
- pg 50-3: 8-15, 19-22

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