Lesson 1-5

Angle Relationships

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Refer to the figure for questions 1 through 5.

1. Name the vertex of 3.

2. Name a point in the interior of ACB.

3. Name the sides of BAC

4. Name an acute angle with vertex B

5. If BD bisects ABC, m ABD = 2x + 5 and m DBC = 3x – 16, find m ABD.

6. If P is in the interior of MON and m MOP = ½ m MOP, what can you conclude?Standardized Test Practice:

A

C

B

D

PON NOM

A

G B

C

D 3

MON is an acute angle

OP is the angle bisector of MON m MOP > m PON

5-Minute Check on Lesson 1-45-Minute Check on Lesson 1-45-Minute Check on Lesson 1-45-Minute Check on Lesson 1-4 Transparency 1-5

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Refer to the figure for questions 1 through 5.

1. Name the vertex of 3.

2. Name a point in the interior of ACB.

3. Name the sides of BAC

4. Name an acute angle with vertex B

5. If BD bisects ABC, m ABD = 2x + 5 and m DBC = 3x – 16, find m ABD.

6. If P is in the interior of MON and m MOP = ½ m MOP, what can you conclude?Standardized Test Practice:

A

C

B

D

PON NOM

A

G B

C

D 3

MON is an acute angle

OP is the angle bisector of MON m MOP > m PON

C

G

BA, AC

ABD or DBC

2x + 5 = 3x – 16 x = 21 m ABD = 47

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°

Angles

Ray VA

Ray VBVertex (point V)

Interior of angle AVB or V

Exterior of angle

Circle

360º

Angles measured in degreesA degree is 1/360th around a circle

Acute Right Obtuse

mA < 90º

A

B

mA = 90º 90º < mA < 180º

A A A

V

Names of angles: Angles have 3 letter names (letter on one side, letter of the vertex, letter on the other side) like AVB or if there is no confusion, like in most

triangles, then an angle can be called by its vertex, V

Name two angles that form a linear pair.

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

Answer: The angle pairs that satisfy this definition are

Name two acute vertical angles.

There are four acute angles shown. There is one pair of vertical angles.

Answer: The acute vertical angles are VZY and XZW.

Name an angle pair that satisfies each condition.

a. two acute vertical angles

b. two adjacent angles whose sum is less than 90

Answer: BAC and CAD or EAF and FAN

Answer: BAC and FAE, CAD and NAF, or BAD and NAE

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

Explore The problem relates the measures of two supplementary angles. You know that the sum of the measures of supplementary angles is 180.

Plan Draw two figures to represent the angles.

Let the measure of one angle be x.

Solve

Given

Simplify.

Add 6 to each side.

Divide each side by 6.

Use the value of x to find each angle measure.

Examine Add the angle measures to verify that the angles are supplementary.

Answer: 31, 149

ALGEBRA Find x and y so that and are perpendicular.

Answer: x = 10° y = 15°

AD CE 4 right angles

sum of parts = whole

4x° + 5x° = 90°

9x° = 90°

x = 10°

(7y – 15)° = 90°

7y° = 105°

y = 15°

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 41-43; 11-16, 21, 32-33, 44-49