Special Pairs of Special Pairs of AnglesAngles

Special Pairs of Special Pairs of AnglesAngles

2-42-4

EXAMPLE 1 Identify complements and supplements

SOLUTION

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Because 122° + 58° = 180°, CAD and RST are supplementary angles.

Because BAC and CAD share a common vertex and side, they are adjacent.

Because 32°+ 58° = 90°, BAC and RST are complementary angles.

GUIDED PRACTICE for Example 1

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

1.

Because FGK and HGK share a common vertex and side, they are adjacent.

Because 49° + 131° = 180°, HGK and GKL are supplementary angles.

Because 41° + 49° = 90°, FGK and GKL are complementary angles.

GUIDED PRACTICE for Example 1

Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain.

2.

KGH and LKG do not share a common vertex , they are not adjacent.

FGK and FGH have common interior points, they are not adjacent.

EXAMPLE 2 Find measures of a complement and a supplement

SOLUTION

a. Given that 1 is a complement of 2 and m 1 = 68°, find m 2.

m 2 = 90° – m 1 = 90° – 68° = 22

a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.

EXAMPLE 2 Find measures of a complement and a supplement

b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°

SOLUTION

b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3.

EXAMPLE 3 Find angle measures

Sports

When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

SOLUTION

EXAMPLE 3 Find angle measures

STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°.

Write equation.

(4x+ 8)° + (x + 2)° = 180° Substitute.

5x + 10 = 180 Combine like terms.

5x = 170

x = 34

Subtract 10 from each side.

Divide each side by 5.

mBCE + m ∠ECD = 180°

EXAMPLE 3 Find angle measures

STEP 2

Evaluate: the original expressions when x = 34.

m BCE = (4x + 8)° = (4 34 + 8)° = 144°

m ECD = (x + 2)° = ( 34 + 2)° = 36°

The angle measures are 144° and 36°.ANSWER

GUIDED PRACTICE for Examples 2 and 3

3. Given that 1 is a complement of 2 and m 2 = 8° , find m 1.

m 1 = 90° – m 2 = 90°– 8° = 82°

You can draw a diagram with complementary adjacent angle to illustrate the relationship

SOLUTION

12 8°

GUIDED PRACTICE for Examples 2 and 3

4. Given that 3 is a supplement of 4 and m 3 = 117°, find m 4.

You can draw a diagram with supplementary adjacent angle to illustrate the relationship

m 4 = 180° – m 3 = 180°– 117° = 63°

SOLUTION

3 4117°

GUIDED PRACTICE for Examples 2 and 3

5. LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)° and m PQR = (9x + 1)°.

m LMN + m PQR = 90°

(4x – 2 )° + ( 9x + 1 )° = 90°

13x – 1 = 90

13x = 91

x = 7

Complementary angle

Substitute value

Combine like terms

Add 1 to each side

Divide 13 from each side

SOLUTION

GUIDED PRACTICE for Examples 2 and 3

Evaluate the original expression when x = 7

m LMN = (4x – 2 )° = (4·7 – 2 )° = 26°

m PQR = (9x – 1 )° = (9·7 + 1)° = 64°

ANSWER m LMN = 26° m PQR = 64°

SOLUTION

EXAMPLE 4 Identify angle pairs

To find vertical angles, look or angles formed by intersecting lines.

To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.

Identify all of the linear pairs and all of the vertical angles in the figure at the right.

1 and 5 are vertical angles.ANSWER

1 and 4 are a linear pair. 4 and 5 are also a linear pair.

ANSWER

SOLUTION

EXAMPLE 5 Find angle measures in a linear pair

Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.

ALGEBRA

EXAMPLE 5 Find angle measures in a linear pair

x + 5x = 180°

6x = 180°

x = 30°

Write an equation.

Combine like terms.

Divide each side by 6.

The measures of the angles are 30° and 5(30)° = 150°.

ANSWER

GUIDED PRACTICE

ANSWER

For Examples 4 and 5

No, adjacent angles have their non common sides as opposite rays, 1 1 and 4 , 2 and 5, 3 and 6, these pairs of angles have sides that from two pairs of opposite rays

Do any of the numbered angles in the diagram below form a linear pair?Which angles are vertical angles? Explain.

6.

GUIDED PRACTICE

7. The measure of an angle is twice the measure of its complement. Find the measure of each angle.

Let x° be the measure of one angle . The measure of the other angle is 2x° then use the fact that the angles and their complement are complementary to write an equation x° + 2x° = 90°

3x = 90x = 30

Write an equation

Combine like terms

Divide each side by 3

ANSWER The measure of the angles are 30° and 2( 30 )° = 60°

For Examples 4 and 5

SOLUTION

EXAMPLE 2 Name the property shown

Name the property illustrated by the statement.

a. If R T and T P, then R P.

b. If NK BD , then BD NK .

SOLUTION

Transitive Property of Angle Congruencea.

b. Symmetric Property of Segment Congruence

GUIDED PRACTICE for Example 2

2. CD CD

3. If Q V, then V Q.

Reflexive Property of Congruence

ANSWER

Symmetric Property of Congruence

ANSWER

EXAMPLE 3Prove the Vertical Angles Congruence Theorem

GIVEN: 5 and 7 are vertical angles.

PROVE:∠5 ∠7

Prove vertical angles are congruent.

EXAMPLE 3Prove the Vertical Angles Congruence Theorem

5 and 7 are vertical angles.1.

STATEMENT REASONS

1.Given

2. 5 and 6 are a linear pair. 6 and 7 are a linear pair.

2.Definition of linear pair, as shown in the diagram

3. 5 and 6 are supplementary. 6 and 7 are supplementary.

3.Linear Pair Postulate

4.∠5 ∠7 Congruent Supplements Theorem

4.

GUIDED PRACTICE for Example 3

In Exercises 3–5, use the diagram.

3. If m 1 = 112°, find m 2, m 3, and m 4.

ANSWER

m 2 = 68°

m 3 = 112°

m 4 = 68°

GUIDED PRACTICE for Example 3

4. If m 2 = 67°, find m 1, m 3, and m 4.

ANSWER

m 1 = 113°

m 3 = 113°

m 4 = 67°

5. If m 4 = 71°, find m 1, m 2, and m 3.

ANSWER

m 1 = 109°

m 2 = 71°

m 3 = 109°

GUIDED PRACTICE for Example 3

6. Which previously proven theorem is used in Example 3 as a reason?

Congruent Supplements Theorem

ANSWER

EXAMPLE 1 Use right angle congruence

GIVEN: ABBC , DC BC

PROVE: B C

Write a proof.

STATEMENT REASONS

1.Given

2.Definition of perpendicularlines

3.Right Angles CongruenceTheorem

2. B and C are right angles.

3. B C

1.ABBC , DC BC

EXAMPLE 2 Prove a case of Congruent Supplements Theorem

GIVEN: 1 and 2 are supplements.3 and 2 are supplements.

PROVE: 1 3

Prove that two angles supplementary to the same angle are congruent.

EXAMPLE 2 Prove a case of Congruent Supplements Theorem

STATEMENT REASONS

1.3 and 2 are supplements.1 and 2 are supplements. Given1.

2. m 1+ m 2 = 180°m 3+ m 2 = 180°

2. Definition of supplementary angles

Transitive Property of Equality

3.3. m 1 + m 2 = m 3 + m 2

4. m 1 = m 3

5. 1 3

Subtraction Property of Equality

4.

Definition of congruent angles

5.

GUIDED PRACTICE for Examples 1 and 2

1. How many steps do you save in the proof in Example 1 by using the Right Angles Congruence Theorem?

2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.

ANSWER

2 Steps

GUIDED PRACTICE for Examples 1 and 2

Write a proof.

Given: 1 and 3 are complements; 3 and 5 are complements.

Prove: ∠1 5

ANSWER

GUIDED PRACTICE for Examples 1 and 2

Statements (Reasons)

1. 1 and 3 are complements; 3 and 5 are complements.

(Given)

2. ∠1 5Congruent Complements Theorem.