3.2 Use Parallel Lines and Transversals

Objectives

• Use the properties of

parallel lines to determine

congruent angles

• Use algebra to find

angle measures

Postulate 15 Corresponding s Postulate

• If 2 lines are cut by a transversal, then each pair of corres. s is .

• i.e. If l m, then 12.

l

m

1

2

Theorem 3.1 Alternate Interior s Theorem

• If 2 lines are cut by a transversal, then each pair of alternate interior s is .

• i.e. If l m, then 12.

l

m

1

2

Theorem 3.2 Alternate Exterior s Theorem

• If 2 lines are cut by a transversal, then the pairs of alternate exterior s are .

• i.e. If l m, then 12.

l m

1

2

Theorem 3.3 Consecutive Interior s Theorem

• If 2 lines are cut by a transversal, then each pair of consecutive int. s is supplementary.

i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°.

l

m 1

2

• If a transversal is to one of 2 lines, then it is to the other.

• i.e. If l m, & t l, then t m.

1

2

Theorem 3.11 Transversal Theorem

l

m

t

EXAMPLE 1 Identify congruent angles

SOLUTION

By the Corresponding Angles Postulate, m 5 = 120°. Using the Vertical Angles Congruence Theorem, m 4 = 120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m 8 = 120°.

The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning.

EXAMPLE 2 Use properties of parallel lines

ALGEBRA Find the value of x.

SOLUTION

By the Vertical Angles Congruence Theorem, m 4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines.

Consecutive Interior Angles Theoremm 4 + (x+5)° = 180°

Substitute 115° for m 4.115° + (x+5)° = 180°

Combine like terms.x + 120 = 180

Subtract 120 from each side.x = 60

GUIDED PRACTICE for Examples 1 and 2

Vertical Angles Congruence Theorem.

Corresponding Angles Postulate.m 5 =105°

Alternate Exterior Angles Theoremm 8 =105°

Use the diagram.

1. If m 1 = 105°, find m 4, m 5, and m 8. Tell which postulate or theorem you use in each case.

m 4 = 105°

ANSWER

GUIDED PRACTICE for Examples 1 and 2

Use the diagram.

2. If m 3 = 68° and m 8 = (2x + 4)°, what is the value of x? Show your steps.

m 3 = m 7

68 + 2x + 4 = 180

2x + 72 = 180

2x = 108

x = 54

m 7 + m 8 = 180ANSWER

EXAMPLE 3 Prove the Alternate Interior Angles Theorem

Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

SOLUTION

Draw a diagram. Label a pair of alternate interior angles as 1 and 2. You are looking for an angle that is related to both 1 and 2. Notice that one angle is a vertical angle with 2 and a corresponding angle with 1. Label it 3.

GIVEN : p q

PROVE :∠ 1 ∠ 2

EXAMPLE 3 Prove the Alternate Interior Angles Theorem

STATEMENTS REASONS

p q1. 1. Given

2. 1 ∠ 3 2. Corresponding Angles Postulate

3. 3 ∠ 2 3. Vertical Angles Congruence Theorem

4. 1 ∠ 2 Transitive Property of Congruence

4.

EXAMPLE 4 Solve a real-world problem

Science

When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m 2 = 40°. What is m 1? How do you know?

EXAMPLE 4 Solve a real-world problem

SOLUTION

Because the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 1 2. By the definition of congruent angles, m 1 = m 2 = 40°.

GUIDED PRACTICE for Examples 3 and 4

3. In the proof in Example 3, if you use the third statement before the second statement, could you still prove the theorem? Explain.

Yes; 3 and 2 congruence is not dependent on the congruence of 1 and 3.

SAMPLE ANSWER

GUIDED PRACTICE for Examples 3 and 4

Suppose the diagram in Example 4 shows yellow light leaving a drop of rain. Yellow light leaves the drop at an angle of 41°. What is m 1 in this case? How do you know?

4. WHAT IF?

41°; 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 1 2. By the definition of congruent angles, m 1 = m 2 = 41°.

ANSWER

Assignment

Pages 181–183 12-22 even; 24 – 29 all; 31-34,36,38,39

Draw diagrams for 38 and 39