3.23.2 Use Parallel Lines and TransversalsBell Thinger

ANSWER alternate interior

1. 5, 7

2. 3, 6

3. 1, 8

ANSWER alternate exterior

ANSWER corresponding

Identify the type of angles.

3.2

3.2Example 1

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.

3.2

3.2Example 2ALGEBRA

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

3.2Guided Practice

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.

Vertical Angles Congruence Theorem.

Corresponding Angles Postulate.m 5 =105°

Alternate Exterior Angles Theoremm 8 =105°

m 4 = 105°

ANSWER

3.2

Use the diagram.

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

2.

Guided Practice

m 3 = m 7

68 + 2x + 4 = 180

2x + 72 = 180

2x = 108

x = 54

m 7 + m 8 = 180

ANSWER

3.2Example 3

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

3.2

GIVEN : p q

PROVE : 1 2

STATEMENTS REASONS

p q1. 1. Given

2. Corresponding Angles Postulate

3. Vertical Angles Congruence Theorem

Transitive Property of Congruence

4.

Example 3

2. 1 3

3. 3 2

4. 1 2

3.2Example 4 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?

Science

SOLUTIONBecause 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°.

3.2Guided Practice

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

3.2

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?

Guided Practice

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

3.2Exit Slip

What theorem justifies each statement?

2. 4 and 6 are supplementary.

3. If m 2 = 115 , find m 7.o

115ANSWERo

Alt. Interior Thm.sANSWER

Consec. Interior Thm.s

ANSWER

1. 3 6

3.24. Find the values of x and y.

17.5, 35ANSWER

Exit Slip

3.2

82ANSWERo

5. The figure shows a plant trellis .

If m 1 = 82o , find m 2.

Exit Slip

3.26. If m 1 = 105o , find all other angle measures.

3.2

Homework

Pg 161-164#22, 23, 24, 31, 41