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Properties Properties of of Parallel Lines Parallel Lines 3-2 3-2

Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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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.

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Page 1: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

Properties Properties ofof

Parallel LinesParallel Lines3-23-2

Page 2: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical
Page 3: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

EXAMPLE 1 Identify congruent angles

SOLUTIONBy 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.

Page 4: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical
Page 5: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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

Page 6: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical
Page 7: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical
Page 8: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

GUIDED PRACTICE for Examples 1 and 2

SOLUTION

Vertical Angles Congruence Theorem.m 4 = 105°

Corresponding Angles Postulate.m 5 =105°

Alternate Exterior Angles Theoremm 8 =105°

Use the diagram at the right.1. If m 1 = 105°, find m 4, m 5, and m 8.

Tell which postulate or theorem you use in each case.

Page 9: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

GUIDED PRACTICE for Examples 1 and 2

Use the diagram at the right.2. If m 3 = 68° and m 8 = (2x + 4)°, what is the

value of x? Show your steps.

Page 10: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

GUIDED PRACTICE for Examples 1 and 2

SOLUTION7 and 5 are supplementary.m 7 + m 8 = 180°

Corresponding Angles Postulate.m 3 = m 7

m 3 = 68°

Substitute 68° for m 7 and (2x + 4)for 8.68° + 2x + 4 = 180°

Combine like terms.72 + 2x = 180°

Subtract 72 from each side. 2x = 108

Divide each side by 2. x = 54

Page 11: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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

Page 12: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical
Page 13: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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.

Page 14: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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?

Page 15: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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°.

Page 16: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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.

SOLUTION

Yes still we can prove the theorem.As 3 and 2 congruence is not dependent on the congruence of 1 and 3.

Page 17: Properties of Parallel Lines 3-2. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120. Using the Vertical

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?

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 = 41°.

4. WHAT IF?