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Properties of Parallel Lines

Properties of Parallel Lines. 3-2. The measure of three of the numbered angles is 120° . Identify the angles. Explain your reasoning.

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Properties of Parallel Lines

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  1. Properties ofParallel Lines 3-2

  2. The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning. By the Corresponding Angles Postulate, m5=120°. Using the Vertical Angles Congruence Theorem, m4=120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m8 = 120°. EXAMPLE 1 Identify congruent angles SOLUTION

  3. ALGEBRA Find the value of x. By the Vertical Angles Congruence Theorem, m4=115°. Lines aand bare parallel, so you can use the theorems about parallel lines. m4 + (x+5)° 180° = 115° + (x+5)° 180° = Substitute 115° for m4. x + 120 = 180 x = 60 EXAMPLE 2 Use properties of parallel lines SOLUTION Consecutive Interior Angles Theorem Combine like terms. Subtract 120 from each side.

  4. 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. m 4 = m 5 = m 8 = 105° 105° 105° for Examples 1 and 2 GUIDED PRACTICE SOLUTION Vertical Angles Congruence Theorem. Corresponding Angles Postulate. Alternate Exterior Angles Theorem

  5. 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. for Examples 1 and 2 GUIDED PRACTICE

  6. 180° m 7 + m 8 = m 7 m 3 = 68° 68° + 2x + 4 = 180° Substitute 68° for m 7 and (2x + 4)for 8. 7 and 5 are supplementary. 72 + 2x = 180° m 3 = 2x = 108 x = 54 for Examples 1 and 2 GUIDED PRACTICE SOLUTION Corresponding Angles Postulate. Combine like terms. Subtract 72 from each side. Divide each side by 2.

  7. Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 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. pq GIVEN : ∠ 1∠ 2 PROVE : EXAMPLE 3 Prove the Alternate Interior Angles Theorem SOLUTION

  8. REASONS STATEMENTS 1. Given pq 1. Corresponding Angles Postulate 2. 2. 3∠ 2 1∠ 3 1∠ 2 Vertical Angles Congruence Theorem 3. 3. 4. 4. Transitive Property of Congruence EXAMPLE 3 Prove the Alternate Interior Angles Theorem

  9. 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, m2 =40°. What is m1? How do you know?

  10. Because the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 12. By the definition of congruent angles, m1 = m2 =40°. EXAMPLE 4 Solve a real-world problem SOLUTION

  11. 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 still we can prove the theorem. As 3 and 2 congruence is not dependent on the congruence of 1 and 3. for Examples 3 and 4 GUIDED PRACTICE SOLUTION

  12. 4. WHAT IF? 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? Because the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 12. By the definition of congruent angles, m1 = m2 =41°. for Examples 3 and 4 GUIDED PRACTICE SOLUTION

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