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EXAMPLE 3

Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. ∠ 4 ∠ 5. GIVEN :. g h. PROVE :. EXAMPLE 3. Prove the Alternate Interior Angles Converse. SOLUTION. REASONS. STATEMENTS. 1 . . g h. 1 . . Given. 2 . .

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EXAMPLE 3

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  1. Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. ∠ 4∠ 5 GIVEN : gh PROVE : EXAMPLE 3 Prove the Alternate Interior Angles Converse SOLUTION

  2. REASONS STATEMENTS 1. gh 1. Given 2. 2. 4∠ 5 1∠ 4 1∠ 5 Vertical Angles Congruence Theorem 3. 3. Transitive Property of Congruence 4. 4. Corresponding Angles Converse EXAMPLE 3 Prove the Alternate Interior Angles Converse

  3. In the figure, rsand 1 is congruent to 3. Prove pq. EXAMPLE 4 Write a paragraph proof SOLUTION Look at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time.

  4. a. Look at 1 and 2. Look at 2 and 3. b. 12 because rs. EXAMPLE 4 Write a paragraph proof Plan for Proof If 23 then pq.

  5. It is given that rs, so by the Corresponding Angles Postulate, 1 2. a. b. It is also given that 13. Then 23 by the Transitive Property of Congruence for angles. Therefore, by the Alternate Interior Angles Converse, pq. EXAMPLE 4 Write a paragraph proof Plan in Action

  6. U.S. Flag The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe. EXAMPLE 5 Use the Transitive Property of Parallel Lines

  7. The stripes from top to bottom can be named s1, s2, s3, . . . , s13. Each stripe is parallel to the one below it, so s1s2, s2s3, and so on. Then s1s3 by the Transitive Property of Parallel Lines. Similarly, because s3s4, it follows that s1s4. By continuing this reasoning, s1s13. So, the top stripe is parallel to the bottom stripe. EXAMPLE 5 Use the Transitive Property of Parallel Lines SOLUTION

  8. If you use the diagram at the right to prove the Alternate Exterior Angles Converse, what GIVEN and PROVE statements would you use? GIVEN :∠ 1 8 PROVE :j k for Examples 3, 4, and 5 GUIDED PRACTICE SOLUTION

  9. Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 3. It is given that 4 5. By the ? , 1 4. Then by the Transitive Property of Congruence, ? .So, by the ? , g h. ANSWER It is given that 4 5. By the Vertical Angle Congruence, 1 4. Then by the Transitive Property of Congruence, 1 5.So, by the Corresponding Angle Converse Postulate , g h. for Examples 3, 4, and 5 GUIDED PRACTICE

  10. Each step is parallel to the step immediately above it. The bottom step is parallel to the ground. Explain why the top step is parallel to the ground. ANSWER All of the steps are parallel. Since the bottom step is parallel to the ground, the Transitive Property of Parallel Lines applies, and the top step is parallel to the ground. for Examples 3, 4, and 5 GUIDED PRACTICE

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