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Big Ideas Math Integrated I, 2016 View details

3. Proofs with Parallel Lines

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Exercise 35 Page 516

Consider the Alternate Interior Angles Converse.

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Let's have a look at the given diagram.

If we prove that ∠ 1 and ∠ 4 are congruent, we know by the Alternate Interior Angles Converse that AB and CD are parallel since these angles are alternate interior angles in the diagram below.

Note that ∠ 2 and ∠ 3 are vertical angles. Therefore they are congruent according to the Vertical Angles Theorem. ∠ 2 ≅ ∠ 3 Because these angles are congruent, we can use the Transitive Property of Congruence to claim that ∠ 1 ≅ ∠ 4. ∠ 1 ≅ ∠ 2, ∠ 3 ≅ ∠ 4 ⇕ ∠ 1 ≅ ∠ 4 Now we have enough information to prove that AB and CD are parallel.

Finally, we will do a two-column proof.

Statement	Reason
1. ∠ 1 ≅ ∠ 2, ∠ 3≅ ∠ 4	1. Given
2. ∠ 2≅ ∠ 3	2. Vertical Angles Theorem
3. ∠ 1≅ ∠ 3	3. Transitive Property of Congruence
4. ∠ 1≅ ∠ 4	4. Transitive Property of Congruence
5. AB ∥ CD	5. Alternate Interior Angles Converse

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Communicate Your Answer p.509

Monitoring Progress p.510-513

Exercises p.514-516

Exercise 35 Page 516

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