# Honors Geometry Companion Book, Volume 1

3.1.3 Proving that Lines are Parallel (continued)

Example 3 Proving that Lines are Parallel

The figure given in this example includes four lines instead of three, as was the case in the previous examples. To prove that a || b , consider the given information: ∠ 4 ≅ ∠ 8 and ∠ 7 ≅ ∠ 16. Notice that ∠ 4 and ∠ 8 are corresponding angles formed by transversal line a . Therefore, it can be concluded that c || d by the Converse of the Corresponding Angles Postulate. Since c || d , it can be concluded that any corresponding, alternate interior, or alternate exterior angles formed by either transversal a or by transversal b are congruent. There are many different ways to complete the proof. Instead of using the Alternative Interior Angles Theorem to conclude that ∠ 16 ≅ ∠ 9, the Corresponding Angles Theorem could be used to conclude that ∠ 16 ≅ ∠ 12. And since ∠ 16 ≅ ∠ 12 and ∠ 7 ≅ ∠ 16 (Given), by the Transitive Property of Congruence, ∠ 7 ≅ ∠ 12. It follows that a || b by the Converse of the Corresponding Angles Postulate.

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