Question

consider parallel lines cut by a transversal. explain which theorems, definitions, or combinations of both can be used to prove that alternate exterior angles are congruent.

Explanation:

• First, use the Corresponding Angles Theorem. If two parallel lines \(q\) and \(s\) are cut by a transversal \(r\), then corresponding angles are congruent. For example, \(\angle1\) and \(\angle5\) are corresponding angles, so \(\angle1\cong\angle5\).
• Then, use the Vertical Angles Theorem. Vertical angles are congruent. \(\angle5\) and \(\angle7\) are vertical angles, so \(\angle5\cong\angle7\). Also, \(\angle1\) and \(\angle3\) are vertical angles, so \(\angle1\cong\angle3\).
• By the Transitive Property of Congruence (if \(a = b\) and \(b = c\), then \(a = c\)), since \(\angle1\cong\angle5\) and \(\angle5\cong\angle7\), we have \(\angle1\cong\angle7\). Similarly, for the other pair of alternate - exterior angles (\(\angle2\) and \(\angle8\)), we can use the same combination of the Corresponding Angles Theorem (e.g., \(\angle2\) and \(\angle6\) are corresponding angles, so \(\angle2\cong\angle6\)), the Vertical Angles Theorem (\(\angle6\cong\angle8\)), and the Transitive Property of Congruence (\(\angle2\cong\angle8\)).

Answer:

Use the Corresponding Angles Theorem (to relate an exterior angle to an interior - angle that is related to the other exterior angle) and the Vertical Angles Theorem (to relate the interior - angle to the second exterior angle), then apply the Transitive Property of Congruence.