Question

making a conjecture about alternate exterior angles
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:

1. Corresponding Angles Theorem:

• When two parallel lines \(q\) and \(s\) are cut by a transversal \(r\), corresponding angles are congruent. For example, \(\angle1\cong\angle5\) and \(\angle2\cong\angle6\).

1. Vertical Angles Theorem:

• Vertical angles are congruent. So, \(\angle1\cong\angle3\) (since they are vertical angles) and \(\angle5\cong\angle7\) (since they are vertical angles).
• If \(\angle1\cong\angle5\) (from the Corresponding Angles Theorem) and \(\angle1\cong\angle3\), \(\angle5\cong\angle7\), then by the Transitive Property of Congruence:
• For alternate - exterior angles \(\angle1\) and \(\angle7\), since \(\angle1\cong\angle5\) (Corresponding Angles Theorem) and \(\angle5\cong\angle7\) (Vertical Angles Theorem), we have \(\angle1\cong\angle7\).
• Similarly, for alternate - exterior angles \(\angle2\) and \(\angle8\), \(\angle2\cong\angle6\) (Corresponding Angles Theorem) and \(\angle6\cong\angle8\) (Vertical Angles Theorem), so \(\angle2\cong\angle8\).

Answer:

The Corresponding Angles Theorem (which states that if two parallel lines are cut by a transversal, then corresponding angles are congruent) and the Vertical Angles Theorem (which states that vertical angles are congruent) can be used. By the Corresponding Angles Theorem, an exterior angle is congruent to a non - adjacent interior angle. Then, using the Vertical Angles Theorem (which gives congruence between pairs of vertical angles), through the Transitive Property of Congruence, alternate exterior angles are congruent.