Question

given: g || h, ∠1 = ∠2. prove: p || r. statements reasons 1. g || h 1. given 2. ∠1 = ∠2 2. given 3. ∠1 = ∠3 3. 4. ∠2 = ∠3 4. 5. p || r 5. corresponding angles postulate converse. 1. corresponding angles postulate 4. transitive property of congruence 2. vertical angles theorem 4. transitive property of congruence 1. corresponding angles postulate 4. reflexive property 3. vertical angles theorem 4. reflexive property

Explanation:

Step1: Identify angle - pair relationship

\(\angle1\) and \(\angle3\) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for \(\angle1=\angle3\) is the Vertical Angles Theorem.

Step2: Use transitive property

We know \(\angle1 = \angle2\) (given) and \(\angle1=\angle3\) (from the Vertical - Angles Theorem). By the Transitive Property of Congruence, if \(a = b\) and \(a = c\), then \(b = c\). So since \(\angle1=\angle2\) and \(\angle1=\angle3\), we can conclude \(\angle2=\angle3\).

Answer:

1. Vertical Angles Theorem
2. Transitive Property of Congruence