Question

given: g || h, < 1 ≅ < 2 prove: p || r 1. statement: < 1 ≅ < 3 reason: corresponding angles are congruent because g || h 2. statement: g || h, < 1 ≅ < 2 reason: given 3. statement: p || r reason: converse of corresponding angles 4. statement: < 3 ≅ < 2 reason: transitive/substitution property

Explanation:

Step1: Use given parallel - lines property

Since \(g\parallel h\), by the corresponding - angles postulate, \(\angle1\cong\angle3\) because corresponding angles are congruent when two parallel lines are cut by a transversal.

Step2: Use given angle - congruence

We are given that \(\angle1\cong\angle2\).

Step3: Apply transitive property

Since \(\angle1\cong\angle3\) and \(\angle1\cong\angle2\), by the transitive property of congruence (if \(a = b\) and \(a = c\), then \(b = c\)), we can conclude that \(\angle3\cong\angle2\).

Step4: Prove parallel lines

To prove \(p\parallel r\), we note that if \(\angle3\cong\angle2\), then by the converse of the corresponding - angles postulate (if corresponding angles are congruent, then the lines are parallel), \(p\parallel r\).

Answer:

The proof is completed as above, showing that if \(g\parallel h\) and \(\angle1\cong\angle2\), then \(p\parallel r\) using the corresponding - angles postulate, given angle - congruence, and the transitive property of congruence.