Question

4.) given: g || h and ∠1 ≅ ∠2 prove: p || r statements reasons 1.) g || h given 2.) def of correspond ∠’s 3.) ∠1 ≅ ∠3 4.) given 5.) alternate ext ∠’s thm 6.) ∠3 and ∠2 are alt ext ∠’s 7.)

Explanation:

Step1: State given parallel lines

Since \(g\parallel h\) (given), by the definition of corresponding - angles, \(\angle1\cong\angle3\).

Step2: Use given angle - congruence

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

Step3: Apply transitive property of congruence

Since \(\angle1\cong\angle3\) and \(\angle1\cong\angle2\), then \(\angle2\cong\angle3\) (transitive property of congruence).

Step4: Apply alternate - exterior angles theorem

\(\angle2\) and \(\angle3\) are alternate - exterior angles. If alternate - exterior angles are congruent, then the lines are parallel. So, \(p\parallel r\) (alternate exterior angles theorem).

Answer:

1. \(g\parallel h\) - Given
2. \(\angle1\cong\angle3\) - Def of Corresponding \(\angle\)'s
3. \(\angle1\cong\angle3\) - (Already filled in the table correctly)
4. \(\angle1\cong\angle2\) - Given
5. \(p\parallel r\) - Alternate Ext \(\angle\)'s Thm
6. \(\angle3\) and \(\angle2\) are alt ext \(\angle\)'s - Definition of alternate exterior angles
7. \(p\parallel r\) - Since \(\angle2\cong\angle3\) (from steps 2, 3 and 4 and transitive property) and they are alt ext \(\angle\)'s, by alternate exterior angles theorem.