Question

geometry hon - garcia (ums)
2 - 2: teacher_s_ prove parallel lines_p8 (ums graded)
a. p || q because ∠2 ≅ ∠3
b. p || q because ∠5 ≅ ∠7
c. r || s because ∠2 ≅ ∠4
d. r || s because ∠5 ≅ ∠6
e. r || s because ∠5 ≅ ∠7

Explanation:

Step1: Recall parallel - line postulates

If two lines are cut by a transversal, then:

1. Corresponding angles are congruent when the lines are parallel.
2. Alternate - interior angles are congruent when the lines are parallel.
3. Alternate - exterior angles are congruent when the lines are parallel.
4. Same - side interior angles are supplementary when the lines are parallel.

Step2: Analyze option A

\(\angle2\) and \(\angle3\) are not corresponding, alternate - interior, alternate - exterior or same - side interior angles for lines \(p\) and \(q\), so \(p\parallel q\) cannot be proved by \(\angle2\cong\angle3\).

Step3: Analyze option B

\(\angle5\) and \(\angle7\) are not corresponding, alternate - interior, alternate - exterior or same - side interior angles for lines \(p\) and \(q\), so \(p\parallel q\) cannot be proved by \(\angle5\cong\angle7\).

Step4: Analyze option C

\(\angle2\) and \(\angle4\) are corresponding angles for lines \(r\) and \(s\). If \(\angle2\cong\angle4\), then by the corresponding - angles postulate, \(r\parallel s\).

Step5: Analyze option D

\(\angle5\) and \(\angle6\) are not corresponding, alternate - interior, alternate - exterior or same - side interior angles for lines \(r\) and \(s\), so \(r\parallel s\) cannot be proved by \(\angle5\cong\angle6\).

Step6: Analyze option E

\(\angle5\) and \(\angle7\) are not corresponding, alternate - interior, alternate - exterior or same - side interior angles for lines \(r\) and \(s\), so \(r\parallel s\) cannot be proved by \(\angle5\cong\angle7\).

Answer:

C. \(r\parallel s\) because \(\angle2\cong\angle4\)