Question

select all the true statements. 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 angle - based parallel - line criteria

If corresponding angles are equal, then lines are parallel. If alternate - interior angles are equal, then lines are parallel. If alternate - exterior angles are equal, then lines are parallel. If same - side interior angles are supplementary, then lines are parallel.

Step2: Analyze each option

• Option A: $\angle2$ and $\angle3$ are not corresponding, alternate - interior, or alternate - exterior angles for lines $p$ and $q$. So, $p\parallel q$ is not true because $\angle2=\angle3$.
• Option B: $\angle5$ and $\angle7$ are vertical angles. They are equal, but vertical - angle equality does not imply parallelism of $p$ and $q$.
• Option C: $\angle2$ and $\angle4$ are not related in a way (corresponding, alternate - interior, etc.) that would imply $r\parallel s$.
• Option D: $\angle5$ and $\angle6$ are alternate - interior angles for lines $r$ and $s$. If $\angle5=\angle6$, then by the alternate - interior angles postulate, $r\parallel s$.
• Option E: $\angle5$ and $\angle7$ are vertical angles. Their equality does not imply $r\parallel s$.

Answer:

D. $r\parallel s$ because $\angle5=\angle6$.