Question

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, alternate - interior angles are congruent if and only if the lines are parallel. Alternate - exterior angles are congruent if and only if the lines are parallel. Corresponding angles are congruent if and only if the lines are parallel.

Step2: Analyze option A

$\angle2$ and $\angle3$ are not related in a way that would prove $p\parallel q$. $\angle2$ and $\angle3$ are neither alternate - interior, alternate - exterior, nor corresponding angles for lines $p$ and $q$.

Step3: Analyze option B

$\angle5$ and $\angle7$ are vertical angles. Vertical - angle congruence does not prove that $p\parallel q$.

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 same - side interior angles for lines $p$ and $q$, not for lines $r$ and $s$. Their congruence does not prove $r\parallel s$.

Step6: Analyze option E

$\angle5$ and $\angle7$ are vertical angles. Their congruence does not prove that $r\parallel s$.

Answer:

C. $r\parallel s$ because $\angle2\cong\angle4$