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 parallel - line postulates

If two lines are cut by a transversal, alternate - interior angles are congruent if and only if the lines are parallel, and corresponding angles are congruent if and only if the lines are parallel.

Step2: Analyze option A

$\angle2$ and $\angle3$ are alternate - interior angles for lines $p$ and $q$ cut by a transversal. If $\angle2\cong\angle3$, then $p\parallel q$ by the alternate - interior angles postulate.

Step3: Analyze option B

$\angle5$ and $\angle7$ are not related in a way that would prove $p\parallel q$. $\angle5$ and $\angle7$ are not corresponding, alternate - interior, or alternate - exterior angles for lines $p$ and $q$.

Step4: Analyze option C

$\angle2$ and $\angle4$ are vertical angles. Vertical angles are always congruent, but they do not prove that $r\parallel s$.

Step5: Analyze option D

$\angle5$ and $\angle6$ are same - side interior angles for lines $r$ and $s$. If $\angle5+\angle6 = 180^{\circ}$, then $r\parallel s$, but just $\angle5\cong\angle6$ does not prove $r\parallel s$.

Step6: Analyze option E

$\angle5$ and $\angle7$ are corresponding angles for lines $r$ and $s$ cut by a transversal. If $\angle5\cong\angle7$, then $r\parallel s$ by the corresponding angles postulate.

Answer:

A. $p\parallel q$ because $\angle2\cong\angle3$; E. $r\parallel s$ because $\angle5\cong\angle7$