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, then:

1. Alternate interior angles are congruent if and only if the lines are parallel.
2. Alternate exterior angles are congruent if and only if the lines are parallel.
3. 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 transversal $s$. 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 can 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 corresponding angles for lines $r$ and $s$ cut by transversal $q$. If $\angle2\cong\angle4$, then $r\parallel s$ by the corresponding - angles postulate.

Step5: Analyze option D

$\angle5$ and $\angle6$ are not related in a way that can prove $r\parallel s$. $\angle5$ and $\angle6$ are not corresponding, alternate interior, or alternate exterior angles for lines $r$ and $s$.

Step6: Analyze option E

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

Answer:

A. $p\parallel q$ because $\angle2\cong\angle3$
C. $r\parallel s$ because $\angle2\cong\angle4$
E. $r\parallel s$ because $\angle5\cong\angle7$