Question

which of the following statements are true? select all that apply.
a. ( pparallel q ) if ( angle 2congangle 3 ).
b. ( pparallel q ) if ( angle 5congangle 7 ).
c. ( rparallel s ) if ( angle 2congangle 4 ).
d. ( rparallel s ) if ( angle 5congangle 6 ).

Explanation:

Step1: Check option A

$\angle2$ and $\angle3$ are vertical angles. Vertical angles are always equal, but this does not imply $p\parallel q$. So option A is incorrect.

Step2: Check option B

$\angle5$ and $\angle7$ are not corresponding angles, alternate - interior angles, or alternate - exterior angles. So $\angle5\cong\angle7$ does not imply $p\parallel q$. Option B is incorrect.

Step3: Check option C

$\angle2$ and $\angle4$ are corresponding angles. If $\angle2\cong\angle4$, then by the converse of the corresponding angles postulate, $r\parallel s$.

Step4: Check option D

$\angle5$ and $\angle6$ are vertical angles. Vertical angles are always equal, but this does not imply $r\parallel s$.

Answer:

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