Question

given: $overline{rt} parallel overline{sp}$, $overline{rq} cong overline{qp}$, $overline{rp}$ bisects $overline{st}$ at $q$
prove: $\triangle rqt cong \triangle pqs$
tamir is working to prove the triangles congruent using sas. after stating the given information, he states that $overline{tq} cong overline{qs}$ by the definition of segment bisector. now he wants to state that $angle rqt cong angle pqs$. which reason should he use?
$\bigcirc$ alternate interior angles theorem
$\bigcirc$ corresponding angles theorem
$\bigcirc$ linear pair postulate
$\bigcirc$ vertical angles theorem

Explanation:

To determine the reason for \( \angle RQT \cong \angle PQS \), we analyze the angles:

• \( \angle RQT \) and \( \angle PQS \) are vertical angles (formed by the intersection of lines \( RP \) and \( ST \) at \( Q \)).
• The Vertical Angles Theorem states that vertical angles are congruent.

Other options are incorrect:

• Alternate Interior Angles Theorem applies to parallel lines cut by a transversal (not vertical angles).
• Corresponding Angles Theorem also applies to parallel lines cut by a transversal (not vertical angles).
• Linear Pair Postulate states adjacent angles forming a line are supplementary (not relevant here).

Answer:

D. vertical angles theorem (assuming the last option is labeled D; if labels differ, the correct option is the one stating "vertical angles theorem").