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?
$circ$ alternate interior angles theorem
$circ$ corresponding angles theorem
$circ$ linear pair postulate
$circ$ vertical angles theorem

Explanation:

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

• \( \angle RQT \) and \( \angle PQS \) are formed by the intersection of lines \( RP \) and \( ST \) at point \( Q \).
• Vertical angles are opposite angles formed by two intersecting lines, and the vertical angles theorem states that vertical angles are congruent.
• The alternate interior angles theorem applies to parallel lines cut by a transversal (for non - vertical angles here), the corresponding angles theorem also applies to parallel lines cut by a transversal (not relevant here), and the linear pair postulate is about adjacent supplementary angles (not the case here). So the vertical angles theorem is the correct reason.

Answer:

D. vertical angles theorem