Question

identifying the reason in proving congruent triangles
given: rt || sp, rq ≅ qp, rp bisects st at q
prove: △rqt ≅ △pqs

tamir is working to prove the triangles congruent using sas. after stating the given information, he states that tq ≅ qs by the definition of segment bisector. now he wants to state that ∠rqt ≅ ∠pqs. which reason should he use?
options:

• alternate interior angles theorem
• linear pair postulate
• vertical angles theorem
• corresponding angles theorem

Explanation:

Step1: Recall Vertical Angles

Vertical angles are formed when two lines intersect, and they are congruent. In the diagram, $\angle RQT$ and $\angle PQS$ are vertical angles (formed by the intersection of $RT$ and $SP$ at $Q$? Wait, no, the intersection is of $TR$ and $PS$? Wait, the lines are $TR$ and $PS$ intersecting at $Q$. So $\angle RQT$ and $\angle PQS$ are vertical angles.

Step2: Identify the Theorem

The vertical angles theorem states that vertical angles are congruent. So to prove $\angle RQT \cong \angle PQS$, we use the vertical angles theorem.

Answer:

vertical angles theorem (the option labeled "vertical angles theorem")