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:

• The vertical angles theorem states that vertical angles (opposite angles formed by two intersecting lines) are congruent.
• In this case, ∠RQT and ∠PQS are vertical angles.
• The alternate interior angles theorem is about angles formed by a transversal and parallel lines. Here, we are not using that context for these specific angles.
• The corresponding angles theorem also relates to parallel lines and a transversal.
• The linear pair postulate is about adjacent angles that form a straight line (sum to 180°), which is not relevant here.

Answer:

vertical angles theorem