Question

part 2 of 3
complete the proof.
given ( angle p cong angle s ) and ( overline{tq} cong overline{rq} )
prove ( \triangle qrs cong \triangle qtp )
complete the proof below.
statement | reason
( angle p cong angle s ) and ( overline{tq} cong overline{rq} ) | given
( angle rqs cong angle tqp ) |
(options for reason: vertical angles theorem, corresponding angles theorem, triangle exterior angle theorem, triangle angle - sum theorem, same - side interior angles postulate, alternate interior angles theorem)

Explanation:

To determine the reason for \( \angle RQS \cong \angle TQP \), we analyze the angles at point \( Q \). \( \angle RQS \) and \( \angle TQP \) are vertical angles (formed by the intersection of two lines, \( TR \) and \( PS \)). The Vertical Angles Theorem states that vertical angles are congruent. The other theorems (Corresponding Angles, Triangle Exterior Angle, etc.) do not apply here as they relate to different angle relationships (e.g., parallel lines, exterior angles of triangles). Thus, the correct reason is the Vertical Angles Theorem.

Answer:

Vertical Angles Theorem