Question

part 3 of 3
complete the proof
given ∠p ≅ ∠s and tq ≅ rq
prove △qrs ≅ △qtp
complete the proof below
statement reason
∠p ≅ ∠s and tq ≅ rq given
∠rqs ≅ ∠tqp vertical angles theorem
dropdown aas
options:
△qrp ≅ △qst
△qsr ≅ △qtp
△qrs ≅ △qtp
△qrp ≅ △qts

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the triangles are congruent.
We know $\angle P\cong\angle S$ (given), $\angle RQS\cong\angle TQP$ (vertical angles), and $TQ\cong RQ$ (given).

Step2: Identify the Congruent Triangles

We need to match the angles and side to the triangles $\triangle QRS$ and $\triangle QTP$. The angles $\angle S\cong\angle P$, $\angle RQS\cong\angle TQP$, and the side $RQ\cong TQ$. So by AAS, $\triangle QRS\cong\triangle QTP$.
Looking at the options, the correct statement for the AAS reason is $\triangle QRS\cong\triangle QTP$.

Answer:

$\triangle QRS\cong\triangle QTP$