Question

complete the proof by filling in the blanks.
given: \\( \angle t \cong \angle r, \overline{pq} \cong \overline{pv} \\)
prove: \\( \angle pqt \cong \angle pvr \\)
complete the proof by filling in the blanks.

Explanation:

Step1: Recall triangle congruence criteria

We have two angles ($\angle T\cong\angle R$ and $\angle TPQ\cong\angle RPV$) and a non - included side ($\overline{PQ}\cong\overline{PV}$).

Step2: Apply AAS Theorem

The AAS (Angle - Angle - Side) Theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. Here, in $\triangle TPQ$ and $\triangle RPV$, $\angle T\cong\angle R$, $\angle TPQ\cong\angle RPV$, and $\overline{PQ}\cong\overline{PV}$, so $\triangle TPQ\cong\triangle RPV$ by AAS Theorem.

Answer:

AAS Theorem