Question

use the given information to prove that $\triangle pqr\cong\triangle str$.

given: $\overline{qr}\cong\overline{tr}$
$\overline{pr}\cong\overline{sr}$
prove: $\triangle pqr\cong\triangle str$

1 $\overline{qr}\cong\overline{tr}$ given
2 $\overline{pr}\cong\overline{sr}$ given
3 $\angle prq\cong\angle srt$ reason?
4 $\triangle pqr\cong\triangle str$ reason?

Explanation:

Step1: Vertical angles

Vertical angles are equal. $\angle PRQ$ and $\angle SRT$ are vertical angles, so $\angle PRQ\cong\angle SRT$.

Step2: SAS congruence

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
We have $\overline{QR}\cong\overline{TR}$, $\overline{PR}\cong\overline{SR}$, and $\angle PRQ\cong\angle SRT$. So, by SAS, $\triangle PQR\cong\triangle STR$.

Answer:

1. Reason: Vertical angles are congruent.
2. Reason: SAS (Side - Angle - Side) congruence criterion.