Question

use the given information to prove that $\triangle pqr\cong\triangle tsr$.
given: $\overline{qr}\cong\overline{sr}$
$\angle qpr\cong\angle str$
prove: $\triangle pqr\cong\triangle tsr$

Explanation:

Step1: State the given information

Given that \(\overline{QR}\cong\overline{SR}\) (by the problem statement).

Step2: Use the property of vertical angles

\(\angle PRQ\) and \(\angle TRS\) are vertical angles. By the Vertical Angles Theorem, \(\angle PRQ\cong\angle TRS\).

Step3: Apply the AAS (Angle - Angle - Side) congruence criterion

We have \(\angle QPR\cong\angle STR\) (given), \(\angle PRQ\cong\angle TRS\) (Vertical Angles Theorem), and \(\overline{QR}\cong\overline{SR}\) (given). By the AAS (Angle - Angle - Side) congruence criterion for triangles, 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. So, \(\triangle PQR\cong\triangle TSR\).

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Vertical Angles Theorem
4. Reason: AAS (Angle - Angle - Side) Congruence Criterion