Question

use the given information to prove that $\triangle pqr \cong \triangle tsr$.
given: $\overline{qr} \cong \overline{sr}$
$\overline{pr} \cong \overline{tr}$
prove: $\triangle pqr \cong \triangle tsr$
(there is a table with statement and reason columns, rows: 1. $\overline{qr} \cong \square$ (given); 2. $\overline{pr} \cong \overline{tr}$ (reason?); 3. $\angle prq \cong \angle \square$ (vertical angles property); 4. $\triangle pqr \cong \triangle tsr$ (reason?))

Explanation:

Step1: Fill QR ≅ SR

Given that \(\overline{QR} \cong \overline{SR}\), so fill \(\overline{SR}\) in the first blank.

Step2: Reason for PR ≅ TR

The reason for \(\overline{PR} \cong \overline{TR}\) is "Given" (as it's provided in the problem statement).

Step3: Vertical Angles

By Vertical Angles Property, \(\angle PRQ \cong \angle TRS\) (since \(\angle PRQ\) and \(\angle TRS\) are vertical angles). So fill \(\angle TRS\) in the third blank.

Step4: SAS Congruence

To prove \(\triangle PQR \cong \triangle TSR\), we use the Side - Angle - Side (SAS) Congruence Criterion. We have two sides (\(\overline{QR} \cong \overline{SR}\), \(\overline{PR} \cong \overline{TR}\)) and the included angle (\(\angle PRQ \cong \angle TRS\)) congruent. So the reason for \(\triangle PQR \cong \triangle TSR\) is "SAS Congruence Criterion".

Answer:

1. \(\overline{SR}\)
2. Given
3. \(\angle TRS\)
4. SAS Congruence Criterion