Question

use the given information to prove that $\triangle pqr \cong \triangle str$.
given: $\angle pqr$ is a right angle
$\angle str$ is a right angle
$\overline{qr} \cong \overline{tr}$
prove: $\triangle pqr \cong \triangle str$

statement	reason
2. $\angle str$ is a right angle	given
3. $\angle pqr \cong \angle str$	reason ?
4. $\overline{qr} \cong \square$	given
5. $\angle prq \cong \angle \square$	vertical angles property
6. $\triangle pqr \cong \triangle str$	reason ?

Explanation:

Step1: Analyze Reason for Step 3

All right angles are congruent. Since $\angle PQR$ and $\angle STR$ are both right angles, the reason for $\angle PQR \cong \angle STR$ is "All right angles are congruent".

Step2: Fill in Step 4

From the given, $\overline{QR} \cong \overline{TR}$, so the blank in step 4 is $\overline{TR}$.

Step3: Fill in Step 5

By Vertical Angles Property, $\angle PRQ \cong \angle SRT$ (vertical angles formed by intersecting lines are congruent). So the blank is $\angle SRT$.

Step4: Analyze Reason for Step 6

We have two angles and a non - included side (AAS) congruent: $\angle PQR \cong \angle STR$, $\overline{QR} \cong \overline{TR}$, $\angle PRQ \cong \angle SRT$. So the reason for $\triangle PQR \cong \triangle STR$ is "AAS (Angle - Angle - Side) Congruence Criterion".

Answer:

1. Reason: All right angles are congruent; 4. $\overline{TR}$; 5. $\angle SRT$; 6. Reason: AAS (Angle - Angle - Side) Congruence Criterion