Question

given: r is the midpoint of \\(\overline{qs}\\), \\(\angle qpr \cong \angle str\\), \\(\angle q\\) and \\(\angle s\\) are right angles. prove: \\(\triangle pqr \cong \triangle tsr\\)\
\

statements	reasons	\
---	---	\
1. r is the midpoint of \\(\overline{qs}\\)	1.	\
2.	2. given	\
3.	3.	\
4.	4. all right angles are congruent.	\
5.	5. the midpoint separates the segment into two congruent segments	\
6.	6.	\
6.	6.

Explanation:

Step1: Identify Given Information

We know \( R \) is the midpoint of \( \overline{QS} \), \( \angle QPR \cong \angle STR \), and \( \angle Q \) and \( \angle S \) are right angles. So, \( \angle Q=\angle S = 90^\circ \).

Step2: State Midpoint Definition

Since \( R \) is the midpoint of \( \overline{QS} \), by the definition of a midpoint, \( \overline{QR}\cong\overline{SR} \).

Step3: Identify Angle - Angle - Side (AAS) Congruence

We have \( \angle Q=\angle S \) (right angles), \( \angle QPR=\angle STR \) (given), and \( \overline{QR}=\overline{SR} \) (from midpoint). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle PQR\cong\triangle TSR \).

Now, filling the table:

Statements:

1. \( R \) is the midpoint of \( \overline{QS} \)
2. \( \angle QPR\cong\angle STR \), \( \angle Q \) and \( \angle S \) are right angles (Given)
3. \( \angle Q=\angle S = 90^\circ \) (Definition of right angle)
4. \( \overline{QR}\cong\overline{SR} \) (Midpoint divides segment into congruent parts)
5. \( \triangle PQR\cong\triangle TSR \) (AAS Congruence Criterion)

Reasons:

1. Given
2. Given
3. Definition of right angle
4. The midpoint separates the segment into two congruent segments
5. AAS (Angle - Angle - Side) Congruence Criterion

Answer:

The completed proof table (statements and reasons) is filled as above with the final conclusion that \( \triangle PQR\cong\triangle TSR \) by AAS congruence.