Question

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question 6 of 10 (1 point) | question attempt: 1 of unlimited
use the given information to prove that △pqr ≅ △psr.

given: ∠prq is a right angle
∠prs is a right angle
∠qpr ≅ ∠spr
prove: △pqr ≅ △psr

statement	reason
2 ∠prs is a right angle	given
3 ∠prq ≅ ∠□	all right angles are ≅
4 ∠qpr ≅ ∠spr	reason?
5 \\(\overline{pr} \cong \overline{pr}\\)	reason?
6 △pqr ≅ △psr	reason?

Explanation:

Step1: Fill in ∠PRS for step 3

Since ∠PRQ and ∠PRS are both right angles, by the property that all right angles are congruent, ∠PRQ ≅ ∠PRS. So the angle to fill in is ∠PRS.

Step2: Reason for step 4

The reason for ∠QPR ≅ ∠SPR is "Given" (as it is provided in the problem statement).

Step3: Reason for step 5

The reason for \(\overline{PR} \cong \overline{PR}\) is "Reflexive Property of Congruence" (a segment is congruent to itself).

Step4: Reason for step 6

To prove \(\triangle PQR \cong \triangle PSR\), we use the "Angle - Angle - Side (AAS)" Congruence Theorem. We have two angles (\(\angle PRQ \cong \angle PRS\), \(\angle QPR \cong \angle SPR\)) and a non - included side (\(\overline{PR} \cong \overline{PR}\)) congruent, so by AAS, \(\triangle PQR \cong \triangle PSR\).

Answer:

1. \(\angle PRS\)
2. Given
3. Reflexive Property of Congruence
4. AAS (Angle - Angle - Side) Congruence Theorem