Given: ∠P ≅ ∠Q and bisects ∠PQR.

Question

Prove:

Supply the missing reason in Statement 1 of the proof of the the Converse of the Isosceles Triangle Theorem.

Begin with isosceles ∆PRQ with ∠P ≅ ∠Q. Construct , a bisector of ∠PRQ.

AAS Theorem

Reflexive Property of Congruence

Given

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Answer 1

Given the information, let's break down the problem and the proof leading to the missing reason.

**Problem Recap:**
We have an isosceles triangle \(\Delta PRQ\) with \(\angle P \cong \angle Q\), and there is a bisector of \(\angle PRQ\).

**Proof of the Converse of the Isosceles Triangle Theorem:**
1. **Given** \(\angle P \cong \(\angle Q\) and \( \text{ bisects } \angle PQR.\)
2. Segment PR is congruent to segment QR. (By definition of the converse of the Isosceles Triangle Theorem, if two angles of a triangle are congruent, then the sides opposite those angles are congruent.)
3. Construct line segment \( \overline{PS}\) where \( S\) lies on \( QR\) such that \( \angle PRS \cong \angle QRS\). (Making \( PS\) the bisector of \( \angle PRQ\).
4. \(\overline{PS}\) is congruent to \(\overline{PS}\) (Reflexive Property of Congruence).
5. \(\angle PRS \cong \angle QRS\) (Construction).
6. \(\Delta PRS \cong \Delta QRS\) (AAS Theorem, because we have two congruent angles and a side s between them).
7. \(\overline{PR} \cong \overline{RQ}\) (CPCTC - Corresponding parts of congruent triangles are congruent).

**Missing reason for statement 1**:
Statement 1 mentions the given information, which essentially sets up our initial conditions for the problem.

So the full statement 1 with the reason included would look like this:

1. \(\angle P \cong \(\angle Q\) and \( \text{ bisects }\angle PQR\). (**Given**)