Given: ∠P ≅ ∠Q and bisects ∠PQR.

Question

Prove:

Supply the missing statement in Statement 5 of the proof of the the Converse of the Isosceles Triangle Theorem.

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

∆PRS ≅ ∆SRQ

∆PRS ≅ ∆QRS

∆PRS ≅ ∆PRS

∆PRS ≅ ∆PRQ

Answer 1

The Isosceles Triangle Theorem states that if a triangle has two equal sides, then the angles opposite those sides are equal. The converse of this theorem states that if a triangle has two equal angles, then the sides opposite those angles are equal.

Given: ∠P ≅ ∠Q and \( \overline{RS} \) bisects ∠PQR.
We are to prove that \( \Delta PRS \cong \Delta SQR \).

Here's the step-by-step proof with the missing statement supplied:

1. **Given:** ∆PQR where ∠P ≅ ∠Q.
2. **Construct:** \( \overline{RS} \) is the bisector of ∠PQR.
3. **Statement 1:** ∠P ≅ ∠Q. **(Given)**
4. **Statement 2:** \( \overline{PS} \cong \overline{QS} \). **(Definition of an angle bisector, as it bisects ∠PQR.)**
5. **Statement 3:** \( \overline{RS} \cong \overline{RS} \). **(Reflexive Property.)**
6. **Statement 4 (to be supplied):** ΔPRS ≅ ΔQRS. **(Angle-Side-Angle (ASA) Postulate. Both the angles at P and Q are congruent (given), \( \overline{RS} \) is common, and the bisected angles ensure that ∠PRS ≅ ∠QRS.)**

Thus, the missing statement in step 5 is:

ΔPRS ≅ ΔQRS