Question

Given: ∠P ≅ ∠Q and bisects ∠PQR.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.

Answer 1

The missing reason involves defining an isosceles triangle with congruent angles at P and Q and using an angle bisector to create congruent triangles as per the Converse of the Isosceles Triangle Theorem.

Step-by-step explanation:

The student has asked to provide the missing reason in Statement 1 of the proof of the Converse of the Isosceles Triangle Theorem. The correct reasoning for beginning with an isosceles ΔPRQ, where ∠P ≅ ∠Q, and constructing a bisector of ∠PRQ, lies in the definition of an isosceles triangle and the properties of angle bisectors.

In an isosceles triangle, two sides are congruent, and the angles opposite those sides are also congruent. By constructing a bisector, we create two congruent triangles, which will be crucial to prove that the sides opposite to the congruent angles are also congruent, as stated by the Converse of the Isosceles Triangle Theorem.