Question

In ΔABC, AB = AC and AP = AQ. Prove that CP = BQ.

Answer 1

Given: In triangle ABC, AB = AC and AP = AQ.

To Prove: CP = BQ.

Proof:

1. Since AB = AC, triangle ABC is isosceles with AB = AC.

2. Given AP = AQ.

3. Connect points B and Q and points C and P.

4. Consider triangles APC and AQB.

5. In triangles APC and AQB,

AP = AQ   ...(Given)

AB = AC   ...(Given)

Angle at A is common.

6. By the Side-Angle-Side (SAS) congruence criterion.

Triangle APC ≅ Triangle AQB.

7. Therefore, corresponding parts of congruent triangles are equal (CPCTC).

Hence, CP = BQ.