Question

ABCD is a square. P and Q are points on AB and BC such that AQ = DP. Prove that ΔAPD ≅ ΔBQA.

Answer 1

Given:

ABCD is a square.

P lies on AB and Q lies on BC such that AQ = DP.

To prove: ΔAPD ≅ ΔBQA

Proof:

1. Since ABCD is a square.

All sides are equal. 

So, AB = BC = CD = DA.

2. Given AQ = DP   ...(By the problem statement)

3. AD = AB   ...(Sides of the square)

4. Angle ADP and Angle BQA are right angles   ...(Since ABCD is a square, all angles are 90°)

5. Consider triangles APD and BQA:

AD = AB   ...(Sides of the square)

AQ = DP   ...(Given)

Angle ADP = Angle BQA = 90°   ...(Angles in the square)

6. By RHS (Right angle-Hypotenuse-Side) congruence rule:

ΔAPD ≅ ΔBQA.

Hence, ΔAPD is congruent to ΔBQA.