Question

In the adjoining figure, AB = PQ, BR = CQ, ∠ABC = ∠PQR = 90°. Prove that AC = PR.

Answer 1

Given:

• AB = PQ.
• BR = CQ.
• ∠ABC = ∠PQR = 90°.
• Points B, R, C, Q lie on the same straight line in that order as shown in the figure.

To Prove:

• AC = PR.

Proof [Step-wise]:

1. From the figure B,R,C,Q are collinear.

So BC = BR + RC and QR = QC + CR.

2. Given BR = CQ.

Substitute into the expressions in step 1:

BC = BR + RC = CQ + RC = QR.

Hence, BC = QR.

3. We now have in ΔABC and ΔPQR:

AB = PQ   ...(Given)

BC = QR  ...(From step 2).

∠ABC = ∠PQR = 90°   ...(Given) 

Thus two sides and the included angle of ΔABC are equal respectively to two sides and the included angle of ΔPQR.

4. By the SAS congruence criterion.

ΔABC ≅ ΔPQR

5. From corresponding parts of congruent triangles (CPCTC).

AC = PR