Question

In the adjoining figure, BA ⊥ AC, PQ ⊥ PR, such that BA = PQ and BR = CQ. Show that : ΔΑΒC ≅ ΔΡQR.

Answer 1

Given: 

BA ⟂ AC

PQ ⟂ PR

BA = PQ

BR = CQ

To Prove: ΔABC ≅ ΔPQR

Proof (Step-wise):

1. From BA ⟂ AC, ∠BAC = 90°.

From PQ ⟂ PR, ∠QPR = 90°.

Hence, ΔABC and ΔPQR are right-angled at A and P respectively.

2. AB = PQ.   ...(Given) 

So one leg of ΔABC equals the corresponding leg of ΔPQR.

3. From the figure B, R, C, Q are collinear in that order, so BC = BR + RC and QR = QC + RC.

Given BR = CQ, replace BR by CQ to get BC = CQ + RC = QR.

Thus, the hypotenuse BC of ΔABC equals the hypotenuse QR of ΔPQR.

4. We now have, for the two right triangles ΔABC and ΔPQR:

• right angles at A and P,
• equal hypotenuses (BC = QR),
• one pair of corresponding legs equal (AB = PQ). By the RHS (Right angle–Hypotenuse–Side) congruence criterion for right triangles, ΔABC ≅ ΔPQR. (See congruence examples for right triangles in the provided material.)

ΔABC ≅ ΔPQR (by RHS).