Question

ABC is an isosceles triangle with AB = AC. If AP ⊥ BC, then show that: ∠B = ∠C.

Answer 1

Given: In triangle ABC, AB = AC and AP ⟂ BC (P is the foot of the perpendicular from A to BC).

To Prove: ∠B = ∠C

Proof [Step-wise]:

1. Draw AP ⟂ BC (Given), so ∠APB = ∠APC = 90°.
2. Consider triangles APB and APC.
3. AB = AC.   ...(Given)
4. AP = AP.   ...(Common side)
5. Thus, in ΔAPB and ΔAPC we have: hypotenuse AB = hypotenuse AC, one leg AP = AP and right angles at P, so ΔAPB ≅ ΔAPC by RHS (right-angle–hypotenuse–side) congruence.
6. Corresponding angles of congruent triangles are equal, hence ∠B = ∠C.

Therefore, ∠B = ∠C, as required.