Question

Suppose ABC is an isosceles triangle with AB = AC; BD and CE are bisectors of ∠B and ∠C. Prove that BD = CE.

Answer 1

Given: AB = CA, ∠1 = ∠2 and ∠3 = ∠4

To Prove: CE = BD

Proof:

In ΔABC:

AB = CA (Given)

∠B = ∠C (Angles opposite to equal sides are equal)

∠1 + ∠2 = ∠3 + ∠4

2∠1 = 2∠3 (1 = 2 and 3 = 4)

∠1 = ∠3 … (1)

In ΔABD and ΔACE:

(Common)

∠1 = ∠3 (From (1))

AB = CA (Given)

ΔABD ≅ ΔACE (ASA congruency)

∴ CE = BD (Corresponding parts of congruent triangles)