Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is an isosceles triangle.

Given: Bisector of ∠BAC of ΔABC is perpendicular to side BC

To Prove: ΔABC is an isosceles triangle.

Proof:

In ΔABD and ΔACD

Since, AD is the angle Bisector of ΔABC

∴ ∠BAD = ∠CAD

AD = AD ……….[Common Side]

∠ADB = ∠ADC ……[Both equal to 90°]

So, by ASA congruency test

ΔABD ≅ ΔACD

Therefore,

AB = AC ………………. corresponding sides of congruent triangles.

∠ABD = ∠ACD ……………… corresponding angles of congruent triangles.

∴ ∠ABC = ∠ACB

Since, AB = AC and ∠ABC = ∠ACB so, ΔABC is an isosceles triangle.