Question

Prove that a triangle ABC is isosceles, if: altitude AD bisects angles BAC.

Answer 1

In ΔABC, let the altitude AD bisects ∠BAC.
Then we have to prove that the ΔABC is isosceles.

In triangles ADB and ADC,
∠BAD = ∠CAD    ...(AD is bisector of ∠BAC)
AD = AD             ...(common)
∠ADB = ∠ADC    ....(Each equal to 90°)
⇒ ΔADB ≅ ΔADC ...(by ASA congruence criterion)
⇒ AB = AC           ...(cpct)
Hence, ΔABC is an isosceles.