Question

Prove that a triangle ABC is isosceles, if: bisector of angle BAC is perpendicular to base BC.

Answer 1

In Δ ABC, the bisector of ∠ BAC is perpendicular to the base BC. 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 isosceles.