Mathematics

Prove that a triangle ABC is isosceles, if :
(i) altitude AD bisects angle BAC or,
(ii) bisector of angle BAC is perpendicular to base BC.

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Let AD be the altitude on side BC.

We know that,

Altitude from a point is always perpendicular to other side.

Let altitude AD bisect angle BAC.

Hence, segment AD satisfies both the conditions.

In △ ADB and △ ADC,

⇒ AD = AD (Common side)

⇒ ∠BAD = ∠CAD (Since, AD bisects angle BAC)

⇒ ∠ADB = ∠ADC (Since, AD is altitude to side BC)

∴ Δ ADB ≅ Δ ADC (By A.S.A. axiom)

We know that,

Corresponding parts of congruent triangle are equal.

⇒ AB = AC.

Hence, proved that ABC is an isosceles triangle.

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