Question

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that :

(i) OB = OC

(ii) AO bisects ∠ A

Answer 1

(i) It is given that in triangle ABC, AB = AC

∴ ∠ACB = ∠ABC (Angles opposite to equal sides of a triangle are equal)

∴$\frac{1}{2}$ ∠ACB= $\frac{1}{2}$ ∠ABC

∴ ∠OCB =∠OBC

∴ OB = OC (Sides opposite to equal angles of a triangle are also equal)

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(ii) In ∆OAB and ∆OAC,

AO =AO (Common)

AB = AC (Given)

OB = OC (Proved above)

Therefore, ∆OAB ∆OAC (By SSS congruence rule)

∠BAO = ∠CAO (CPCT)

∴ AO bisects A.