1. 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
Solution: Given: AB = AC and the bisectors of ∠B and ∠C intersect each other at O (i) Since ABC is an isosceles with AB = AC, ∠B = ∠C ½ ∠B = ½ ∠C ⇒ ∠OBC = ∠OCB (Angle bisectors) ∴ OB = OC (Side opposite to the equal angles are equal.) (ii) In ΔAOB and ΔAOC, AB = AC (Given in the question) AO = AO (Common arm) OB = OC (As Proved Already) So, ΔAOB ≅ ΔAOC by SSS congruence condition. BAO = CAO (by CPCT) Thus, AO bisects ∠A.
