Question

1. In an isosceles triangle ABC, with AB = AC, the bisectors of Z B and C intersect
each other at 0. Join A to 0. Show that:
A
(i) OB=OC
(ii) AO bisects ZA​

Answer 1

Answer:

(i) In △ABC, we have

AB=AC

⇒∠C=∠B ∣ Since angles opposite to equal sides are equal

⇒

2

1

∠B=

2

1

∠C

⇒∠OBC=∠OCB

⇒∠ABO=∠ACO …(1)

⇒OB=OC ∣ Since sides opp. to equal ∠s are equal …(2)

(ii) Now, in △ABO and △ACO, we have

AB=AC ∣ Given

∠ABO=∠ACO ∣ From (1)

OB=OC ∣ From (2)

∴ By SAS criterion of congruence, we have

△ABO≅△ACO

⇒∠BAO=∠CAO ∣ Since corresponding parts of congruent triangles are equal

⇒ AO bisects ∠A