Question

EXERCISE 7.2
B
In an isosceles triangle ABC, with AB = AC, the bisectors of B and Z C intersect
each other at O. Join A to O. Show that:
A
(1) 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

Answer 2

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..

Thanks..!!