Question

"Question 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

Class 9 - Math - Triangles Page 123"

Answer 1

Use the result that angles opposite to equal sides of a triangle are equal and its Converse to show part (i ) & show ΔAOB ≅ ΔAOC by using SAS Congruent rule & then use CPCT for part (ii).

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[fig. is in the attachment]

Given:

ΔABC is an isosceles∆ with AB = AC, OB & OC are the bisectors of ∠B and ∠C intersect each other at O.
i.e, ∠OBA= ∠OBC
& ∠OCA= ∠OCB

To Prove:
i) OB=OC
ii) AO bisects ∠A.


Proof:

(i) In ∆ABC is an isosceles with AB = AC,
∴ ∠B = ∠C
[Since , angles opposite to equal sides are equal]

⇒ 1/2∠B = 1/2∠C
[Divide both sides by 2]

⇒ ∠OBC = ∠OCB
& ∠OBA= ∠OCA.......(1)
[Angle bisectors]

⇒ OB = OC .......(2)
[Side opposite to the equal angles are equal]

(ii) In ΔAOB & ΔAOC,
AB = AC (Given)
∠OBA= ∠OCA (from eq1)
OB = OC. (from eq 2)

Therefore, ΔAOB ≅ ΔAOC
( by SAS congruence rule)

Then,
∠BAO = ∠CAO
(by CPCT)

So, AO is the bisector of ∠BAC.

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Hope this will help you.....

Answer 2

Answer:Hope it's helpful