In an isoscales triangle ABC,with AB=AC, the bisectors of angle B and angle C intersect each other at O. Join A at O . Show that 1.) OB=OC 2.) AO bisects angle A
Answers
Answer:
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.