Question

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In an isosceles ∆ with AB=AC the bisectors of angle B and angle C intersect each other at O. Show that (1) OB=OC (2) AO bisects angle A​

Answer 1

[Diagram Attached.]

Given

• ABC is an isosceles triangle.
• AB = AC
• Bisectors of ∠B and ∠C intersect each other at O.

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To Prove

(i) OB = OC

(ii) AO bisects ∠A

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Proof

(i) According to the figure,

In ΔABC,

AB = AC   [Given]

∠B = ∠C [Angles opposite to the equal sides of an isosceles triangle are equal as well]

∴ $\bf \dfrac{1}{2}B =\dfrac{1}{2}C$

∴ ∠CBO =∠BCO  

∴ ∠ABO = ∠ACO    - (i)

Using the property of angles and sides of an isosceles triangle, which states that sides opposite to the equal angles are equal as well, we can say that -:

OB = OC

Hence, proved.

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(ii) According to the figure,

In ΔABO and ΔACO,

(S) AB = AC     [Given]

(A) ∠ABO = ∠ACO  [From (i) ; OC bisects ∠C and OB bisects ∠B]

(S) AO = OA    [Common Side]

∴ ΔABO ≅ ΔACO by SAS Congruency.

By C.P.C.T,

∠BAO = ∠OAC

∴ OA bisects ∠A.

Hence proved.

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Answer 2

Answer:

Correct Question :-

✯ 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

Given :-

✯ In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O.

Find Out :-

☯ i) OB = OC

☯ ii) AO bisects ∠A

Solution :-

i) OB = OC

➸ AB = AC [given]

➸ ∠ACB = ∠ABC [Angles opposite to equal sides of an isosceles triangle are equal]

➙ $\sf \dfrac{1}{2}$ ∠ACB = $\sf \dfrac{1}{2}$ ∠ABC

➙ ∠OCB = ∠OBC [Since OB and OC are the angle bisectors of ∠ABC and ∠ACB]

∴ OB = OC [Sides opposite to equal angles of an isosceles triangle are also equal]

HENCE PROVED :-

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

ii) AO bisects ∠A

➸ In ΔOAB and ΔOAC,

➸ AO = AO [Common]

➸ AB = AC [Given]

➸ OB = OC [We already prove this above :)]

Hence,

➸ ΔOAB ≅ ΔOAC [By SSS congruence rule]

➙ ∠BAO = ∠CAO [CPCT (Corresponding parts of congruent triangles) ]

∴ AO bisects ∠A or AO is the angle bisector of ∠A.

HENCE PROVED :-

[Note :- Please refer that attachment for your figure]

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