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In ABC, AB = AC and the bisectors B and C meet at a point O

Question

In $△ A BC, A B = A C$ and the bisectors $∠ B$ and $∠ C$ meet at a point $O$ . Prove that $BO = CO$ and the ray $A O$ is the bisector of $∠ A$ .

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△ A BC, A B = A C

and the bisectors

∠ B

and

∠ C

meet at a point

O

. Prove that

BO = CO

and the ray

A O

is the bisector of

∠ A

7 mins ago

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7 mins ago

Text solutionVerified

It is given that

A B = A C

and the bisectors

∠ B

and

∠ C

meet at a point

O

Consider

△ BOC

So we get

∠ BOC = \frac{1}{2} ∠ B

and

∠ OCB = \frac{1}{2} ∠ C

It is given that

A B = A C

so we get

∠ B = ∠ C

So we get

∠ OBC = ∠ OCB

We know that if the base angles are equal even the sides are equal

So we get

OB = OC . (1)

∠ B

and

∠ C

has the bisectors

OB

and

OC

so we get

∠ A BO = \frac{1}{2} ∠ B

and

∠ A CO = \frac{1}{2} ∠ C

So we get

∠ A BO = ∠ A CO .. (2)

Considering

△ A BO

and

△ A CO

and equation (1) and (2)

It is given that

A B = A C

S A S

congruence criterion

△ A BO ≅ △ A CO

∠ B A O = ∠ C A O (c . p . c . t)

Therefore, it is proved that

BO = CO

and the ray

A O

is the bisector of

∠ A

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Question Text	In $△ A BC, A B = A C$ and the bisectors $∠ B$ and $∠ C$ meet at a point $O$ . Prove that $BO = CO$ and the ray $A O$ is the bisector of $∠ A$ .
Answer Type	Text solution:1
Upvotes	151

In △ABC,AB=AC and the bisectors ∠B and ∠C meet at a point O. Prove that BO=CO and the ray AO is the bisector of ∠A.

Text solutionVerified

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In $△ A BC, A B = A C$ and the bisectors $∠ B$ and $∠ C$ meet at a point $O$ . Prove that $BO = CO$ and the ray $A O$ is the bisector of $∠ A$ .