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SAS Criteria for Congruency

In an isoscel...

Question

In an isosceles traingle ABC, with $A B = A C$ , the bisectors of $∠ B$ and $∠ C$ intersect each other at the point $O$ . Join $A$ to $O$ . Then:

∠ O A B = ∠ O A C

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O B = O C

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A O

bisects

∠ A

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None of these

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Solution

The correct options are
A $∠ O A B = ∠ O A C$
C $A O$ bisects $∠ A$
D $O B = O C$
$A B = A C$ .... (Given)

$∠ B = ∠ C$ .... (angles opposite to equal sides of triangle are equal)

$\frac{1}{2} ∠ B = \frac{1}{2} ∠ C$ .... ( $B O$ and $C O$ are bisectors of $∠ B$ and $∠ C$ , respectively)

$∴ O B C = ∠ O C B$

$∴ O B = O C$ .... (Sides opposite to equal angles of triangle are equal).

In $△ O A B$ and $△ O S C$ ,

$A B = A C$ .... (Given)

$O B = O C$ .... (Proved)

$∠ B = ∠ C$ ..... (Angles opposite to equal sides of triangle are equal)

$\frac{1}{2} ∠ B = \frac{1}{2} ∠ C$

$∴ ∠ A B O = ∠ A C O$ .

$∴ △ O A B ≅ △ O A C$ .... (by $S A S$ Rule)

$∴ A O$ bisects $∠ A$ .... (By $C P C T$ ).

Hence, option $A$ , $B$ and $C$ are correct.

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

In an isosceles traingle ABC, with $A B = A C$ , the bisectors of $∠ B$ and $∠ C$ intersect each other at the point $O$ . Join $A$ to $O$ . Then: