Question

In the given figure, if $O$ is the center of circle and $AD=BC$, then choose the correct option.

• A. $\angle AOC=\angle BOD$
• B. $AC=BD$
• C. $\angle BAD=\angle ABC$
• D. All of the above.

Answer 1

The correct option is D All of the above.


As $AD=BC$
So,$\angle AOD=\angle BOC$
$\text{(angle subtended by equal chords}\text{are equal at center of circle)}$

Now, subtracting $\angle COD$ from both the angles, we get
$\angle AOD-\angle COD=\angle BOC-\angle COD$
$\angle AOC=\angle BOD$
So,
$AC=BD\text{(chords subtending equal angles}\text{at center are equal in length)}$

We know that
$\angle BAD=\frac{1}{2}\angle BOD\angle ABC=\frac{1}{2}\angle AOC$

Since it is proved that $\angle AOC=\angle BOD$
Hence, $\angle BAD=\angle ABC$

So, option $\left(d\right)$ correct.