Question

In the given figure, O is the centre of the circle. Which of the following option is correct?

• A. $\angle x+\angle y=\angle z$
• B. $\angle y=\angle z$
• C. $\angle x+2\angle y=\angle z$
• D. $2\angle x=\angle z$

Answer 1

The correct option is A $\angle x+\angle y=\angle z$

Since, the angle subtended by an arc of a circle at the centre is twice the angle subtended by it at any point on the remaining part of the circle
$\therefore \angle EBF=\frac{1}{2}\angle EOF=\frac{1}{2}\angle z$
$\therefore \angle ABC={180}^{\circ }-\frac{1}{2}\angle z$ (Sum of all angles on a line is ${180}^{\circ }$)$\dots$(i)
Similarly, $\angle EDF=\frac{1}{2}\angle EOF=\frac{1}{2}\angle z$
$\therefore \angle ADC={180}^{\circ }-\frac{1}{2}\angle z$ (Sum of all angles on a line is ${180}^{\circ }$)$\dots$(ii)
Now, in quadrilateral ABCD
$\angle ABC+\angle BCD+\angle CDA+\angle BAD={360}^{\circ }$ (Angle sum property of a quadrilateral)$\dots$(iii)
$\Rightarrow {180}^{\circ }-\frac{1}{2}\angle z+\angle y+{180}^{\circ }-\frac{1}{2}\angle z+\angle x={360}^{\circ }$
$\Rightarrow \angle x+\angle y=\angle z$