Question

In the given figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of $\left(\angle ACD+\angle BED\right)$ (in degrees)

Answer 1

Join $AE$.

$ACDE$ is a cyclic quadrilateral since all four points lies on circle.

$\Rightarrow$ $\angle ACD+\angle AED={180}^o$ [ Sum of opposite angles of a cyclic quadrilateral is supplementary ]

$AB$ is a diameter of a circle.

$\Rightarrow$ $\angle AEB={90}^o$ [ Angle in a semi-circle ] ------ ( 2 )

Adding ( 1 ) and ( 2 ) we get,

$\Rightarrow$ $\angle ACD+\angle AED+\angle AEB={180}^o+{90}^o$

$\Rightarrow$ $\angle ACD+\angle BED={270}^o.$