Chapter 9.5, Problem 25WE

To determine

To prove that point $D$ lies on circle $O$ .

Explanation of Solution

Given:

Vertices $A,B$ and $C$ of quadrilateral $ABCD$ lie on circle $O$ ;

  $m\angle A+m\angle C={180}^0$; $m\angle B+m\angle D={180}^0$

  

Since,

  $m\angle A+m\angle C={180}^0$ ….. (i)

  $m\angle B+m\angle D={180}^0$ ….. (ii)

Now adding equation (i) and (ii)

  $m\angle A+m\angle B+m\angle C+m\angle D={180}^0+{180}^0$

  $\Rightarrow m\angle A+m\angle B+m\angle C+m\angle D={360}^0$

Now since points $A,B,$ and $C$ already lies on the circle $O$ and the sum of all angles is ${360}^0$ ,

Therefore $ABCD$ is a cyclic quadrilateral.

Hence, point $D$ lies on the circle $O$ .

Conclusion:

Therefore, point $D$ lies on the circle $O$ .