Chapter 9, Problem 18CT

To determine

To prove: that $ABCD$ is a rectangle.

Explanation of Solution

Given information:

The parallelogram $ABCD$ is inscribed in a circle.

Proof:

The parallelogram $ABCD$ is cyclic.

The opposite angles of cyclic quadrilateralare supplementary.

Thus,

  $A+C=180$

  $B+D=180$

But in parallelogram opposite sides are equal.

Thus,

  $A=C=90$

  $B=D=90$

In rectangle all angles are rectangles.

Hence, $ABCD$ is a rectangle.