Question

Prove that a cyclic parallelogram is a rectangle.

Answer 1

Hint: - A cyclic parallelogram is a parallelogram which is inside a circle that has all its four vertices on the circle itself.

Complete step-by-step answer:

Sum of the opposite angles of a cyclic parallelogram is equal to $${180}^{\circ }$$ .

Opposite angles of a parallelogram are equal.

$$\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ }...\left(a\right)$$
(As angle A and angle C are opposite angles of a cyclic parallelogram and as we know that sum of the opposite angles of a cyclic parallelogram is equal to $${180}^{\circ }$$ )
$$\mathrm{\angle }A=\mathrm{\angle }CAND\mathrm{\angle }B=\mathrm{\angle }D...\left(b\right)$$
(As angle A and angle C and angle B and angle B are pairs of opposite angles of a parallelogram and as we already know that opposite angles of a parallelogram are equal)
Now, on using the above equations that are mentioned as equation (a) and equation (b), we get
$$\mathrm{\angle }A+\mathrm{\angle }C={180}^{\circ }$$ (From equation (a))
$$\mathrm{\angle }A+\mathrm{\angle }A={180}^{\circ }$$ (Using equation (b), we get that angle A and angle C are equal)
$$\begin{array}{rl}& 2\mathrm{\angle }A={180}^{\circ }\\ & \mathrm{\angle }A={90}^{\circ }\end{array}$$
As we now get from the above equation that is equation (c) that angle A of the cyclic parallelogram is equal to $${90}^{\circ }$$ and we already know the property of a parallelogram if it is a rectangle is that one of its angle’s equals to $${90}^{\circ }$$ .
 Here, as one of the angle’s of the cyclic parallelogram is a rectangle.
Hence proved.
NOTE: -
Another way of proving the above theorem is that:-
The diameter of the circle runs through opposite vertices of the cyclic parallelogram. By using the property of a circle that its diameter subtends a $${90}^{\circ }$$ angle at the circumference. Therefore, by using this we can prove that the cyclic parallelogram is a rectangle.