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.

\[\angle A+\angle C={{180}^{\circ }}\ \ \ \ \ \ ...\left( a \right)\]
(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 \[{{180}^{\circ }}\] )
\[\angle A=\angle C\ \ \ \ \ \ AND\ \ \ \ \ \ \angle B=\angle D\ \ \ \ \ \ \ \ \ \ \ \ \ ...\left( b \right)\]
(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
\[\angle A+\angle C={{180}^{\circ }}\] (From equation (a))
\[\angle A+\angle A={{180}^{\circ }}\] (Using equation (b), we get that angle A and angle C are equal)
\[\begin{align}
  & 2\angle A={{180}^{\circ }} \\
 & \angle A={{90}^{\circ }} \\
\end{align}\]
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.