# Ex.10.5 Q12 Circles Solution - NCERT Maths Class 9

## Question

Prove that cyclic parallelogram is a rectangle.

## Text Solution

**What is given ?**

Cyclic quadrilateral is a parallelogram.

**What is unknown?**

Prove that cyclic parallelogram is a rectangle.

**Reasoning:**

The sum of either pair of opposite angles of a cyclic quadrilateral is \(180^\circ.\) By using this fact we can show each angle of cyclic parallelogram as \(90^\circ\) which will prove the statement it is a rectangle.

**Steps:**

Let \({ABCD}\) be the cyclic parallelogram.

We know that opposite angles of a parallelogram are equal.

\[\begin {align}\angle {A}=\angle {C} \text { and } \angle {B}=\angle {D} \ldots .(1) \end {align}\]

We know that the sum of either pair of opposite angles of a cyclic quadrilateral is \(180^{\circ}.\)

\[\begin {align} \angle {A}+\angle {C}=180^{\circ} \ldots .(2) \end {align}\]

Substituting (**\(1\)**) in (**\(2\)**),

\[\begin{align}\angle {A}+\angle {C}&=180^{\circ} \\ \angle {A}+\angle {A}&=180^{\circ} \\ 2 \angle {A}&=180^{\circ} \\ \angle {A}&=90^{\circ}\end{align}\]

We know that if one of the interior angles of a parallelogram is \(90^{\circ},\) all the other angles will also be equal to \(90^{\circ}.\)

Since all the angles in the parallelogram is \(90^{\circ},\) we can say that parallelogram \({ABCD}\) is a rectangle.