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Ex.10.5 Q12 Circles Solution - NCERT Maths Class 9

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Question

Prove that cyclic parallelogram is a rectangle.

 Video Solution
Circles
Ex 10.5 | Question 12

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.

 Video Solution
Circles
Ex 10.5 | Question 12
  
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