Ex.8.1 Q2 Quadrilaterals Solution - NCERT Maths Class 9

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Text Solution

What is known/?

The diagonals of a parallelogram are equal.

What is unknown?

How we can show that it is a rectangle.

Reasoning:

To show that given parallelogram is a rectangle, we have to prove that one of its interior angles is and this can be done by showing two triangles congruent.

Steps:

Let \(ABCD\) be a parallelogram. To show that \(ABCD\) is a rectangle, we have to prove that one of its interior angles is \(\begin{align}90^{\circ}\end{align}\) .

In \(\Delta ABC\) and  \(\Delta DCB\)

\[\begin{align}  & AB=DC\, \\  & \left(\!\begin{array}  & \text{In a parallelogram, } \\ \text{opposite sides are equal}\text{.} \\ \end{array}\!\right) \\  &  \\  & BC=BC\,\left( \text{Common} \right) \\  & AC=DB\,\left( \text{Given} \right) \\  & \therefore \Delta ABC\,\cong \,\Delta DCB \\  & \left( By\,SSS\,\text{Congruence rule} \right) \\  &  \\  & \Rightarrow \,\,\,\angle ABC=\angle DCB \\ \end{align}\]

It is known that the sum of the measures of angles on the same side of transversal is \(180^\circ\).

\[\begin{align}\angle &{\rm{ABC}} + \angle {\rm{DCB}} = {\rm{18}}{0^0}\;\left( {{\rm{AB }}||{\rm{ CD}}} \right)\\&\Rightarrow {\rm{ }}\angle {\rm{ABC}} + \angle {\rm{ABC}} = {\rm{18}}{0^0}\\&\Rightarrow {\rm{  2}}\angle {\rm{ABC}} = {\rm{18}}{0^0}\\& \Rightarrow {\rm{ }}\angle {\rm{ABC}} = {\rm{9}}{0^0}{\rm{ }}\end{align}\]

Since \(ABCD\) is a parallelogram and one of its interior angles is \(\begin{align}90^{\circ},\end{align}\) \(ABCD\) is a rectangle.