Question

Which of these is a condition for a parallelogram to be a rectangle?

• A. One angle is ${90}^{\circ }$
• B. Diagonals are equal
• C. Quadrilateral obtained by joining midpoints of adjacent sides is a rhombus
• D. All of the above

Answer 1

The correct option is D

All of the above

Each of the given conditions taken individually will ensure that a parallelogram is a rectangle.
When one angle is ${90}^{\circ }$, then the opposite angle will also be ${90}^{\circ }$. Adjacent angle is ${180}^{\circ }$ - ${90}^{\circ }$ = ${90}^{\circ }$.
When all the angles of a parallelogram are ${90}^{\circ }$, it becomes a rectangle.
Diagonals are equal -
Suppose ABCD is a parallelogram with diagonals equal.
In $△$ ABC and $△$ DCB -
AB = DC (Opposite sides of a parallelogram)
BC = BC (Opposite sides of a parallelogram)
AC = BD (Diagonals are given to be equal)
$\Rightarrow$ $△$ ABC $\cong$$△$ DCB
$\Rightarrow$ $\angle$ B = $\angle$ C = ${90}^{\circ }$
$\Rightarrow$ $ABCD$ is a rectangle.