Question

The bisectors of interior angles of a parallelogram forms a ___________.

• A. Square
• B. Rectangle
• C. Trapezium
• D. Rhombus

Answer 1

The correct option is B Rectangle

$ABCD$ is a parallelogram

$AE$ bisects $\angle BAD$

$BF$ bisects $\angle ABC$

$CG$ bisects $\angle BCD$

$DH$ bisects $\angle ADC$
$\angle BAD+\angle ABC=180{}^o$ (because adjacent angles of a parallelogram are supplementary)

$\angle BAJ=\frac{1}{2}\angle BAD$ because $AE$ bisects $\angle BAD$

$\angle ABJ=\frac{1}{2}\angle ABC$ because $DH$ bisects $\angle ABC$

$\angle BAJ$ + $\angle ABJ$ = ${90}^o$ [ halves of supplemetary angles are complementary]

$△ABJ$ is a right triangle because its acute interior angles are complementary with $\angle J={90}^o$
Similarly, $\angle K=\angle L=\angle I={90}^o$

In quadrilateral $IJLK$, since all angles are ${90}^o$, it must be a rectangle.
Therefore, internal angle bisectors of a parallelogram forms a rectangle.