The Angle Bisectors of a Parallelogram Form a Rectangle


 
 
Concept Explanation
 

The Angle Bisectors of a Parallelogram Form a Rectangle

The Angle Bisectors of a Parallelogram Form a Rectangle :  A parallelogram is a quadrilateral in which both the opposite pair of sides are parallel and equal to each other. The angle bisectors of a parallelogram form a rectangle

Theorem: The Angle Bisectors of a Parallelogram Form a Rectangle.

GIVEN A parallelogram ABCD in which bisectors of angles A, B, C, D intersect at P, Q, R, S to form a quadrilateral PQRS.

To Prove PQRS is a rectangle.

Proof Since ABCD is a parallelogram. Therefore,   AD large parallel BC

Now, AD large parallel BC and transversal AB intersects them at A and B respectively. Therefore,

   large angle A +angle B=180^{circ}            [large because Sum of consecutive interior angles is large 180^{circ}]

large Rightarrow ;;frac{1}{2}angle A+frac{1}{2}angle B=90^{circ}

large Rightarrow ;;angle BAS+angle ABS=90^{circ}     ....(i)     

[large because AS and BS are bisectors  of    large angle A;and;angle B respectively]

But, in large Delta ABS, we have

   large angle BAS+angle ABS+angle ASB=180^{circ}   [Sum of the angles of a triangle is large 180^{circ}]

large Rightarrow ;;;90^{circ}+angle ASB=180^{circ}

large Rightarrow ;;;angle ASB=90^{circ}

large Rightarrow ;;;angle RSP=90^{circ}       [large because angle ASB;and;angle RSP  are vertically opposite  angles largetherefore angle RSP=angle ASB ]

Similarly, we can prove that

  large angle SRQ=90^{circ},angle RQP=90^{circ};and;angle SPQ=90^{circ}

Hence, PQRS is a rectangle.

ILLUSTRATION: LMOP is a quadrilateral in which LM = OP   and angle MLP= 70^0 and angle LPO= 110^0. The angle bisectors of LMOP form a quadrilateral ABCD. Find the measure of angle A

Solution: In the quadrilateral LMOP

angle MLP+ angle LPO= 70^0+ 110^0=180^0

But they are cointerior angles when LM  and PO are two straight lines and LP is the transversal

As they are supplementary. LM || PO

Also LM = PO     [ Given]

therefore LMOP is a parallelogram as its one opposite pair of sides is parallel and equal.

As we know that the angle bisectors of a parallelogram form a rectangle.

Rightarrow    ABCD is a rectangle.

Hence angle A = 90^0              [ Each angle od a rectangle is a right angle]

 

 
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