Question

Prove that the bisectors of the angles of a parallelogram form a rectangle.



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Answer 1

Parallelogram PQRS is RECTANGLE

Step-by-step explanation:

In ΔADS,

$\angle DAS + \angle ADS = \frac{1}{2}\angle A - \frac{1}{2}\angle D$

= $\frac{1}{2}(\angle A - \angle D)$

= $\frac{1}{2}(180)$

= 90 °

($\angleA$ & $\angleD$are interior angles on the same side of transversal)

Also in ΔADS,

 $\angle DAS + \angle ADS + \angle DSA = 180$ (angle sum property of triangles)

$90 + \angle DSA = 180$

$\angle DSA = 180 - 90$

$\angle DSA = 90 = \angle PSR$ (Vertically Opposite Angles)

Similarly, it can be shown that \angle APB or \angle SPQ = 90 °

Also, $\angle SRQ = \angle RQP = 90$

Hence, PQRS is a Rectangle.