Question
Wed March 04, 2015 By: Akansha Banerjee

In the figure, PQRS is a parallelogram PA & RB bisects the angles P & R respectively. Show that PAIn the figure, PQRS is a parallelogram PA & RB bisects the an, Question no 88994, MathematicsBR.

Expert Reply
Vimala Ramamurthy
Fri March 06, 2015
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
S i n c e space P Q R S space i s space a space p a r a l l e log r a m comma space o p p o s i t e space a n g l e s space a r e space e q u a l. T h u s comma space angle P equals angle R space a n d space angle Q equals angle S A l s o space t h e space a d j a c e n t space a n g l e s space a r e space s u p p l e m e n t a r y. rightwards double arrow angle P plus angle Q equals 180 degree comma space angle Q plus angle R equals 180 degree comma space angle R plus angle S equals 180 degree space a n d space angle S plus angle P equals 180 degree G i v e n space t h a t space P A space a n d space B R space a r e space a n g l e space b i s e c t o r s space o f space t h e space a n g l e s space angle P space a n d space angle R space r e s p e c t i v e l y. T h u s comma angle A P S equals angle A P B equals fraction numerator angle P over denominator 2 end fraction S i m i l a r l y comma angle Q R B equals angle B R S equals fraction numerator angle R over denominator 2 end fraction S i n c e space angle P equals angle R comma space fraction numerator angle P over denominator 2 end fraction equals fraction numerator angle R over denominator 2 end fraction rightwards double arrow angle A P B equals angle B R S T h u s comma space o p p o s i t e space a n g l e s space o f space t h e space q u a d r i l a t e r a l space A R B P space a r e space e q u a l. I f space t h e space o p p o s i t e space a n g l e s space o f space a space q u a d r i l a t e r a l space a r e space e q u a l comma space t h e n space t h e space q u a d r i l a t e r a l space i s space a a space p a r a l l e log r a m. T h u s comma space A R B P space i s space a space p a r a l l e log r a m. I n space a space p a r a l l e log r a m comma space t h e space o p p o s i t e space s i d e s space a r e space p a r a l l e l. H e n c e space P A parallel to B R space a n d space P B parallel to A R.
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