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 Prove that a quadrilateral is a parallelogram if its opposite angles are equal.

Asked by araima2001 4th March 2015, 11:04 AM
Answered by Expert
Answer:
I n space a space q u a d r i l a t e r a l comma space s u m space o f space a l l space t h e space a n g l e s space i s space 360 degree S u p p o s e space A B C D space i s space a space q u a d r i l a t e r a l. angle A plus angle B plus angle C plus angle D equals 360 degree.... left parenthesis 1 right parenthesis G i v e n space t h a t 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 A equals angle C space a n d space angle B equals angle D T h u s space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma comma 2 open parentheses angle A plus angle B close parentheses equals 360 degree rightwards double arrow open parentheses angle A plus angle B close parentheses equals fraction numerator 360 degree over denominator 2 end fraction rightwards double arrow open parentheses angle A plus angle B close parentheses equals 180 degree space a n d space angle C plus angle D equals 180 degree rightwards double arrow A B space i s space t h e space t r a n s v e r s a l space m e e t i n g space A D space a n d space B C space a t space A space a n d space B space s u c h space t h a t space space space space space space space S u m space o f space c o n s e c u t i v e space i n t e r i o r space a n g l e s space i s space 180 degree T h u s space A D parallel to B C A g a i n space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma 2 open parentheses angle A plus angle D close parentheses equals 360 degree rightwards double arrow open parentheses angle A plus angle D close parentheses equals fraction numerator 360 degree over denominator 2 end fraction rightwards double arrow open parentheses angle A plus angle D close parentheses equals 180 degree space a n d space angle B plus angle C equals 180 degree rightwards double arrow A D space i s space t h e space t r a n s v e r s a l space m e e t i n g space A B space a n d space D C space a t space A space a n d space D space s u c h space t h a t space space space space space space space S u m space o f space c o n s e c u t i v e space i n t e r i o r space a n g l e s space i s space 180 degree rightwards double arrow A B parallel to C D A n d space h e n c e space A B C D space i s space a space p a r a l l e log r a m.
Answered by Expert 5th March 2015, 12:03 PM
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