Show that each angle of a rectangle is right angle.

Asked by abhinavkumar281 | 1st Nov, 2014, 03:53: PM

Expert Answer:

Consider the following figure.
A B C D space i s space a space r e c tan g l e space w i t h space A B equals C D space a n d space A D equals B C W e space n e e d space t o space p r o v e space t h a t space e a c h space a n g l e space o f space A B C D space i s space a space r i g h t space a n g l e. O space i s space t h e space m i d p o i n t space o f space t h e space d i a g o n a l s space A C space a n d space B D. C o n s i d e r space t h e space t r i a n g l e s space A O D space a n d space B O C. angle A O D equals angle B O C space space space space open square brackets v e r t i c a l l y space o p p o s i t e space a n g l e s close square brackets angle A D B equals angle D B C space space space space open square brackets A C space a n d space B D space a r e space t h e space t r a n s v e r s a l space a n d space close square brackets A D equals B C space space space space space space space space space space space space space space space space open square brackets g i v e n close square brackets space space space space T h u s comma space b y space A n g l e minus A n g l e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space triangle A O D identical to triangle B O C T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g u e n t. T h e r e f o r e comma space angle D A C equals angle A C B space S i m i l a r l y comma space triangle A O B identical to triangle C O D S i n c e space t h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g u e n t comma space angle B D C equals angle A B D C o n s i d e r space angle A D C : angle A D C equals angle A D B plus angle B D C equals angle D B C plus angle A B D equals angle A B C I n 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 s u p p l e m e n t a r y H e n c e comma space angle A D C plus angle A B C equals 180 degree rightwards double arrow angle A B C plus angle A B C equals 180 degree rightwards double arrow space angle A D C equals angle A B C equals 90 degree S i m i l a r l y comma space w e space c a n space p r o v e space t h a t space angle D A B equals angle B C D equals 90 degree

Answered by Vimala Ramamurthy | 3rd Nov, 2014, 12:59: PM