ABCD is a parallelogram of which no angle is 600 . Equilateral triangles ADE and DCF are drawn outwardly on the sides AD and DC. Show that triangle ABE is congruent to triangle CFB

 

Asked by Paresh | 10th Dec, 2015, 10:31: AM

Expert Answer:

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.
A B C D space i s space a space p a r a l l e log r a m. H e n c e space A B equals C D space a n d space A D equals B C triangle A D E space a n d space triangle D C F space a r e space e q u i l a t e r a l space t r i a n g l e s space d r a w n space o n space A D space a n d space D C. T h e r e f o r e comma space A D equals D E equals A E space a n d space D C equals C F equals D F  N o w space c o n s i d e r space t h e space t r i a n g l e s space triangle A B E space a n d space triangle C F B colon A B equals C D space space space space space space open square brackets s i d e s space o f space a space p a r a l l e log r a m close square brackets space space space space space space equals C F space space space space space space open square brackets s i d e s space o f space a n space e q u i l a t e r a l space t r i a n g l e close square brackets  angle E A B equals angle E A D plus angle D A B space space space space space space space space space space space equals 60 degree plus angle D A B space space space space space open square brackets triangle A D E space i s space a n space e q u i l a t e r a l space t r i a n g l e comma space angle E A D equals 60 degree close square brackets space space space space space space space space space space space equals 60 degree plus angle D C B space space space space space space space space open square brackets o p p o s i t e space a n g l e s space o f space t h e space p a r a l l e log r a m space a r e space e q u a l comma space angle D A B equals angle D C B close square brackets space space space space space space space space space space space equals angle F C B   A E equals A D space space space space space space space open square brackets s i d e s space o f space a n space e q u i l a t e r a l space t r i a n g l e close square brackets space space space space space equals B C space space space space space space space space space open square brackets s i d e s space o f space a space p a r a l l e l o g r a m close square brackets  T h u s comma space b y space S i d 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 w e space h a v e comma space triangle A B E approximately equal to triangle C F B.

Answered by Vimala Ramamurthy | 10th Dec, 2015, 12:27: PM