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CBSE Class 9 Answered

ABCD is a rhombus with angle A = 670. If ∆DEC is an equilateral triangle, then calculate i)angle CBE and ii)angle DBE

Asked by geethasree.vani5 | 15 Feb, 2015, 12:59: AM
answered-by-expert Expert Answer
Consider the following figure.
G i v e n space t h a t space A B C D space i s space a space r h o m b u s space a n d space angle A equals 67 degree S i n c e space a l l space r h o m b u s e s space a r e space p a r a l l e log r a m s comma space t h e space p r o p e r t i e s space o f space p a r a l l e log r a m space h o l d space g o o d space f o r r h o m b u s space a l s o. 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 r h o m b u s space a r e space e q u a l space a n d 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. T h u s comma space sin c e space angle C space i s space o p p o s i t e space t o space angle A comma space w e space h a v e comma space angle C equals 67 degree. A n d comma angle A plus angle B equals 180 degree rightwards double arrow angle B equals 180 degree minus angle A equals 180 degree minus 67 degree equals 113 degree A l s o space g i v e n space t h a t space triangle D E C 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. T h u s space a l l space t h e space a n g l e s space o f space t h e space t r i a n g l e space a r e space 60 degree rightwards double arrow angle D C E equals 60 degree angle B C E equals angle D C B plus angle D C E space space space space space space space space space space equals 67 degree plus 60 degree equals 127 degree A B C D space i s space a space r h o m b u s space a n d space D E C 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 space a n d space h e n c e a s l l space t h e space s i d e s space a r e space e q u a l. 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 space C B E. S i n c e space B C equals C E comma space triangle B C E space i s space a n space i s o s c e l e s space t r i a n g l e. rightwards double arrow angle C E B equals angle C B E B y space a n g l e space s u m space p r o p e r t y comma angle C E B plus angle C B E plus angle B C E equals 180 degree rightwards double arrow angle C E B plus angle C E B plus 127 degree equals 180 degree rightwards double arrow 2 angle C E B equals 180 degree minus 127 degree rightwards double arrow 2 angle C E B equals 53 degree rightwards double arrow angle C E B equals angle C B E equals fraction numerator 53 degree over denominator 2 end fraction equals 26.5 degree D i a g o n a l s space o f space t h e space r h o m b u s space b i s e c t s space t h e space v e r t e x space a n g l e T h u s comma space angle A B D equals angle D B C equals fraction numerator angle B over denominator 2 end fraction equals fraction numerator 113 degree over denominator 2 end fraction equals 56.5 degree A l s o space w e space h a v e comma angle D B E equals angle D B C minus angle C B E equals 56.5 degree minus 26.5 degree equals 30 degree
Answered by Vimala Ramamurthy | 15 Feb, 2015, 09:50: AM

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