ABCD Is a square and on side DC an equilateral triangle is constructed. Prove that AE =BE AND Angle DAE = 15 degrees

Asked by avantikadandapani | 17th Nov, 2014, 06:17: PM

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.
G i v e n space t h a t space D C 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. T h e r e f o r e comma space angle D E C equals angle D C E equals angle E D C equals 60 degree A B C D space i s space a space s q u a r e comma space a n d space h e n c e space angle A D C equals 90 degree T h e r e f o r e comma space angle A D E equals angle A D C minus angle E D O rightwards double arrow angle A D E equals 90 degree minus 60 degree rightwards double arrow angle A D E equals 30 degree space S i n c e space triangle D C 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 D C equals D E equals C E S i n c e space A B C D space i s space a space s q u a r e comma space w e space h a v e comma space A D equals D C therefore D E equals A D T h e r e f o r e comma space triangle A D E space i s c space a n space i s o s c e l e space t r i a n g l e. H e n c e space t h e space b a s e space a n g l e s space a r e space e q u a l. angle D A E equals angle D E A B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle D A E plus angle D E A plus angle A D E equals 180 degree rightwards double arrow 2 angle D A E plus 30 degree equals 180 degree rightwards double arrow 2 angle D A E equals 180 degree minus 30 degree rightwards double arrow 2 angle D A E equals 150 degree rightwards double arrow angle D A E equals fraction numerator 150 degree over denominator 2 end fraction rightwards double arrow angle D A E equals 75 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 comma space angle C B E equals 75 degree angle D A B equals 90 degree space a n d space angle C B A equals 90 degree rightwards double arrow angle E A B equals 90 degree minus angle D A E space a n d space space angle E B A equals 90 degree minus angle C B E rightwards double arrow angle E A B equals 90 degree minus 75 degree space a n d space space angle E B A equals 90 degree minus 75 degree rightwards double arrow angle E A B equals 15 degree space a n d space space angle E B A equals 15 degree I n space t r i a n g l e space A B E comma space S i n c e space t h e space b a s e space a n g l e s space a r e space e q u a l comma space w e space h a v e comma space A E equals B E

Answered by Vimala Ramamurthy | 18th Nov, 2014, 09:50: AM

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