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In a square ABCD, its diagonals AC and BD intersect each other at point O. The bisectors of angle DAO meets BD at point M and the bisector of angle ABD meets AC at N and AM at L. Show that: (i)angle ONL+ angle OML =180 degree. (ii) angle BAM =angle BMA. (iii)ALOB is a cyclic quadrilateral.
Asked by ashutoshtripathy2000 | 02 Oct, 2015, 11:18: AM
answered-by-expert 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.
S i n c e space A B C D space i s space a space s q u a r e comma space angle A B C equals angle B C D equals angle C D A equals angle D A B equals 90 degree D i a g o n a l s space A C space a n d space B D space o f space t h e space s q u a r e space b i s e c t space a t space O. T h u s comma space t h e space a n g l e space angle D A O equals 45 degree G i v e n space t h a t space angle D A M equals angle M A O equals fraction numerator 45 degree over denominator 2 end fraction space space space space open square brackets because A M space i s space t h e space a n g l e space b i s e c t o r space o f space angle D A O close square brackets S i n c e space t h e space d i a g o n a l s space o f space a space s q u a r e space b i s e c t space p e r p e n d i c u l a r l y comma space w e space h a v e comma space angle A O D equals angle D O C equals angle C O B equals angle A O B equals 90 degree left parenthesis i right parenthesis S i n c e space angle A O D equals 90 degree space a n d space angle M A O equals fraction numerator 45 degree over denominator 2 end fraction comma space w e space h a v e comma space angle O M L equals 90 degree minus fraction numerator 45 degree over denominator 2 end fraction equals fraction numerator 135 degree over denominator 2 end fraction... left parenthesis 1 right parenthesis S i m i l a r l y comma space w e space h a v e space a n g l e space angle A B O equals 45 degree A n d space angle A B N equals angle N B O equals fraction numerator 45 degree over denominator 2 end fraction space space space space open square brackets because B N space i s space t h e space a n g l e space b i s e c t o r space o f space angle A B D close square brackets S i n c e space angle A O B equals 90 degree space a n d space angle N B O equals fraction numerator 45 degree over denominator 2 end fraction comma space w e space h a v e comma space angle O N B equals 90 degree minus fraction numerator 45 degree over denominator 2 end fraction equals fraction numerator 135 degree over denominator 2 end fraction T h e space a n g l e s space angle O N B space a n d space angle O N L space a r e space l i n e a r space p a i r space a n d space h e n c e space w e space h a v e comma space space angle O N B plus angle O N L equals 180 degree rightwards double arrow fraction numerator 135 degree over denominator 2 end fraction plus angle O N L equals 180 degree rightwards double arrow angle O N L equals 180 degree minus fraction numerator 135 degree over denominator 2 end fraction rightwards double arrow angle O N L equals fraction numerator 225 degree over denominator 2 end fraction... left parenthesis 2 right parenthesis N o w space a d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma angle O M L plus angle O N L equals fraction numerator 135 degree over denominator 2 end fraction plus fraction numerator 225 degree over denominator 2 end fraction rightwards double arrow angle O M L plus angle O N L equals fraction numerator 135 degree plus 225 degree over denominator 2 end fraction rightwards double arrow angle O M L plus angle O N L equals fraction numerator 360 degree over denominator 2 end fraction rightwards double arrow angle O M L plus angle O N L equals 180 degree... left parenthesis 3 right parenthesis left parenthesis i i right parenthesis F r o m space e q u a t i o n space left parenthesis 1 right parenthesis space w e space g o t comma space angle O M L equals fraction numerator 135 degree over denominator 2 end fraction rightwards double arrow angle B M A equals angle O M L equals fraction numerator 135 degree over denominator 2 end fraction... left parenthesis 4 right parenthesis S i n c e space A M space i s space t h e space a n g l e space b i s e c t o r comma space w e space h a v e comma space angle M A O equals fraction numerator 45 degree over denominator 2 end fraction S i n c e space A B C D space i s space a space s q u a r e comma space t h e space d i a g o n a l s space a r e space t h e space a n g l e space b i s e c t o r s space o f space t h e space c o r n e r s. rightwards double arrow angle B A O equals fraction numerator 90 degree over denominator 2 end fraction equals 45 degree A n d space w e space k n o w space t h a t comma space angle B A M equals angle B A O plus angle M A O rightwards double arrow angle B A M equals 45 degree plus fraction numerator 45 degree over denominator 2 end fraction rightwards double arrow angle B A M equals fraction numerator 135 degree over denominator 2 end fraction... left parenthesis 5 right parenthesis F r e o m space e q u a t i o n s space left parenthesis 4 right parenthesis space a n d space left parenthesis 5 right parenthesis comma space w e space h a v e comma space angle B M A equals angle B A M equals fraction numerator 135 degree over denominator 2 end fraction left parenthesis i i i right parenthesis C o n s i d e r space t h e space q u a d r i l a t e r a l space O M L N I n space a space q u a d r i l a t e r a l comma space t h e 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 T h u s comma space angle O N L plus angle N L M plus angle O M L plus angle M O N equals 360 degree space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 3 right parenthesis space a n d space angle M O N equals angle D O A equals 90 degree close square brackets rightwards double arrow space 180 degree plus angle N L M plus 90 degree equals 360 degree rightwards double arrow space 270 degree plus angle N L M equals 360 degree rightwards double arrow space angle N L M equals 360 degree minus 270 degree rightwards double arrow space angle N L M equals 90 degree T h e space a n g l e s space angle N L M space a n d space angle A L B space a r e space l i n e a r space p a i r T h u s comma space angle N L M plus angle A L B equals 180 degree rightwards double arrow 90 degree plus angle A L B equals 180 degree rightwards double arrow angle A L B equals 180 degree minus 90 degree rightwards double arrow angle A L B equals 90 degree A n d space angle A L B equals angle A O B equals 90 degree rightwards double arrow angle A L B plus angle A O B equals 180 degree A g a i n. space i n space a space q u a d r i l a t e r a l comma space t h e 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 T h u s comma space t h e space q u a d r i l a t e r a l space A L O B space i s space c y c l i c space q u a d r i l a t e r a l.
Answered by Vimala Ramamurthy | 03 Oct, 2015, 09:48: AM

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