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ICSE Class 9 Answered

In a quadrilateral ABCD , AB=CD & angle B=angle C. Prove that :- (i) AC=DB , (ii) AD is parallel to BC.
Asked by ashutoshtripathy2000 | 02 Feb, 2015, 10:10: PM
answered-by-expert Expert Answer
Consider the following diagram.
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 C space a n d space triangle D B C colon A B equals C D space space space space space 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 angle A B C equals angle B C D space space space space space space space space space..... open square brackets g i v e n close square brackets B C equals B C space space space space space space space space space space space space space space space space space space space space space.... open square brackets c o m m o n 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 triangle A B C approximately equal to triangle D B 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 r r e s p o n d i n g space t r i a n g l e s space a r e space c o n g r u e n t. T h e r e f o r e space w e space h a v e comma space A C equals B D S i m i l a r l y comma space angle C A B equals angle B D C space a n d space angle A C B equals angle D B C rightwards double arrow angle O A B equals angle O D C space a n d space angle O C B equals angle O B C T h e r e f o r e comma space i n space t r i a n g l e space C O B comma space sin 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 triangle C O B 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 O B equals O C S i n c e space A C equals B D space a n d space O B equals O C comma space w e space h a v e comma space O A equals O D therefore fraction numerator O A over denominator O C end fraction equals fraction numerator O D over denominator O B end fraction angle B O C space a n d space angle A O D space a r e space 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 space a n d space h e n c e comma space angle B O C equals angle A O D H e n c e comma space triangle A O D space a n d space triangle B O C space a r e space s i m i l a r space t o space e a c h space o t h e r. T h u s comma space angle O D A equals angle O B C space a n d space angle O A D equals angle O C B T h u s comma space t h e space l i n e s space A D space a n d space B C space a r e space p a r a l l e l space l i n e s.
Answered by Vimala Ramamurthy | 03 Feb, 2015, 09:14: AM

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