PQRS is a parallelogram. QA bisects angle pqr and meet PS at A. A straight line through R and parallel to QA meets PQ produced at B and PS produced at C. Prove that traingle PBC is an isosceles triangle. Please give it fast. Tommorow is my exam.

Asked by vatsalchoudhary41 | 13th Mar, 2015, 09:02: AM

Expert Answer:

$C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.$

$S i n c e space Q A space i s space t h e space a n g l e space b i s e c t o r space o f space angle P Q R comma space w e space h a v e comma angle P Q S equals angle S Q R... left parenthesis 1 right parenthesis P Q R S space i s space a space p a r a l l e log r a m space a n d space h e n c e comma space P S parallel to Q R rightwards double arrow P C parallel to Q R rightwards double arrow angle C S Q space a n d space angle R Q S space a r e space c o i n t e r i o r space a n g l e s rightwards double arrow angle C S Q plus angle S Q R equals 180 degree rightwards double arrow angle C S Q equals 180 degree minus angle S Q R... left parenthesis 2 right parenthesis B u t comma space angle P S Q space a n d space angle C S Q space a r e space l i n e a r space p a i r. H e n c e space angle P S Q plus angle C S Q equals 180 degree S u b s t i t u t i n g space t h e space v a l u e space o f space angle C S Q space f r o m space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma angle P S Q plus 180 degree minus angle S Q R equals 180 degree rightwards double arrow angle P S Q minus angle S Q R equals 180 degree minus 180 degree rightwards double arrow angle P S Q minus angle S Q R equals 0 degree rightwards double arrow angle P S Q equals angle S Q R rightwards double arrow angle P S Q equals angle P Q S space space space space space space space left square bracket f r o m space e q u a t i o n space left parenthesis T h e space a n g l e s space angle P S Q space a n d space angle P Q S space a r e space t h e space b a s e space a n g l e s space o f space t h e space t r i a n g l e space triangle P Q S T h u s comma space t h e space t r i a n g l e space triangle P Q S space i s space a n space i s o s c e l e s space t r i a n g l e.$

All the best for your exams!!!!

Answered by Vimala Ramamurthy | 13th Mar, 2015, 09:31: PM

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