Question

if 2 opposite sides of cyclic quadrilateral are equal prove that the other two sides are parallel

Answer 1

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Let ABCD be quadrilateral with ab||cd
Join be. 
In triangle abd and CBD,
Angle abd=angle cdb(alternate angles) 
Anglecbd=angle adb(alternate angles) 
Bd=bd(common) 
Abd=~CBD by asa test
Ad=BC by cpct
Since ad =bc ABCD is a isosceles trapezium
Angle d= angle c - - - - - - (1)
Join ac. 
In triangle adc and bcd
Cd=cd(common) 
Angle d =angle c(from 1) 
Ad =bc(proved above) 
Triangle adc=~bcd. by sas test
Ac=bd. by cpct

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