Question

If two opposite sides of a cyclic quadrilateral are parallel then prove that remaining two sides are equal and both diagonals are equal.

Answer 1

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
h. p..