Question

prove that cyclic parallelogram is a rectangle

Answer 1

Let ABCD be a cyclic quadrilateral such that its diagonals AC and BD are the diameters of the circle through the vertices A, B, C, and D. 

Since AC is a diameter and angle in a semi-circle is a right angle, 

angle ADC = 900 and angle ABC = 900

 Similarly, BD is a diameter. 

Therefore, angle BCD = 900 and angle BAD = 900

Thus, ABCD is a rectangle 

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Answer 2

Consult the picture:-

PROOF:-
Since,ABCD is a cyclic quadrilateral
Therefore, Angle 1 + Angle 2 =180°---------eq.(i)
Because opposite angles of a cyclic quadrilateral are supplementary

Thus, ABCD is a parallelogram

Therefore, angle1 = angle2------------eq.(ii)
Because opposite angles of parallelogram are equal

From eq(i) and eq.(ii)

Angle1=Angle2=90°

Parallelogram is a rectangle