Question

prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180. using the above. ABCD is a cyclic quadrilateral in which angle A=(x+y+10),angle B= (Y+20),ANGLE D=(X+Y),ANGLE c= (x +y-30) find X,Y

Answer 1

Construction : Join AC and BD
Proof:
angleACB = angleADB (angles in the same segment are equal)......(I)
and. angle BAC = angle BDC ( angles in the same segment are equal)......(II)
Adding I and II
=> angle ACB + angle BAC = angle ADB + angle BDC
=> angle ACB + angle BAC = angle ADC
=> angle ABC + angle ACB + angle BAC = angle ABC + angle ADC
=> 180° = angle ABC + angle ADC (Angle sum property of a triangle)
=> Angle ABC + angle ADC = 180°
=> angle B + angle D = 180°

But,
angle A + angle B + angle C + angle D = 360°
=>
angle A + angle C = 360° - ( angle B + angle D)
=> angle A + angle C = 360° - 180°
=> angle A + angle C = 180°

Hence ,
angle A + angle C = 180° and
angle B + angle D = 180°