Opposite angles of a cyclic quadrilateral are supplementary - Proof

A quadrilateral is said to be “inscribed in a circle” if its four corners(vertices) lie on the circumference of the circle. Any inscribed quadrilateral in a circle is termed as cyclic quadrilateral. 

A cyclic quadrilateral has a property that both pairs of its opposite angles are supplementary(their sum is 180 degrees). 

In the figure, ∠A + ∠C = 180°

          and     ∠B + ∠D = 180°

So how do we prove this property? 

We just need to prove any one pair of opposite angles to be supplementary. ABCD is a cyclic quadrilateral. But even before that it is a quadrilateral first. Therefore, sum total of all four of its interior angles is 360 degrees. 

∠A + ∠B + ∠C + ∠D = 360°

If we can prove ∠B + ∠D = 180°, it follows from the above equation that the other pair of opposite angles will have to be supplementary too. 

∠A + ∠C = 360° - (∠B + ∠D)  ...(1)

= 360° - 180° 

= 180°

So let’s figure how we can prove ∠B + ∠D = 180°. 

We observe that A and C are the endpoints of arc ABC. Arc ABC subtends ∠D at point D which is also one of the angles of our concern. Another angle of our concern is ∠B, which is the angle subtended by the major arc ADC at point B. Let x and y be the angles that arc ABC and arc ADC subtend at the center.

Arc ABC and arc ADC together form the entire boundary(circumference) of the circle. So together they will subtend 360 degrees angle at the center of the circle. 

In the figure, 

x + y = 360° …(2)

Now what? 

x is the central angle of arc ABC. An important theorem in circles states that the central angle of an arc is twice the measure of the angle subtended by the arc at any point on the circumference. 

D is a point on the circumference where arc ABC subtends ∠D. By this theorem, the central angle x must be twice the ∠D. 

x = 2(∠D)

B is the point on the circumference where arc ADC subtends ∠B, and y is the central angle of arc ADC. So y must be twice ∠B. 

y = 2(∠B)

From (2), 

x + y = 2 (∠D) + 2 (∠B) = 360°

Therefore, ∠B + ∠D = 180°

Also from (1), 

∠C + ∠D = 360° - (∠B + ∠D)  

= 360° - 180° 

= 180°

Hence proved.