Question

Quadrilateral OPQR is inscribed inside a circle as shown below. Write a proof showing that angle O and Q are supplementary

Answer 1

In the circle, we can see that angle O and angle Q are both inscribed angles which intercepts an arc. Angle O intercepts arc RQP while angle Q intercepts arc ROP. The measure of each angle is measured to be one half of its intercepted arc, that is: 

m ∠ O = m (arc PQR/2)
m ∠ Q = m (arc POR/2)

 

However we can also see that arcs RQP and ROP together make up the entire circle which has 360 degrees.

arc PQR + arc POR = 360°

 

therefore dividing both sides by 2:
arc PQR/2 + arc POR/2 = 180°

Then:
m ∠ O + m ∠ Q =  m (arc PQR/2) + m (arc POR/2) = 180°

 

∠ O + ∠ Q = 180° are supplementary