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

1 answer:

7 0

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 = 18<span>0° are supplementary</span>

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