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
