Circumscribed Angle Theorem

• • The measures of a circumscribed angle and central angle that intersect at the same points on a circle are supplementary.

May sound complicated, but it's actually pretty easy with a picture

Here we have a circle with central angle ∠CAB, and circumscribed angle ∠CDB. We'd like to know the measure of the circumscribed angle. It's easy to find.

Notice that CDBA makes a 4-sided polygon. We know that, in any 4-sided polygon, the angles must add up to 360°. Well, what do we have so far

• • ∠A = 60° (given)

  • ∠B = 90° (tangent-radius theorem)

  • ∠C = 90° (tangent-radius theorem)

• ∠D = x (unknown)

60 + 90 + 90 + x = 360

240 + x = 360

x = 120°

Interesting. So the circumscribed angle (120°) and the central angle (60°) add to 180°. Will it always be this way? Yes.

Why? Well, let's let the central be any random angle, y

• ∠A = y

  • ∠B = 90° (tangent-radius theorem)

  • ∠C = 90° (tangent-radius theorem)

• ∠D = x (unknown)

y + 90 + 90 + x = 360

y + 180 + x = 360

y + x = 180°

Well look at that! No matter what the angle, when we add them they must equal 180°. Remember, two angles that add to 180° are said to be supplementary. So, that means

• • The measures of a circumscribed angle and central angle that intersect at the same points on a circle are supplementary.