Inscribed Angle Theorem

    • The measure of an inscribed angle is half the measure of the central angle that intersects a the circle at the same points.

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

Here we have a circle with an arc (), a central angle (∠CAD), and an inscribed angle (∠CED).

  • m∠CAD = 90°

Also, since EA and CA are both radii, they are the same length. That means that triangle ECA would be an isosceles right triangle. We'll, remember that's a special right triangle, 45-45-90. So that means ∠CED is 45°.

  • m∠CED = 45°

Interesting. So the inscribed angle is half the measure of the central angle. Will it always be this way? Yes.

If you ever need to remember the relationship between a central angle and an inscribed angle, this picture makes it easy to remember. It's worth taking a good look at it.

Special Case: 90° Inscribed Angle

This is just a special case that comes up a lot. We know that the central angle is always 2x bigger than the inscribed angle.

  • If the inscribed angle is 90°, the central angle must be 180°.

  • A 180° angle is a straight line

  • A straight line through the center of the circle is a diameter

Therefore, if an inscribed angle is a right angle, then it intersects the circle through a diameter.

This video shows that, no matter how I move my inscribed angle or my diameter, the angle will always be 90°