In this video I go over the Inscribed Angle Theorem (or Central Angle Theorem) as well as go over its proof. The inscribed angle theorem states that the inscribed angle on the major arc of a circle which subtends 2 points on the circle is half of the central angle suspended on the same arc and 2 points on the circle. I first prove the theorem for the case where one of the cords of the inscribed angle crosses through the center of the circle and thus form the diameter. I use the proof of this first case to prove the other 2 cases where the center of the circle is located inside the inscribed angle and the case where the center of the circle is on the outside of the inscribed angle. This theorem is very important because it allows the angles of triangles in many applications to be greatly simplified.
This theorem is only for when the inscribed angle is on the major arc but in my later video I prove that if the inscribed angle is on the minor arc then the inscribed angle is supplementary of the half of the central angle.
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Inscribed Angle Theorem
Inscribed Angle: The angle subtended at a point on the circle by two given points on the circle.
The Inscribed Angle Theorem (or Central Angle Theorem) states that:
The angle inscribed in a circle is half of the central angle that subtends that same arc on the circle.
- This relation holds only if the angles are subtended on the major arc
Proof: 3 Cases to Consider
(1) One of the subtended cords pass through the center of the circle and form the diameter
(2) The center of the circle is in the interior of the inscribed angle
(3) The center of the circle is in the exterior of the inscribed angle
