Definition: An inscribed angle has its vertex on a circle and its sides are chords; its measure is half the measure of the arc it intercepts.
What is the inscribed angle theorem?
The angle measures half of its intercepted arc. So an inscribed angle is half of the central angle that cuts the same arc. All inscribed angles that intercept the same arc are equal.
What are useful consequences?
An angle inscribed in a semicircle is a right angle. Opposite angles of a quadrilateral inscribed in a circle are supplementary.
Example. An arc AB measures 118 degrees. An inscribed angle that intercepts it measures 59 degrees. In a cyclic quadrilateral with one angle of 103 degrees, the opposite angle is 180 – 103 = 77 degrees. If an inscribed angle is (3x + 4) degrees and intercepts an arc of 98 degrees, then 3x + 4 = 49, so x = 15.
What mistakes occur?
Forgetting to halve the arc, or doubling the angle in the wrong direction. Remember: arc = 2 × inscribed angle. Students also mix up the vertex position. If the vertex is at the center, it is a central angle; if it is on the circle, it is inscribed.