The circle cuts the measure in half
How the intercepted arc matters
The central relationship is that an inscribed angle measures half the measure of its intercepted arc in the standard minor-arc setting. This is also why an angle inscribed in a semicircle is always a right angle.
This page keeps the angle point and intercepted arc visible together so the theorem can be read from the geometry instead of memorised without context.
If the intercepted arc measures 100 degrees, the inscribed angle is 50 degrees. The angle is smaller because its vertex sits on the circle and sees the arc from the edge. Moving the vertex along the same arc can preserve the intercepted arc and preserve the angle measure.
When the intercepted arc is a semicircle, half of 180 degrees is 90 degrees. That is the reason a triangle with a diameter as one side has a right angle at any point on the circle. The theorem grows naturally from the half-arc rule.
- An inscribed angle has its vertex on the circle and intercepts an arc.
- The sides of an inscribed angle are chords, not radii.
- An inscribed angle equals half the measure of its intercepted arc in the usual case.
- If the intercepted arc is a semicircle, the inscribed angle measures 90 degrees.
Angle reading from the circumference
Where inscribed angles help
- Use inscribed angles in arc-measure and missing-angle problems.
- Use them when a circle theorem links an edge point on the circle to an arc across the figure.
- Use them when comparing the same endpoints under both a central angle and an inscribed angle.
Follow both chords
Inscribed-angle errors to catch
- Do not confuse the inscribed-angle rule with the central-angle rule; the measures are not the same.
- Do not choose the wrong intercepted arc when tracing the angle's sides to the circle.
- Do not place the vertex inside the circle and still call the angle inscribed.
