Central Angles, Inscribed Angles, and Arcs

Objective: Relate a central angle to its intercepted minor arc.

Prerequisite skills: circle arcs, triangle angle sums, and degrees. A central angle has its vertex at the center. Its degree measure equals the degree measure of its intercepted arc. A full circle has 360°, so complementary major and minor arcs total 360°. Common error: halving a central angle as though it were inscribed. Restrictions and interpretation: An inscribed angle is half its intercepted arc not containing the vertex; a central angle equals its arc measure in degrees.

Objective: Double an inscribed angle to recover its intercepted arc.

An inscribed angle has its vertex on the circle and sides along chords. Its measure is half the intercepted arc not containing that vertex. Angles intercepting the same arc are equal. Common error: using equality of angle and arc for an inscribed angle. Restrictions and interpretation: An inscribed angle is half its intercepted arc not containing the vertex; a central angle equals its arc measure in degrees.

Objective: Use a semicircle to establish a right angle.

An angle inscribed in a semicircle intercepts a 180° arc and therefore measures 90°. A triangle with one side as a circle diameter is right at the third circle point. Use the triangle sum for its remaining angles. Common error: placing the right angle at a diameter endpoint instead of the third point. Restrictions and interpretation: An inscribed angle is half its intercepted arc not containing the vertex; a central angle equals its arc measure in degrees. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.