Apply the Inscribed Angle Theorem and its corollaries, and the tangent-chord angle relationship.
An inscribed angle has its vertex ON the circle, with both sides as chords. The Inscribed Angle Theorem is the key relationship: an inscribed angle measures HALF of its intercepted arc -- different from a central angle, which equals its arc exactly. Two corollaries follow immediately: inscribed angles that intercept the SAME arc are always congruent to each other, and an angle inscribed in a semicircle is always a right angle (half of a 180-degree arc is 90 degrees).
A cyclic quadrilateral (one whose four vertices all lie on a circle) has one more guaranteed property: each pair of opposite angles is supplementary, since the two opposite inscribed angles intercept arcs that together add up to the whole circle, and half of 360 is 180. One more angle relationship rounds out the set: a tangent-chord angle (formed where a tangent line meets a chord at the point of tangency) also measures half its intercepted arc, exactly like an inscribed angle, even though one side is a tangent instead of a chord.
Mastery looks like: They apply the Inscribed Angle Theorem and its corollaries (same-arc angles congruent, semicircle angles are right angles, cyclic quadrilateral opposite angles supplementary) to find unknown angle and arc measures.
Common stumbles: Forgetting to divide (or multiply) by 2 when converting between an inscribed angle and its intercepted arc; confusing an inscribed angle's rule with a central angle's rule.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.