Inscribed Angles

An inscribed angle has its vertex *on* the circle rather than at the centre. Its measure is half the arc it intercepts.

So a 100° arc gives a 50° inscribed angle. Compare that to a central angle, which gets the full 100°.

Two consequences do a lot of work:

Any inscribed angle in a semicircle is a right angle. If a triangle's longest side is a diameter, the angle opposite it is 90°.

Inscribed angles intercepting the *same* arc are congruent, no matter where their vertices sit on the circle.

And for a quadrilateral inscribed in a circle, opposite angles are supplementary.

The habit that prevents errors: check where the vertex is. On the circle means halve the arc. At the centre means do not.

The Inscribed Angle Theorem states that an inscribed angle measures half its intercepted arc. The proof splits into three cases by where the centre lies relative to the angle, with the case where one side is a diameter proved via the isosceles triangle formed by two radii and the exterior angle theorem; the other cases follow by adding or subtracting.

Corollaries: an angle inscribed in a semicircle is right (Thales' theorem); inscribed angles intercepting the same arc are congruent; and opposite angles of a cyclic quadrilateral are supplementary, since their intercepted arcs together make the full 360°.

The tangent-chord angle, formed by a chord and a tangent at its endpoint, is also half its intercepted arc — the limiting case of an inscribed angle as one endpoint approaches the vertex.

These unify with the vertex-position rule that governs all circle angles: a vertex at the centre gives the full arc, on the circle gives half the arc, inside gives half the *sum* of the two intercepted arcs, and outside gives half their *difference*.