Two Chords Meeting Inside a Circle

When two chords cross inside a circle, the angle they make is set entirely by the two arcs they cut off — it is exactly half their sum.

Theorem Two chords meeting inside

If two chords intersect inside a circle, the angle formed equals half the sum of the two arcs it intercepts.

The angle at P intercepts arcs AC and BD, so:

\angle P = \frac{\overset{\frown}{AC} + \overset{\frown}{BD}}{2}

Note Which arcs, and vertical angles

Use the two opposite arcs — the ones the angle and its vertical partner open into — not the arcs beside them.

The two vertical angles at the crossing point are always equal, so they share the same pair of arcs and the same measure.

Example Three cases

Finding the angle

Arc AC = 80° and arc BD = 120°.

\angle P = \dfrac{80^\circ + 120^\circ}{2} = \dfrac{200^\circ}{2}

⟹ ∠P = 100°.

Working backwards

The angle is 65° and arc AC = 50°.

65^\circ \times 2 = 130^\circ

, so arc BD

= 130^\circ - 50^\circ

⟹ arc BD = 80°.

A right angle

Arc PR = 70° and arc QS = 110°, so the arcs sum to 180°.

\angle = \dfrac{70^\circ + 110^\circ}{2} = \dfrac{180^\circ}{2}

⟹ the angle is 90°.

Reference Inside vs outside

Summary

Two chords crossing inside a circle make an angle equal to half the sum of the two opposite arcs.
Inside the circle you add the arcs; outside the circle you subtract them.
The vertical angles at the crossing point are always equal.
Special case: if the two arcs add up to 180°, the angle is a right angle.