Angles Formed by Chords Inside a Circle

Two chords crossing inside a circle make an X shape, and that X creates two pairs of vertical angles. Each angle you see at the crossing point has an arc sitting directly across from it, on the far side of the circle, plus a second arc intercepted by the opposite rays of the same two chords. Because the crossing point sits inside the circle rather than on its edge, the angle is pulled toward both arcs at once. It cannot depend on just one of them, since the chords lean toward both ends of the circle simultaneously.

The rule: the angle equals half the sum of the two intercepted arcs. Each chord alone would give an inscribed-angle relationship worth half its own arc, but here two chords meet away from the circle's edge, so their combined pull averages the two arcs together. Picture the two arcs as two separate pulls on the crossing point, one from each chord pair, and the angle splits the difference between them equally.

This sits next to the inscribed angle rule, where a single angle on the circle is half of one arc, and the exterior angle rule, where two secants meeting outside the circle give half the difference of their arcs. The interior case is the middle ground: instead of one arc or a difference of arcs, you get a sum. Watching where the vertex sits, on the circle, inside it, or outside it, tells you instantly whether to take half of one arc, half a sum, or half a difference.