The Tangent–Chord Angle

A tangent–chord angle is formed at the point where a tangent meets a circle, between that tangent and a chord drawn from the same point. Its size is set entirely by the arc the chord cuts off.

Theorem Half the intercepted arc

A tangent–chord angle equals half the measure of the arc it intercepts — the arc lying inside the angle.

If the bold arc TA measures $x$ , then the angle α is:

\alpha = \frac{1}{2}\,\overset{\frown}{TA}

Note Two angles at the point

A tangent and a chord actually make two angles at the point of tangency — one on each side. Each equals half of the arc that lies inside it.

The two arcs are the minor and major arcs, so they add to 360°, and the two angles add to 180°.

Reference Circle angles compared

Angle	Vertex	Measure
Central	the centre	the arc
Inscribed	on the circle	half the arc
Tangent–chord	at the point of tangency	half the arc

Example Three cases

From the arc to the angle

The intercepted arc measures 100°.

\alpha = \tfrac{1}{2}\times 100^\circ

⟹ α = 50°.

From the angle to the arc

The tangent–chord angle is 35°.

Arc

= 2 \times 35^\circ

⟹ the intercepted arc is 70°.

When the chord is a diameter

A diameter cuts off a semicircle, an arc of 180°.

\alpha = \tfrac{1}{2}\times 180^\circ = 90^\circ

— matching the tangent ⊥ radius rule.

⟹ the angle is 90°.

Summary

A tangent–chord angle is formed by a tangent and a chord meeting at the point of tangency.
Its measure is half of the arc it intercepts: $\alpha = \tfrac{1}{2}\,\overset{\frown}{arc}$ .
This matches the inscribed angle rule — both are half their arc.
If the chord is a diameter, the angle is 90°, consistent with the tangent being perpendicular to the radius.