the tangent-chord angle

Draw a tangent grazing a circle at a point T, and from that same point draw a chord into the circle. The wedge between the tangent line and the chord is the tangent-chord angle. It behaves like an inscribed angle whose vertex has been slid until one of its arms lies flat along the circle — and it obeys a matching half-the-arc rule.

Precisely, the tangent-chord angle is the angle formed at the point of tangency T between a tangent line and a chord TA drawn from T. Its measure equals half the intercepted arc — specifically half of the arc TA that lies 'inside' the angle (cut off by the chord on the side the angle opens toward). You can see why by thinking of the tangent as the limiting position of a secant whose far intersection point has slid all the way to T; the inscribed angle theorem (inscribed angle = half its arc) then passes to the limit. As a special check: if the chord is a diameter, the tangent-chord angle is 90 degrees (half of the 180-degree semicircular arc), recovering tangent-radius perpendicularity.

This angle is the bridge between tangents and inscribed angles, and it powers many angle-chases and the 'alternate segment theorem' (the tangent-chord angle equals the inscribed angle in the alternate segment, since both are half the same arc). The caveat: 'half the arc' refers to the arc cut off by the chord on the side toward which the angle opens — a single chord at T makes two tangent-chord angles (one on each side) intercepting the two complementary arcs, so they are halves of arcs that sum to 360 degrees and therefore are themselves supplementary. Identify which arc the angle 'faces' before halving.