One tangent, one chord, one arc
The half-arc relationship
Its key theorem says that the measure of the angle equals half the measure of the intercepted arc. That makes the angle a close relative of the inscribed angle, but the construction of the angle is different.
This page keeps the touch point and intercepted arc visible together so you can see why the tangent-chord angle must be read from the arc opposite the opening, not from the small corner alone.
Suppose the intercepted arc measures 120 degrees. The angle formed by the tangent and chord is 60 degrees because the theorem uses half the arc. The tangent's direction may look different when the circle is moved, but the relationship stays tied to the touch point and the same intercepted arc.
- A tangent-chord angle is formed by a tangent and a chord through the point of tangency.
- The vertex of a tangent-chord angle is the point of tangency.
- Its measure is half the measure of its intercepted arc.
- One side is tangent to the circle and the other side is a chord through the tangency point.
A theorem built at the edge
Where tangent-chord angles help
- Use tangent-chord angles in circle theorems that mix tangency with intercepted-arc reasoning.
- Use them when solving for unknown angles formed on the edge of a circle by a tangent line.
- Use the concept in proof problems where a tangent introduces a half-arc measure relationship.
Keep the touch point fixed
Tangent-chord errors to catch
- Do not use the wrong arc; the angle depends on the intercepted arc opened by the tangent and chord together.
- Do not treat the figure as an inscribed angle if one side is clearly tangent rather than a chord.
- Do not move the vertex away from the point of tangency and keep the same name.
