Angle Formed by a Tangent and a Secant

Angle Formed by a Tangent and a Secant

How to solve problems about the angle formed by a tangent and a secant: formula, proof, example, and its solution.

Formula

m(angle formed by a tangent and a secant) = (1/2)*(m[intercepted arc 1] - m[intercepted arc 2])

m∠(red) = (1/2)⋅(m[purple arc] - m[blue arc])

∠(red): angle formed by a tangent and a secant
purple & blue arcs: intercepted arcs

Proof

Angle Formed by a Tangent and a Secant: Proof of the Formula

Draw a chord
whose endpoints are the endpoints of the purple arc.

m∠(purple) = (1/2)⋅m[purple arc]

Angle formed by a tangent and a chord

m∠(blue) = (1/2)⋅m[blue arc]

Inscribed angle

See the formed triangle.

The purple angle is the exterior angle of the triangle.

So m∠(red) + (1/2)⋅m[blue arc] = (1/2)⋅m[purple arc].

Move (1/2)⋅m[blue arc] to the right side.

Then m∠(red) = (1/2)⋅(m[purple arc] - m[blue arc]).

Example

Find the value of x. The measure of the angle formed by a tangent and a secant: x. The measures of the intercepted arcs: 63, 143.

m∠(red) = x, m[purple arc] = 143, m[blue arc] = 63

x = (1/2)⋅(143 - 63)