Angle Formed by Two Intersecting Tangents

Angle Formed by Two Intersecting Tangents

How to solve problems about the angle formed by two intersecting tangents: formula, proof, example, and its solution.

Formula

m(angle formed by two intersecting tangents) = (1/2)*(m[intercepted arc 1] - m[intercepted arc 2])

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

∠(red): angle formed by two intersecting tangents
purple & blue arcs: intercepted arcs

Proof

Angle Formed by Two Intersecting Tangents: Proof of the Formula

Draw a chord
whose endpoints are the endpoints of the arcs.

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

Angle formed by a tangent and a chord

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 two intersecting tangents: x. The measures of the interior intercepted arc: 130.

The blue arc and the purple arc form a circle.
So m[blue arc] + m[purple arc] = 360.

130 + m[purple arc] = 360

m[purple arc] = 230

m∠(red) = x, m[purple arc] = 230, m[blue arc] = 130

x = (1/2)⋅(230 - 130)