Angle Formed by Two Intersecting Secants

Angle Formed by Two Intersecting Secants

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

Formula

m(angle formed by two intersecting secants) = (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 secants
purple & blue arcs: intercepted arcs

Proof

Angle Formed by Two Intersecting Secants: Proof of the Formula

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

m∠(purple) = (1/2)⋅m[purple arc]
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 two intersecting secants: x. The measures of the intercepted arcs: 44, 108.

m∠(red) = x, m[purple arc] = 108, m[blue arc] = 44

x = (1/2)⋅(108 - 44)