Angle Formed by Two Intersecting Chords

Angle Formed by Two Intersecting Chords

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

Formula

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

m∠(red) = (1/2)⋅(m[blue arc] + m[green arc])

∠(red): angle formed by two chords
blue & green arcs: intercepted arcs

Proof

Angle Formed by Two Intersecting Chords: Proof of the Formula

Draw a chord to make a triangle.
(gray trangle)

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

Inscribed angle

See the gray triangle.
The red angle is the exterior angle of the gray triangle.

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

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

Example

Find the value of x. The measure of the angle formed by two intersecting chords: x. The measures of the arcs: 32, 74.

m∠(red) = x, m[blue arc] = 32, m[green arc] = 74

x = (1/2)⋅(32 + 74)