# Angle Formed by a Tangent and a Chord

How to solve problems about the angle formed by a tangent and a chord: formula, proof, examples, and their solutions.

## Formula

m∠(red) = (1/2)⋅m[blue arc]

∠(red): angle formed by a tangent and a chord

blue arc: intercepted arc

## Proof

Draw a diameter

that starts from the tangent point.

Then the upper red angle

is the inscribed angle of the blue arc.

So m∠(red) = (1/2)⋅m[blue arc].

See the inscribed triangle.

The side of the inscribed triangle

passes through the center of the circle.

So the interior purple angle is a right angle.

And the triangle is a right triangle.

Inscribed right triangle

The right triangle is a 'triangle'.

So m∠(purple) + m∠(green) + m∠(red) = 180.

Interior angles of a triangle

The tangent and the diameter are perpendicular.

So the left side angle of the tangent point

is the purple angle (the right angle).

Tangent to a circle

Since m∠(purple) + m∠(green) + m∠(red) = 180,

the right side angle of the tangent point

is the red angle.

So m∠(right side red angle) = (1/2)⋅m[blue arc]

## Example 1

m∠(red) = *x*, m[blue arc] = 110*x* = (1/2)⋅110

## Example 2

m∠(red) = *x*, m[blue arc]*x* = (1/2)⋅m[blue arc]

80 = (1/2)⋅m[blue arc]

Inscribed angle

Both red angles have the same intercepted arc.

(blue arc)

So *x* = 80.