Angle Formed by Tangents and Secants

How to find the angle formed by tangents and secants of a circle: 3 formulas, 3 examples, and their solutions.

Secant

Definition

A secant is a line
that passes through a circle
at two points.

Formula

Angle formed by Two Secants

θ = [1/2](m[arc BD] - m[arc AC])

θ: Angle formed by two intersecting secants
arc BD: The outer arc
arc AC: The inner arc

Angle formed by a Tangent and a Secant

θ = [1/2](m[arc BC] - m[arc AC])

θ: Angle formed by a tangent and a secant
arc BC: The outer arc
arc AC: The inner arc

Angle formed by Two Tangents

θ = [1/2](m[arc ABC] - m[arc AC])

θ: Angle formed by two intersecting tangents
arc ABC: The outer arc
arc AC: The inner arc

Example 1

Example

Solution

xº: is the angle
formed by two intersecting secants.
The outer arc is 108º.
The inner arc is 44º.

Then x = [1/2](108 - 44).

108 - 44 = 64

[1/2]⋅64 = 32

So x = 32.

Example 2

Example

Solution

xº: is the angle
formed by a tangent and a secant.
The outer arc is 143º.
The inner arc is 63º.

Then x = [1/2](143 - 63).

143 - 63 = 80

[1/2]⋅80 = 40

So x = 40.

Example 3

Example

Solution

See this circle.

Set the outer arc α.
The inner arc is 130º
These two arcs form a circle.

So [α] + [130] = 360.

Move +130 to the right side.

Then α = 230.

Write 230º
next to the outer arc.

xº: is the angle
formed by two intersecting tangents.
The outer arc is 230º.
The inner arc is 130º.

Then x = [1/2](230 - 130).

230 - 130 = 100

[1/2]⋅100 = 50

So x = 50.