Angle from Secants: Vertex Outside Circle

There are some useful formulas to calculate angles in circles, formed by radii, chords, secants, and tangents.

1. Central Angle ========

A central angle is formed by two intersecting radii in a circle. Its vertex must be at the center of the circle.

In this diagram, x is the central angle. In a circle, or congruent circles, congruent central angles have congruent arcs.

Central Angle = Intercepted Arc

Thus,

Angle Measure of Arc AB = Central Angle x (Angle AOB)

1. Inscribed Angle =====

An inscribed angle is an angle with its vertex on the circumference of the circle. It is formed by two intersecting chords.

Inscribed Angle = 1/2 Intercepted Arc

Thus,

1/2*Angle Measure of Arc AC = Angle ABC

1. Tangent Chord Angle =====

An angle formed by an intersecting tangent and chord; the angle has its vertex on the circumference of the circle.

Tangent Chord Angle = 1/2 Intercepted Arc

Angle ABC= 1/2 Measure of Arc AB

1. Angle Formed Inside of a Circle By Two Intersecting Chords ======

When two chords intersect inside a circle, they created two sets of vertical angles (vertical angles are always equal). In the diagram, angle x is on of the vertical angles.

Angle Formed Inside Circle by Intersecting Chords = 1/2 Sum of Intercepted Arcs

Measure of Angle BED = 1/2 (Angle Measure of Arc AC + Angle Measure of Arc BD)

1. Angle Formed By the Intersection of Two Tangents, Two Secants, or a Tangent and a Secant: ========

The formula for ALL THREE of these cases are the same:

Angle Formed Outside Circle = |1/2 Difference of Intercepted Arcs|

a) Two Tangents

Angle ABC is formed by two tangents intersecting outside of the circle at point B. The intercepted arcs are minor arc AC and major arc AC.

b) Two Secants

Angle ACE is formed by two secants intersecting at point C. The intercepted arcs in this diagram are the minor arcs BD and AE.

c) A Secant and a Tangent

Angle ABD is formed by the tangent and secant intersecting at point B. The intercepted arcs in the diagram are the minor arcs AC and AD.

References: "Formulas for Working with Angles in Circles." Geometry Lesson Page. Regentsprep.org, 1988. Web. 6 Sept. 2014.

Blurb/stub article

In the following diagram, we want a simple expression for $\angle{AXB} = \angle{CXD}$.

Diagram (secants intersect outside circle)

The picture below consists of two secants: $AB$ and $BD$. They intersect outside of the circle, producing $\angle{ABC}$.

By this we know that the #degrees in arc $AD$ - the #degrees in arc $EC$ divided by two is the #degrees in $\angle{EBC}$.

For instance, if arc $AD$ measured $140$ degrees and arc $EC$ measured $30$ degrees, the measure of $\angle{ABC}$ would be $(140 - 30)/2$, which would equal $55$ degrees.