<meta http-equiv="refresh" content="1; url=/nojavascript/"> Angles On and Inside a Circle ( Read ) | Geometry | CK-12 Foundation
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Angles On and Inside a Circle

%
Progress
Practice Angles On and Inside a Circle
Practice
Progress
%
Practice Now
Angles On and Inside a Circle

What if you were given a circle with either a chord and a tangent or two chords that meet at a common point? How could you use the measure of the arc(s) formed by those circle parts to find the measure of the angles they make on or inside the circle? After completing this Concept, you'll be able to apply the Chord/Tangent Angle Theorem and the Intersecting Chords Angle Theorem to solve problems like this one.

Watch This

Angles On and Inside a Circle CK-12

Guidance

When we say an angle is on a circle, we mean the vertex is on the edge of the circle. One type of angle on a circle is the inscribed angle (see Inscribed Angles in Circles ). Another type of angle on a circle is one formed by a tangent and a chord.

Chord/Tangent Angle Theorem: The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.

m\angle DBA=\frac{1}{2} m\widehat{AB}

If two angles, with their vertices on the circle, intercept the same arc then the angles are congruent.

An angle is inside a circle when the vertex lies anywhere inside the circle.

Intersecting Chords Angle Theorem: The measure of the angle formed by two chords that intersect inside a circle is the average of the measures of the intercepted arcs.

m\angle SVR & = \frac{1}{2} \left ( m\widehat{SR} + m\widehat{TQ} \right )= \frac{m\widehat{SR}+m\widehat{TQ}}{2}=m\angle TVQ\\m\angle SVT& = \frac{1}{2} \left (m\widehat{ST} + m\widehat{RQ} \right )=\frac{m\widehat{ST}+m\widehat{RQ}}{2} = m\angle RVQ

Example A

Find m\angle BAD .

Use the Chord/Tangent Angle Theorem. m\angle BAD = \frac{1}{2} m \widehat{AB} = \frac{1}{2} \cdot 124^\circ=62^\circ .

Example B

Find a, \ b, and c .

50^\circ + 45^\circ + m\angle a & = 180^\circ \qquad \text{straight angle}\\m\angle a & = 85^\circ

m \angle b & = \frac{1}{2} \cdot m \widehat{AC}\\m \widehat{AC} & = 2 \cdot m\angle EAC = 2 \cdot 45^\circ=90^\circ\\m \angle b & = \frac{1}{2} \cdot 90^\circ=45^\circ

85^\circ + 45^\circ + m\angle c & = 180^\circ \qquad \text{Triangle Sum Theorem}\\m\angle c & = 50^\circ

Example C

Find m \widehat{AEB} .

Use the Chord/Tangent Angle Theorem. m\widehat{AEB} = 2 \cdot m\angle DAB= 2 \cdot 133^\circ=266^\circ .

Angles On and Inside a Circle CK-12

Guided Practice

Find x .

1.

2.

3.

Answers:

Use the Intersecting Chords Angle Theorem to write an equation.

1. x=\frac{129^\circ+71^\circ}{2}=\frac{200^\circ}{2}=100^\circ

2. 40^\circ &= \frac{52^\circ+x}{2}\\80^\circ &= 52^\circ+x\\28^\circ &= x

3. x is supplementary to the angle that is the average of the given intercepted arcs. We call this supplementary angle y .

y=\frac{19^\circ+107^\circ}{2}=\frac{126^\circ}{2}=63^\circ \qquad  x+63^\circ=180^\circ; \ x=117^\circ

Practice

Find the value of the missing variable(s).

  1. y \ne 60^\circ

Solve for x .

  1. Fill in the blanks of the proof for the Intersecting Chords Angle Theorem

Given : Intersecting chords \overline{AC} and \overline{BD} .

Prove : m\angle a=\frac{1}{2} \left (m\widehat{DC}+m\widehat{AB}\right )

Statement Reason
1. Intersecting chords \overline{AC} and \overline{BD} . 1.

2. Draw \overline{BC}

2. Construction
3. m\angle DBC &= \frac{1}{2} m\widehat{DC}\\m\angle ACB &= \frac{1}{2} m\widehat{AB} 3.
4. m\angle a=m\angle DBC+m\angle ACB 4.
5. m\angle a=\frac{1}{2} m\widehat{DC}+\frac{1}{2} m\widehat{AB} 5.

Fill in the blanks.

  1. If the vertex of an angle is _______________ a circle, then its measure is the average of the __________________ arcs.
  2. If the vertex of an angle is ________ a circle, then its measure is ______________ the intercepted arc.

Vocabulary

central angle

central angle

An angle formed by two radii and whose vertex is at the center of the circle.
chord

chord

A line segment whose endpoints are on a circle.
circle

circle

The set of all points that are the same distance away from a specific point, called the center.
diameter

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
inscribed angle

inscribed angle

An angle with its vertex on the circle and whose sides are chords.
intercepted arc

intercepted arc

The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency

point of tangency

The point where the tangent line touches the circle.
radius

radius

The distance from the center to the outer rim of a circle.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Angles On and Inside a Circle.

Reviews

Please wait...
Please wait...

Original text