6.16: Angles On and Inside a Circle

Angles whose vertices are on the circumference of a circle or formed by tangent lines and chords.

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.

Figure \(\PageIndex{1}\)

\(m\angle DBA=\dfrac{1}{2}m\widehat{AB}\)

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.

Figure \(\PageIndex{2}\)

\(\begin{aligned}m\angle SVR&=\dfrac{1}{2}(m\widehat{SR} +m\widehat{TQ} )=m\widehat{SR} +m\widehat{TQ} 2=m\angle TVQ \\ m\angle SVT&=\dfrac{1}{2}(m\widehat{ST} +m\widehat{RQ} )=m\widehat{ST} +m\widehat{RQ} 2=m\angle RVQ\end{aligned}\)

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?

Example \(\PageIndex{1}\)

Find \(x\).

Figure \(\PageIndex{3}\)

Solution

Use the Intersecting Chords Angle Theorem to write an equation.

\(x=\dfrac{129^{\circ}+71^{\circ}}{2}=\dfrac{200^{\circ}}{2}=100^{\circ}\)

Example \(\PageIndex{2}\)

Find \(x\).

Figure \(\PageIndex{4}\)

Solution

Use the Intersecting Chords Angle Theorem to write an equation.

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

\(y=\dfrac{19^{\circ}+107^{\circ}}{2}=\dfrac{126^{\circ}}{2}=63^{\circ} \qquad x+63^{\circ}=180^{\circ}; \: x=117^{\circ}\)

Example \(\PageIndex{3}\)

Find \(m\angle BAD\).

Figure \(\PageIndex{5}\)

Solution

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

Example \(\PageIndex{4}\)

Find \(a\), \(b\), and \(c\).

Figure \(\PageIndex{6}\)

Solution

\(\begin{array}{c}
50^{\circ}+45^{\circ}+m \angle a=180^{\circ} \quad \text { straight angle } \\
m \angle a=85^{\circ} \\
m \angle b=\frac{1}{2} \cdot m \widehat{A C} \\
\qquad \begin{array}{c}
\widehat{A C}=2 \cdot m \angle E A C=2 \cdot 45^{\circ}=90^{\circ} \\
m \angle b=\frac{1}{2} \cdot 90^{\circ}=45^{\circ}
\end{array} \\
85^{\circ}+45^{\circ}+m \angle c=180^{\circ} \quad \text { Triangle Sum Theorem } \\
m \angle c=50^{\circ}
\end{array}\)

Example \(\PageIndex{5}\)

Find \(m\widehat{AEB}\).

Figure \(\PageIndex{7}\)

Solution

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

Review

Find the value of the missing variable(s).

1. 

   Figure \(\PageIndex{8}\)

2. 

   Figure \(\PageIndex{9}\)

3. 

   Figure \(\PageIndex{10}\)

4. 

   Figure \(\PageIndex{11}\)

5. 

   Figure \(\PageIndex{112}\)

6. 

   Figure \(\PageIndex{13}\)

7. \(y\neq 60^{\circ}\)

   

   Figure \(\PageIndex{14}\)

Solve for \(x\).

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

   

   Figure \(\PageIndex{15}\)

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

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

Statement	Reason
1. Intersecting chords \(\overline{AC}\) and \(\overline{BD}\).	1.
2. Draw \(\overline{BC}\) Figure \(\PageIndex{16}\)	2. Construction
3. \(m\angle DBC=\dfrac{1}{2}m\widehat{DC}\) \(m\angle ACB=\dfrac{1}{2}m\widehat{AB}\)	3.
4. \(m\angle a=m\angle DBC+m\angle ACB\)	4.
5. \(m\angle a=\dfrac{1}{2}m\widehat{DC}+\dfrac{1}{2}m\widehat{AB}\)	5.

Fill in the blanks.

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

Vocabulary

Term	Definition
central angle	An angle formed by two radii and whose vertex is at the center of the circle.
chord	A line segment whose endpoints are on a circle.
circle	The set of all points that are the same distance away from a specific point, called the center.
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	An angle with its vertex on the circle and whose sides are chords.
intercepted arc	The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency	The point where the tangent line touches the circle.
radius	The distance from the center to the outer rim of a circle.
Chord/Tangent Angle Theorem	The Chord/Tangent Angle Theorem states that 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.
Intersecting Chords Angle Theorem	The Intersecting Chords Angle Theorem states that the measure of the angle formed by two chords that intersect inside a circle is the average of the measures of the intercepted arcs.

Additional Resources

Interactive Element

Video: Angles On and Inside a Circle Principles - Basic

Activities: Angles On and Inside a Circle Discussion Questions

Study Aids: Arcs and Angles Study Guide

Practice: Angles On and Inside a Circle