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Angles On and Inside a Circle

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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.

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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.

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