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Angles Outside a Circle

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What if you were given a circle with either two secents, two tangents, or one of each that share a common point outside the circle? How could you use the measure of the arcs formed by those circle parts to find the measure of the angle they make outside the circle? After completing this Concept, you'll be able to apply the Outside Angle Theorem to solve problems like this one.

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Angles Outside a Circle CK-12

Guidance

An angle is outside a circle if its vertex is outside the circle and its sides are tangents or secants. The possibilities are: an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants.

Outside Angle Theorem: The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.

m\angle D = \frac{m\widehat{EF}-m\widehat{GH}}{2} , m\angle L =\frac{m\widehat{MPN}-m\widehat{MN}}{2} , m\angle Q =\frac{m\widehat{RS}-m\widehat{RT}}{2}

Example A

Find the value of x .

x=\frac{72^\circ - 22^\circ}{2}=\frac{50^\circ}{2}=25^\circ .

Example B

Find the value of x .

x=\frac{120^\circ - 32^\circ}{2}=\frac{88^\circ}{2}=44^\circ .

Example C

Find the value of x .

First note that the missing arc by angle x measures 32^\circ because the complete circle must make 360^\circ . Then, x=\frac{141^\circ - 32^\circ}{2}=\frac{109^\circ}{2}=54.5^\circ .

Angles Outside a Circle CK-12

Guided Practice

Find the measure of x .

1.

2.

3.

Answers:

For all of the above problems we can use the Outside Angle Theorem.

1. x=\frac{125^\circ-27^\circ}{2}=\frac{98^\circ}{2}=49^\circ

2. 40^\circ is not the intercepted arc. The intercepted arc is 120^\circ, \ (360^\circ-200^\circ-40^\circ) . x=\frac{200^\circ-120^\circ}{2}=\frac{80^\circ}{2}=40^\circ

3. Find the other intercepted arc, 360^\circ-265^\circ=95^\circ x = \frac{265^\circ-95^\circ}{2}=\frac{170^\circ}{2}=85^\circ

Practice

Find the value of the missing variable(s).

Solve for x .

  1. Fill in the blanks of the proof for the Outside Angle Theorem

Given : Secant rays \overrightarrow{AB} and \overrightarrow{AC}

Prove : m\angle a = \frac{1}{2} \left (m\widehat{BC}-m\widehat{DE} \right )

Statement Reason
1. Intersecting secants \overrightarrow{AB} and \overrightarrow{AC} . 1.

2. Draw \overline{BE} .

2. Construction
3. m\angle BEC &= \frac{1}{2} m\widehat{BC}\\m\angle DBE &= \frac{1}{2} m\widehat{DE} 3.
4. m\angle a+m\angle DBE=m\angle BEC 4.
5. 5. Subtraction PoE
6. 6. Substitution
7. m\angle a=\frac{1}{2} \left (m\widehat{BC}-m\widehat{DE} \right ) 7.

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