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6.17: Angles Outside a Circle

  • Page ID
    5042
  • Angles formed by tangents and/or secants.

    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.

    f-d_dc68b26d59aed9d2b06b2108488cb8a0470a72367e8d9437bae3503d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    \(m\angle D=\dfrac{m\widehat{EF}−m\widehat{GH}}{2}\), \(m\angle L=\dfrac{m\widehat{MPN}−m\widehat{MN}}{2}\), \(m\angle Q=\dfrac{m\widehat{RS}−m\widehat{RT}}{2}\)

    What if you were given a circle with either two secants, 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?

    Example \(\PageIndex{1}\)

    Find the measure of \(x\).

    f-d_ef74ef004d0e57715b1ea59776dd836f3e4a72a35270a83031ada232+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Solution

    Use the Outside Angle Theorem.

    \(x=\dfrac{125^{\circ} −27^{\circ}}{2}=\dfrac{98^{\circ}}{2}=49^{\circ}\)

    Example \(\PageIndex{2}\)

    Find the measure of \(x\).

    f-d_d910ae948fa7ac80862d5c4a7bae35204e456457871f68052b5aa41b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Solution

    Use the Outside Angle Theorem.

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

    Example \(\PageIndex{3}\)

    Find the value of \(x\).

    f-d_1863c92a65103991377f12167c3e08119787ce710c9ba77dbe368d0a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    \(x=\dfrac{72^{\circ} −22^{\circ} }{2}=\dfrac{50^{\circ} }{2}=25^{\circ}\).

    Example \(\PageIndex{4}\)

    Find the value of \(x\).

    f-d_b8fd7cf4cc1483e43b1c055717b8d1cbe01fa58aab50c3f3eacee2e0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    \(x=\dfrac{120^{\circ} −32^{\circ}}{2}=\dfrac{88^{\circ} }{2}=44^{\circ}\).

    Example \(\PageIndex{5}\)

    Find the value of \(x\).

    f-d_c0ed93b22a2b3da1049289b3e010f3e72a124b8903231d529c84f5e0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

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

    Review

    Find the value of the missing variable(s).

    1. f-d_7e990f528b7d69cca93a37cd5cd03e5d730c92497595056b5986a280+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. f-d_dac0029f39840159996c1fe303cef130ee5ded06e21e06ab71054a02+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    3. f-d_9980b767c51b3966c1ea65c48338d5eea8fa885c5311b21eb3c29b96+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    4. f-d_699d5ec6e99e739f29da83604b80ec24bdb5d5cf20760b951171ed73+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    5. f-d_0edce2d6716fbb2746e974e455ec99df150420c0ba718db1d886ff20+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    6. f-d_23fe4752b3c7daa2b66538d2b9242881331e158ef1784c0407506b0a+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    7. f-d_960d38d70f6058eccfb689b690c5bfa3fc98f19b35cf04d876fc72fe+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)

    Solve for \(x\).

    1. f-d_788430d8f33f6989a1d2bd08da608937f9d83b29eaea3feba92c7413+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    2. f-d_60493252fb935ce8f24206add0e58ed326f0bf710e0bbf86bbf579b4+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    3. Fill in the blanks of the proof for the Outside Angle Theorem
    f-d_4f27a4fc6bc357f53bd119cd62bc2346de5c968f044ab199d05d2c88+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{16}\)

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

    Prove: \(m\angle a=12(m\widehat{BC}−m\widehat{DE})\)

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

    2. Draw \(\overline{BE}\).

    f-d_ede37a85abab749cc71c2a065bc3ffa5989d3f09c3c82ebfd5294380+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{17}\)
    2. Construction

    3. \(m\angle BEC=\dfrac{1}{2}m\widehat{BC}\)

    \(m\angle DBE=\dfrac{1}{2}m\widehat{DE}\)

    3.
    4. \(m\angle a+m\angle DBE=m\angle BEC\) 4.
    5. 5. Subtraction PoE
    6. 6. Substitution
    7. \(m\angle a=\dfrac{1}{2}(m\widehat{BC}−m\widehat{DE})\) 7.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 9.8.

    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.
    Secant The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
    Tangent The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.
    Outside Angles Theorem The Outside Angles Theorem states that 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.

    Additional Resources

    Interactive Element

    Video: Angles Outside a Circle Principles - Basic

    Activities: Angles Outside a Circle Discussion Questions

    Study Aids: Arcs and Angles Study Guide

    Practice: Angles Outside a Circle