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6.15: Inscribed Quadrilaterals in Circles

  • Page ID
    5040
  • Quadrilaterals with every vertex on a circle and opposite angles that are supplementary.

    An inscribed polygon is a polygon where every vertex is on the circle, as shown below.

    f-d_3b423cece4d07f83fc74f35d7ecac4b7b0f212fe4f6e6236319fde1d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    For inscribed quadrilaterals in particular, the opposite angles will always be supplementary.

    Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary.

    f-d_234aed0c25ab3088ee1539f4f8cbda8d779cd55821268cf106c165f1+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    If \(ABCD\) is inscribed in \(\bigodot E\), then \(m\angle A+m\angle C=180^{\circ}\) and \(m\angle B+m\angle D=180^{\circ}\). Conversely, If \(m\angle A+m\angle C=180^{\circ}\) and \(m\angle B+m\angle D=180^{\circ}\), then \(ABCD\) is inscribed in \(\bigodot E\).

    What if you were given a circle with a quadrilateral inscribed in it? How could you use information about the arcs formed by the quadrilateral and/or the quadrilateral's angle measures to find the measure of the unknown quadrilateral angles?

    Example \(\PageIndex{1}\)

    1. f-d_f24b37126d5836fd55db753eeb521107c3754f643667625dacaf1479+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{3}\)
    2. f-d_4e9fa23995200ff2bce5845a55ac0b9a3feedb20776807da7788843c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{4}\)

    Solution

    1. \(\begin{aligned}
      x+80^{\circ}&=180^{\circ} \qquad& y+71^{\circ}&=180^{\circ} \\
      x&=100^{\circ} & y&=109^{\circ}
      \end{aligned}\)
    1. \(\begin{aligned}
      z+93^{\circ} &=180^{\circ} & x&=\frac{1}{2}\left(58^{\circ}+106^{\circ}\right) & y+82^{\circ}&=180^{\circ} \\
      z &=87^{\circ} & x &=82^{\circ} & y&=98^{\circ}
      \end{aligned}\)

    Example \(\PageIndex{2}\)

    Find \(x\) and \(y\) in the picture below.

    f-d_64f8d45d7bded8e76a6d0c885eca321e6148062fea389c6837ff75dd+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    \(\begin{array}{rlrl}
    (7 x+1)^{\circ}+105^{\circ} & =180^{\circ} & (4 y+14)^{\circ}+(7 y+1)^{\circ} & =180^{\circ} \\
    7 x+106^{\circ} & =180^{\circ} & 11 y+15^{\circ} & =180^{\circ} \\
    7 x & =74 & 11 y & =165 \\
    x & =10.57 & y&=15
    \end{array}\)

    Example \(\PageIndex{3}\)

    Find the values of x and y in \(\bigodot A\).

    f-d_f2ea4fb83ec882d7580bd9e11efb85d1defafd3b1188c5c7fa7f5ec3+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    Use the Inscribed Quadrilateral Theorem. \(x^{\circ}+108^{\circ}=180^{\circ}\) so \(x=72^{\circ}\). Similarly, \(y^{\circ}+88^{\circ}=180^{\circ}\) so \(y=92^{\circ}\).

    Example \(\PageIndex{4}\)

    Quadrilateral \(ABCD\) is inscribed in \(\bigodot E\). Find \(m\angle A\), \(m\angle B\), \(m\angle C\), and \(m\angle D\).


    f-d_28916d1f401e34988707414adb44c915487a4f08050767bfff944c90+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    First, note that \(m\widehat{AD}=105^{\circ}\) because the complete circle must add up to \(360^{\circ}\).

    \(\begin{aligned}m\angle A&=\dfrac{1}{2}m\widehat{BD}=12(115+86)=100.5^{\circ} \\ m\angle B&=\dfrac{1}{2}m\widehat{AC}=12(86+105)=95.5^{\circ} \\ m\angle C&=180^{\circ}−m\angle A=180^{\circ}−100.5^{\circ}=79.5^{\circ} \\ m\angle D&=180^{\circ}−m\angle B=180^{\circ}−95.5^{\circ}=84.5^{\circ}\end{aligned}\)

    Review

    Fill in the blanks.

    1. A(n) _______________ polygon has all its vertices on a circle.
    2. The _____________ angles of an inscribed quadrilateral are ________________.

    Quadrilateral \(ABCD\) is inscribed in \(\bigodot E\). Find:

    f-d_3bb0b500960d9fbd9502dbf49ab2b3b335d590165bbcc4228e01f369+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)
    1. \(m\angle DBC\)
    2. \(m\widehat{BC}\)
    3. \(m\widehat{AB}\)
    4. \(m\angle ACD\)
    5. \(m\angle ADC\)
    6. \(m\angle ACB\)

    Find the value of \(x\) and/or \(y\) in \(\bigodot A\).

    1. f-d_e5c6113c5f81b51a1b335af83993babf54f2020eeaedd687d985c21a+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    2. f-d_6aeedd2059d1d9ebfc81e7df42693c5f93332a5b3dc846355d840c85+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    3. f-d_0652ac7ccf2739e340734a85a42de79c2ce279e0970e3f4844de4115+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)

    Solve for \(x\).

    1. f-d_4c1df032bf17250331001e38694b77100784e74570d46982b9061952+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    2. f-d_2365c18f60fd2ba0361e3eef433fbd5a400423169540c3c1d6abc3e0+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)

    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.
    radius The distance from the center to the outer rim of a circle.
    Inscribed Polygon An inscribed polygon is a polygon with every vertex on a given circle.
    Inscribed Quadrilateral Theorem The Inscribed Quadrilateral Theorem states that a quadrilateral can be inscribed in a circle if and only if the opposite angles of the quadrilateral are supplementary.
    Cyclic Quadrilaterals A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.

    Additional Resources

    Interactive Element

    Video: Inscribed Quadrilaterals in Circles Principles - Basic

    Activities: Inscribed Quadrilaterals in Circles Discussion Questions

    Study Aids: Inscribed in Circles Study Guide

    Practice: Inscribed Quadrilaterals in Circles

    Real World: Sunrise at Stonehenge