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Inscribed Quadrilaterals in Circles

Quadrilaterals with every vertex on a circle and opposite angles that are supplementary.

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Inscribed Quadrilaterals in Circles

Inscribed Quadrilaterals in Circles 

An inscribed polygon is a polygon where every vertex is on a circle. Note, that not every quadrilateral or polygon can be inscribed in a circle. Inscribed quadrilaterals are also called cyclic quadrilaterals. For these types of quadrilaterals, they must have one special property. We will investigate it here.

Investigation: Inscribing Quadrilaterals

Tools Needed: pencil, paper, compass, ruler, colored pencils, scissors

  1. Draw a circle. Mark the center point \begin{align*}A\end{align*}A.
  2. Place four points on the circle. Connect them to form a quadrilateral. Color the 4 angles of the quadrilateral 4 different colors.
  3. Cut out the quadrilateral. Then cut the quadrilateral into two triangles, by cutting on a diagonal.
  4. Line up \begin{align*}\angle B\end{align*}B and \begin{align*}\angle D\end{align*}D so that they are adjacent angles. What do you notice? What does this show?

This investigation shows that the opposite angles in an inscribed quadrilateral are supplementary. By cutting the quadrilateral in half, through the diagonal, we were able to show that the other two angles (that we did not cut through) formed a linear pair when matched up.

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

Solving for Unknown Values

1. Find the value of the missing variable.

\begin{align*}x+80^\circ=180^\circ\end{align*}x+80=180 by the Inscribed Quadrilateral Theorem. \begin{align*}x=100^\circ\end{align*}x=100.

\begin{align*}y+71^\circ=180^\circ\end{align*}y+71=180 by the Inscribed Quadrilateral Theorem. \begin{align*}y=109^\circ\end{align*}y=109.

2. Find the value of the missing variable.

It is easiest to figure out \begin{align*}z\end{align*}z first. It is supplementary with \begin{align*}93^\circ\end{align*}93, so \begin{align*}z=87^\circ\end{align*}z=87. Second, we can find \begin{align*}x\end{align*}x. \begin{align*}x\end{align*}x is an inscribed angle that intercepts the arc \begin{align*}58^\circ + 106^\circ = 164^\circ\end{align*}58+106=164. Therefore, by the Inscribed Angle Theorem, \begin{align*}x=82^\circ\end{align*}x=82. \begin{align*}y\end{align*}y is supplementary with \begin{align*}x\end{align*}x, so \begin{align*}y=98^\circ\end{align*}.Find the value of the missing variables.

3. Find \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the picture below.

The opposite angles are supplementary. Set up an equation for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}(7x+1)^\circ+105^\circ &= 180^\circ && (4y+14)^\circ+(7y+1)^\circ = 180^\circ\\ 7x+106^\circ &= 180^\circ && \qquad \qquad \quad \ 11y+15^\circ = 180^\circ\\ 7x &= 84^\circ && \qquad \qquad \qquad \quad \quad 11y = 165^\circ\\ x &= 12^\circ && \qquad \qquad \qquad \qquad \quad y = 15^\circ\end{align*}


Quadrilateral \begin{align*}ABCD\end{align*} is inscribed in \begin{align*}\bigodot E\end{align*}

First, note that \begin{align*}m\widehat{AD}=105^\circ\end{align*} because the complete circle must add up to \begin{align*}360^\circ\end{align*}

Example 1

Find \begin{align*}m\angle A\end{align*}.

\begin{align*}m\angle A=\frac{1}{2}m\widehat{BD}=\frac{1}{2}(115+86)=100.5^\circ\end{align*}

Example 2

Find \begin{align*}m\angle B\end{align*}.

\begin{align*}m\angle B=\frac{1}{2}m\widehat{AC}=\frac{1}{2}(86+105)=95.5^\circ\end{align*}

Example 3

Find \begin{align*}m\angle C\end{align*}.

\begin{align*}m\angle C=180^\circ-m\angle A=180^\circ-100.5^\circ=79.5^\circ\end{align*}

Example 4

Find \begin{align*}m\angle D\end{align*}.

\begin{align*}m\angle D=180^\circ-m\angle B=180^\circ-95.5^\circ=84.5^\circ\end{align*} 


Fill in the blanks.

  1. A\begin{align*}(n)\end{align*} _______________ polygon has all its vertices on a circle.
  2. The _____________ angles of an inscribed quadrilateral are ________________.

Quadrilateral \begin{align*}ABCD\end{align*} is inscribed in \begin{align*}\bigodot E\end{align*}. Find:

  1. \begin{align*}m\angle DBC\end{align*}
  2. \begin{align*}m \widehat{BC}\end{align*}
  3. \begin{align*}m \widehat{AB}\end{align*}
  4. \begin{align*}m\angle ACD\end{align*}
  5. \begin{align*}m\angle ADC\end{align*}
  6. \begin{align*}m\angle ACB\end{align*}

Find the value of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*} in \begin{align*}\bigodot A\end{align*}.

Solve for \begin{align*}x\end{align*}.

Use the diagram below to find the measures of the indicated angles and arcs in problems 14-19.

  1. \begin{align*}m \angle EBO\end{align*}
  2. \begin{align*}m \angle EOB\end{align*}
  3. \begin{align*}m \widehat{BC}\end{align*}
  4. \begin{align*}m \angle ABO\end{align*}
  5. \begin{align*}m \angle A\end{align*}
  6. \begin{align*}m \angle EDC\end{align*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.6. 

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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.
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.
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.

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