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All courses Math II · G-C.3 31 of 73
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.

Find a missing angle in a cyclic quadrilateral using opposite supplementary angles

Problem
Find the missing angle in cyclic quadrilateral ABCD when the opposite angle is 110 degrees.
Unworked geometry diagram showing only the supplied configuration and givens. Open full size
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Hint

Use the fact that opposite angles in a cyclic quadrilateral are supplementary. Start with 110° + x = 180°.

In any cyclic quadrilateral, opposite angles add to 180°. So the missing opposite angle is found by subtracting the given angle from 180°.

Solution walkthrough

01

Verify the cyclic condition

\[A,B,C,D~\text{all}~\text{lie}~\text{on}~\text{one}~\text{circle}\]

A quadrilateral inscribed in a circle is cyclic, so each pair of opposite interior angles is supplementary.

02

Pair the correct opposite angles

\[\text{angle}~A~\text{opposite}~\text{angle}~C=110~\text{degrees}\]

A and C share no side in quadrilateral ABCD, so they form the relevant opposite pair in the inspected diagram.

03

Write and solve the supplement equation

\[\begin{aligned} m~\text{angle}~A+110~\text{degrees}=180~\text{degrees} \\ m~\text{angle}~A=70~\text{degrees} \end{aligned}\]

Subtracting 110 from 180 gives the missing opposite angle.

04

Check the conclusion

\[70~\text{degrees}+110~\text{degrees}=180~\text{degrees}\]

The sum verifies the cyclic-opposite-angle theorem and the answer diagram's labels.

+

Another way

  1. Use m angle A=180 degrees-m angle C directly.

!

Common mistake

Do not subtract from 360 degrees. Opposite angles of a cyclic quadrilateral form a supplementary pair totaling 180 degrees.

Completed answer diagram with exact construction, relationship, and calculation.
missing angle: 70 degrees