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All courses Geometry · G-C.3 16 of 57
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 \(110^{\circ}\).
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

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

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

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

Subtracting 110 from 180 gives the missing opposite angle.

04

Check the conclusion

\[70^{\circ}+110^{\circ}=180^{\circ}\]

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

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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

Solution walkthrough video