Cyclic quadrilaterals (quadrilateral inscribed in a circle)
Cyclic Quadrilaterals
Cyclic Quadrilaterals is a topic in Circles in the California Common Core State Standards. It is aligned to Standard G-C.3, which requires students to prove properties of angles for a quadrilateral inscribed in a circle.
A cyclic quadrilateral is inscribed in a circle, and its opposite angles are supplementary, summing to \(180^\circ\).
Theory
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on a single circle (it is inscribed in the circle). Its defining property:
The converse is also true: if a quadrilateral's opposite angles are supplementary, it is cyclic.
The opposite-angle rule:
How to use the property
- Confirm the quadrilateral is cyclic (vertices on a circle).
- Pair opposite angles.
- Set each pair's sum to \(180^\circ\) and solve.
Opposite angles of a cyclic quadrilateral are supplementary.
| \(180^\circ-85^\circ\) | \(=\) | \(95^\circ\) |
Set the sum to \(180^\circ\).
| \(2x+(x+30)\) | \(=\) | \(180\) |
| \(3x+30\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(50\) |
The angle opposite the \(70^\circ\) is its supplement.
| \(180^\circ-70^\circ\) | \(=\) | \(110^\circ\) |
Exactly when its opposite angles are supplementary (each pair sums to \(180^\circ\)).
Common pitfalls
Frequently asked questions
What is a cyclic quadrilateral?
A quadrilateral whose four vertices all lie on a single circle.
What is the key property of a cyclic quadrilateral?
Its opposite angles are supplementary — each pair sums to \(180^\circ\).
How do you find a missing angle?
Subtract the opposite angle from \(180^\circ\).
When can a quadrilateral be inscribed in a circle?
Exactly when its opposite angles are supplementary.