Cyclic Quadrilateral Theorem

A proof that opposite angles of a cyclic quadrilateral sum to 180°, using the inscribed angle theorem and arc addition.

Keywords: cyclic quadrilateral, inscribed quadrilateral, opposite angles, supplementary, inscribed angle, arc

Prerequisites: circle-angles-intro, inscribed-angle-proof · Difficulty: intermediate

A cyclic quadrilateral has all four vertices on one circle. Its opposite angles are supplementary:

Strategy: place $ABCD$ on a circle with center $O$. Angle $\angle A$ subtends arc $BCD$ and $\angle C$ subtends arc $DAB$; these arcs make up the full circle. By the inscribed angle theorem each angle is half its arc, so $\angle A+\angle C=\frac{1}{2}(360°)=180°$.

Step 1: A Quadrilateral on a Circle

A cyclic quadrilateral has all four vertices on a circle. We draw quadrilateral $ABCD$ inscribed in a circle with center $O$.

Step 2: Opposite Angles Subtend Complementary Arcs

The key insight: opposite angles subtend arcs that together make the full circle.

• $\angle A$ (at vertex $A$) is an inscribed angle subtending arc $BCD$ — the arc from $B$ to $D$ passing through $C$.
• $\angle C$ (at vertex $C$) is an inscribed angle subtending arc $DAB$ — the arc from $D$ to $B$ passing through $A$.

These two arcs together comprise the entire circle ($360°$).

Step 3: Opposite Angles Sum to $180°$

By the inscribed angle theorem:

• $\angle A=\frac{1}{2}\cdot \text{arc }BCD$
• $\angle C=\frac{1}{2}\cdot \text{arc }DAB$

Since arc $BCD$ + arc $DAB$ = $360°$:

By the same reasoning, $\angle B+\angle D=180°$.

Step 4: The Converse

Converse: If a quadrilateral has opposite angles summing to $180°$, then it is cyclic.

Suppose $\angle A+\angle C=180°$. Construct the circle through $A$, $B$, $D$. If $C$ were not on this circle, the inscribed angle at $C^{\prime }$ on the circle would differ from $\angle C$, contradicting $\angle A+\angle C=180°$. So $C$ must lie on the circle.

In a cyclic quadrilateral $ABCD$, opposite angles are supplementary:

The converse also holds: if a quadrilateral's opposite angles sum to $180°$, then it is cyclic. ∎