Cyclic Quadrilateral (Opposite Angles)

Statement

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles always add up to 180°:

\[ A + C = 180^\circ, \quad B + D = 180^\circ \]

Why it’s true

• The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.
• For opposite angles of a cyclic quadrilateral, each pair subtends supplementary arcs that together make a full circle (360°).
• Therefore, the opposite angles are supplementary: they add to 180°.

Recipe (how to use it)

1. Identify the quadrilateral inscribed in the circle.
2. Pair opposite angles: \(A\) with \(C\), \(B\) with \(D\).
3. Use the property: \(A+C=180^\circ\) or \(B+D=180^\circ\).
4. Solve for the unknown angle(s).

Spotting it

Look for a quadrilateral drawn inside a circle with all four vertices on the circle. The question may explicitly say “cyclic quadrilateral” or show the shape within a circle.

Common pairings

• Circle theorems (angle subtended by same arc, angle at centre = 2× angle at circumference).
• Properties of tangents and chords when combined with cyclic quadrilaterals.

Mini examples

1. In a cyclic quadrilateral, angle \(A=75^\circ\). Then angle \(C=180-75=105^\circ\).
2. In the same figure, if \(B=110^\circ\), then \(D=180-110=70^\circ\).

Pitfalls

• Forgetting that this only works if all four vertices lie on the circle.
• Mixing adjacent and opposite angles — only opposite angles add to 180°.

Exam strategy

• State clearly that “opposite angles in a cyclic quadrilateral add to 180°” when giving reasons.
• Check the diagram: confirm all four vertices lie on the circle before applying the rule.

Summary

In any cyclic quadrilateral, the opposite angles are supplementary: \(A+C=180^\circ\), \(B+D=180^\circ\). This is a direct consequence of circle theorems and is often used to find missing angles in geometry problems.