Grade 11 Euclidean Geometry • Theorem 3

Opposite Angles of a Cyclic Quadrilateral

The Intuition (Plain English)

A "cyclic quadrilateral" is just a 4-sided shape where all 4 corners touch the edge of the same circle. When this happens, the angles that are diagonally opposite each other will always add up to exactly 180 degrees (they are supplementary).

The Golden Rule: ALL four vertices must touch the circumference. If even one corner is inside or outside the circle, it is NOT a cyclic quad, and this theorem does not apply!

The Formal Proof

This is one of the 6 examinable proofs. It relies heavily on the "Angle at Centre" theorem you learned previously.

📐 [Insert GeoGebra Diagram] Cyclic quad ABCD. Centre O. Radii OB and OD drawn.

Given: A circle with centre O and cyclic quadrilateral ABCD on the circumference.

R.T.P: Â + Ĉ = 180° and B̂ + D̂ = 180°

Construction: Draw radii OB and OD.

Statement	Reason
1. Ô₁ = 2Â	∠ at centre = 2 × ∠ at circumference
2. Ô₂ (reflex) = 2Ĉ	∠ at centre = 2 × ∠ at circumference
3. Ô₁ + Ô₂ = 360°	∠s round a pt
4. ∴ 2Â + 2Ĉ = 360°	substitution
5. ∴ 2(Â + Ĉ) = 360°	factorising
6. ∴ Â + Ĉ = 180°	divide by 2
7. Similarly, by drawing radii OA and OC, we can prove B̂ + D̂ = 180°.

How to apply this in an exam (The "Exterior Angle" shortcut)

While the theorem itself is heavily tested, examiners LOVE testing the corollary: The exterior angle of a cyclic quad is equal to the interior opposite angle.

Typical Exam Question:
ABCD is a cyclic quadrilateral. The line BC is extended to E. If ∠DCE = 110°, what is the size of ∠A? Prove it using the main theorem.

Step-by-Step Solution:

First, find the interior angle ∠BCD using the straight line.
∠BCD + 110° = 180° (∠s on a str line)
∠BCD = 70°
Now, use the cyclic quad theorem to find the opposite angle ∠A.
∠A + ∠BCD = 180° (opp ∠s of cyclic quad)
∠A + 70° = 180°
∠A = 110°
Shortcut: Notice that ∠A is exactly the same as the exterior angle ∠DCE. In an exam, if they don't ask you to prove it step-by-step, you can skip straight to: ∠A = 110° (ext ∠ of cyclic quad).

⚠️ Common Exam Traps

The "Almost Cyclic" Trap: Examiners will draw a quadrilateral where 3 vertices are on the circle, but the 4th vertex is at the centre of the circle. This is NOT a cyclic quad! Do not use this theorem unless all 4 corners touch the circumference.
Confusing it with a Parallelogram: In a parallelogram, opposite angles are EQUAL. In a cyclic quadrilateral, opposite angles ADD TO 180. Don't mix them up!