Circles
328. The Cyclic Balance · opposite angles always sum to 180°
The sum of opposite angles ∠ABC + ∠ADC and ∠BAD + ∠BCD always equals 180°.
In a cyclic quadrilateral (all four vertices on one circle), opposite angles are supplementary — each pair adds to 180°: ∠ABC + ∠ADC = 180° and ∠BAD + ∠BCD = 180°.
What this lesson covers
Try to break it
Drag A, B, C, or D around the circle. Opposite interior angles always add to exactly 180°: ∠A + ∠C = 180° and ∠B + ∠D = 180°. Squish a corner close to another — the magic sum still holds. Try to find a cyclic configuration where it doesn't; impossible.
How you build it
Construct a cyclic quadrilateral.
- Draw a circle with centre O using the circle tool.
- Place point A on the circumference of the circle.
- Place point B on the circumference of the circle.
- Place point C on the circumference of the circle.
- Place point D on the circumference of the circle.
- Join A and B to draw side AB.
- Join B and C to draw side BC.
- Join C and D to draw side CD.
- Join D and A to complete the cyclic quadrilateral.
The proof, step by step
Prove that the opposite angles of a cyclic quadrilateral add up to 180°.
- Join OA and OC to form the central angles subtended by arcs ABC and ADC.
- By the Angle at Centre Theorem, reflex ∠AOC = 2∠ABC and minor ∠AOC = 2∠ADC.
- Adding these gives: reflex ∠AOC + minor ∠AOC = 2(∠ABC + ∠ADC).
- Since the angles around the centre sum to 360°, we have 2(∠ABC + ∠ADC) = 360°.
- Dividing by 2 proves that ∠ABC + ∠ADC = 180°.
- Similarly, joining OB and OD proves that ∠BAD + ∠BCD = 180°.
Worked example
In a cyclic quadrilateral ABCD, if ∠A = 75°, then ∠C is equal to:
Opposite angles of a cyclic quadrilateral are supplementary. Thus, ∠A + ∠C = 180°. Substituting ∠A = 75°, we get ∠C = 180° - 75° = 105°.
- 75°
- 105° — correct
- 115°
- 125°