Opposite angles of a cyclic quadrilateral add to 180°. If the vertices A, B, C and D appear in that order around the circle, pair A with C and B with D.
Use the pair that matches the information given. An algebraic expression can help you find one pair, while a ratio can split the other pair into two angles.
- Label the vertices in order around the circle.
- Write a separate 180° equation for each opposite pair.
- Check that all four angles add to 360°.
Worked example | Use an equation for A and C
In cyclic quadrilateral ABCD, angle A is (2x + 10)° and angle C is (3x − 5)°. Set (2x + 10) + (3x − 5) = 180.
This gives 5x + 5 = 180, so x = 35. Angle A is 80° and angle C is 100°. Substituting these angles back confirms they add to 180°.
Use a ratio for B and D
Suppose B:D = 5:7. Together the two opposite angles make 180°, so 12 ratio units represent 180°. One unit is 15°. This gives B = 75° and D = 105°.
Check the four angles by adding the two pairs. A + C = 180° and B + D = 180°, so the total is 360°.
Solve a different opposite pair
In a cyclic quadrilateral, two opposite angles are (3x + 5)° and (x + 15)°. Find both angles.
Answer. Set 4x + 20 = 180, giving x = 40. The two angles are 125° and 55°.
Frequently asked questions
Can I use the 180° rule for adjacent angles?
Use the cyclic quadrilateral rule for opposite angles. Adjacent angles may need another fact from the diagram.
What should I write as the reason?
State that opposite angles of a cyclic quadrilateral are supplementary.
Official information
SEAB, 2027 SEC G3 Mathematics syllabus (K310), Geometry and Measurement, properties of circles. Syllabus checked 11 November 2026. The examples are original.