Grade 11 Cyclic Quadrilaterals: Short Notes

11 ශ්‍රේණිය · 21 පරිච්ඡේදය · වෘත්ත චතුරස්‍ර

52%

of Grade 11 term tests include it (32 of 61)

264

marks in past papers (52 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Recognise a cyclic quadrilateral
Use the fact that opposite angles of a cyclic quadrilateral add to 180°
Use the fact that an exterior angle equals the interior opposite angle
Use the converses to show a quadrilateral is cyclic
Combine these with other circle facts to find angles and give reasons

Lamps round the pond

Four lamp posts stand on the edge of a round lotus pond at a temple. Paths join them into a quadrilateral. Measure two opposite corners: 70° and 110°. At another round pond: 84° and 96°. Always 180°. That's no coincidence; it's the rule of the cyclic quadrilateral.

What is a cyclic quadrilateral?

A quadrilateral is cyclic when all four vertices lie on one circle.All four, not three: any three points (not on a line) fit on a circle, but the fourth may not.Name the vertices in order round the circle: ABCD means A, B, C, D follow each other.

Opposite angles add to 180°

Theorem: the opposite angles of a cyclic quadrilateral are supplementary: a + c = 180° and b + d = 180°.Opposite means across the shape (A with C), not side by side (A with B).

✏️ Find the other two angles

ABCD is a cyclic quadrilateral with ∠A = 75° and ∠B = 110°. Find ∠C and ∠D.

1C is opposite A: ∠C = 180° − 75° = 105°.
2D is opposite B: ∠D = 180° − 110° = 70°.
3Check: 75 + 110 + 105 + 70 = 360° ✓

✅ ∠C = 105°, ∠D = 70°

The exterior angle

Extend BC to E. Theorem: the exterior angle ∠DCE equals the interior opposite angle ∠BAD.Why: ∠DCE = 180° − ∠BCD (straight line), and ∠BAD = 180° − ∠BCD (opposite angles).

⚠️ Which interior angle?

The most repeated error in O/L reports on this chapter: matching the exterior angle with the wrong interior angle. It equals the angle at the opposite vertex, not the one next to it.Tip: the exterior angle at C 'looks across' the shape to A.

✏️ Using the exterior angle

ABCD is cyclic and BC is produced to E. ∠DCE = 82° and ∠ABC = 95°. Find ∠BAD and ∠ADC.

1Exterior angle = interior opposite angle: ∠BAD = ∠DCE = 82°.
2∠ADC is opposite ∠ABC: ∠ADC = 180° − 95° = 85°.

✅ ∠BAD = 82°, ∠ADC = 85°

Is it cyclic? The converses

Both theorems work backwards:• If a pair of opposite angles is supplementary, the quadrilateral is cyclic.• If an exterior angle equals the interior opposite angle, it is cyclic.So rectangles, squares and isosceles trapeziums are always cyclic. A parallelogram or rhombus that is not a rectangle never is.

⚠️ Adjacent angles are not supplementary

Examiners also report misused opposite angles, such as adding neighbouring angles (∠A + ∠B) to 180°. That works in a parallelogram, not in a cyclic quadrilateral.And watch for isosceles triangles hiding inside: equal sides or radii OA = OB give equal base angles. Mark them on the figure.

📌 Remember

• Cyclic: all four vertices on one circle.• Opposite angles add to 180°.• Exterior angle = interior opposite angle.• Converses: either fact proves a quadrilateral is cyclic.• Combine with: angle at centre = 2 × angle at circumference, angle in a semicircle = 90°, isosceles triangles. Always give the reason.

🧾 Quick revision sheet

Cyclic quadrilateral	A quadrilateral whose four vertices all lie on one circle.
Opposite angles of a cyclic quadrilateral	Supplementary: they add to 180°.
Exterior angle of a cyclic quadrilateral	Equals the interior opposite angle.
How do you prove a quadrilateral is cyclic?	Show a pair of opposite angles adds to 180°, or an exterior angle equals the interior opposite angle.
Is every rectangle cyclic?	Yes: opposite angles are 90° + 90° = 180°.
Is a rhombus (not a square) cyclic?	No: its opposite angles are equal but not 90°, so they don't add to 180°.

Now test yourself: 5 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 11-minute lesson free ▶ Practise 5 past-paper questions