The idea, step by step
Four vertices on one circle
A cyclic quadrilateral has all four vertices on the same circle. Opposite interior angles sum to 180°. Adjacent angles do not generally sum to 180°.
The converse is also useful: if opposite angles of a simple quadrilateral are supplementary, its vertices lie on a circle. Ensure the diagram represents a simple quadrilateral rather than crossed edges.
Tangent and chord
The angle between a tangent and a chord equals the circumference angle in the alternate segment. Identify the contact point, the chord endpoints and the vertex on the opposite segment.
There are supplementary angles at the tangent line. Specify which angle is given or use the accompanying segment information. A radius is perpendicular to the tangent, but a general chord is not.
A common trap
Applying the cyclic angle rule to any quadrilateral.
Instead: Check that all four vertices lie on one circle or prove cyclicity first.
See the thinking
Cyclic quadrilateral
ABCD is cyclic in that order. ∠A=80° and ∠B=110°. Find ∠C and ∠D.
- Opposite angles A and C sum to 180°, so C=100°.
- Opposite angles B and D sum to 180°, so D=70°.
Tangent and chord
The acute angle between a tangent at A and chord AB is 52°. C is on the alternate segment. Find ∠ACB.
- The tangent–chord theorem links this tangent angle to the alternate-segment circumference angle.
- Therefore ∠ACB=52°.
Check your understanding
A cyclic quadrilateral has an interior angle 73°. Find its opposite angle.
Show answer
107°
Opposite angles total 180°.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- ABCD is cyclic and ∠A=112°. Find ∠C.2 marks
- A simple quadrilateral has opposite angles 96° and 84°. What can be concluded?2 marks
- A tangent–chord angle is 41° and C lies in its alternate segment. Find the corresponding circumference angle.2 marks
- A cyclic quadrilateral has opposite angles 3 x+10 and 5 x−6 degrees. Find x and both angles.3 marks
- ABCD is cyclic. An exterior angle at A is 65°. Find interior ∠C and explain.3 marks
- A radius meets a tangent at A. Chord AB makes 35° with radius AO. Find the smaller angle between the tangent line and chord line.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- ABCD is cyclic and ∠A=112°. Find ∠C.2 marks
180−112=68.
68° - A simple quadrilateral has opposite angles 96° and 84°. What can be concluded?2 marks
The opposite angles sum to 180°.
It is cyclic. - A tangent–chord angle is 41° and C lies in its alternate segment. Find the corresponding circumference angle.2 marks
Use the alternate-segment theorem.
41° - A cyclic quadrilateral has opposite angles 3 x+10 and 5 x−6 degrees. Find x and both angles.3 marks
8 x+4=180 gives x=22.
3×22+10=76; 5×22−6=104.
x=22; 76° and 104°. - ABCD is cyclic. An exterior angle at A is 65°. Find interior ∠C and explain.3 marks
Interior A=115°.
Opposite C=180−115=65.
65° - A radius meets a tangent at A. Chord AB makes 35° with radius AO. Find the smaller angle between the tangent line and chord line.2 marks
Radius and tangent make 90°.
The remaining angle is 90−35=55.
55°
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗