Grade 11 Maths · Euclidean Geometry

Prove a quadrilateral is cyclic

Prove that four points lie on a circle, for example by showing that opposite angles of the quadrilateral add up to 180°.

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Grade 11 Circle Geometry: To Prove a Quad is Cyclic Page 254 Ex 7Maths Matters

Circle Geometry: Proving Cyclic QuadrilateralsMindset

Euclidean Geometry Grade11: L13 How to prove a quadrilateral is cyclicOnline Maths by Miss Pythagoras

11 Proving a cyclic quadWiskunde_Mathematics_SUM

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Question 4.1.3 · March 2020 · Provincial · 4 marks

In the diagram below, points A, B, C and D lie on the circumference of a circle. FG and FD are tangents to the circle at C and D respectively. CD is produced to meet AE at E. Furthermore,

∠ GC A = 7 8^{\circ}

∠ CB D = 4 1^{\circ}

and

∠ B D A = 3 4^{\circ}

4.1.3 Determine, with reasons, whether CADF is a cyclic quadrilateral or not.(4)

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More past paper questions: prove a quadrilateral is cyclic

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3. Question 9.2.2 · May/June 2025 · Paper 2 · National (DBE) · 5 marks
Provide reasons for your statements in QUESTIONS 9 and 10.
9.2 In the diagram, POL is a diameter of the larger circle with centre O. TMR is a diameter of the smaller circle with centre M. The two circles intersect at L and R.
PLK is a tangent to the smaller circle at L and TR is a tangent to the larger circle at R. OM intersects the smaller circle at N. Straight lines LT, LR, LN and PR are drawn.

Prove, giving reasons, that:
9.2.2 LORM is a cyclic quadrilateral, if it is also given that $L T ∥ OM$ (5)
Show the memo (1 tick)Memo
1. ✓
  Show that the angles subtended by LM are equal $\hat{L}_{1} = \hat{R}_{1}$ [tan chord theorem]
  $\hat{L}_{1} = \hat{O}_{1}$ [corresp $∠$ s; $L T ∥ OM$ ]
  $∴ \hat{R}_{1} = \hat{O}_{1}$
  $∴$ L, O, R and M are concyclic.
  $∴$ LORM is a cyclic quadrilateral [converse $∠$ s in the same seg]
  5 marks · Prove a quadrilateral is cyclic
4. Question 5.2.3 · June 2024 · Paper 2 · Provincial · 2 marks
5.2 In the diagram below, $O$ is the centre of the circle and $A C$ is the diameter.
$A C$ is produced to meet $E D$ produced at $G$ . $OE ∥ C D$ and $\hat{B} = 7 2^{\circ}$ .
5.2.3 Is $OE D C$ a cyclic quadrilateral? Give a reason for your answer.(2)
Show the memo (2 ticks)Memo
1. ✓
  ConclusionNo, $OE D C$ is not a cyclic quadrilateral.
  1 mark · Prove a quadrilateral is cyclic
2. ✓
  Statement and reason $\hat{E}_{2} + \hat{C}_{1} = 5 4^{\circ} + 7 2^{\circ} = 12 6^{\circ} \neq = 18 0^{\circ}$ (opposite angles not supplementary)
  1 mark · Prove a quadrilateral is cyclic
5. Question 10.1.2(a) · November 2024 · Paper 2 · Provincial · 4 marks
10.1 In the diagram, O is the centre of circle ABP. PB is extended to C. AC is drawn. PN is a tangent to the circle at P. OC intersects AP perpendicularly at point O. $\hat{P}_{2} = 37°$

Answer the questions that follow. $Diagram for question 10.1$
10.1.2 Prove that:
10.1.2(a) OACB is a cyclic quadrilateral(4)
Show the memo (4 ticks)Memo
1. ✓
  SA $\hat{B}$ P $= 90°$
  1 mark · Use angle in a semicircle
2. ✓
  R[ $∠$ in a semi-circle]
  $∴ \hat{B}_{1} = 90°$ [ $∠$ s on a str. line]
  1 mark · Use angle in a semicircle
3. ✓
  S $\hat{O}_{3} = 90°$ [given]
  1 mark · Prove a quadrilateral is cyclic
4. ✓
  ROACB is a cyclic quadrilateral [converse $∠$ s same seg]
  1 mark · Prove a quadrilateral is cyclic
6. Question 8.1 · March 2022 · Provincial · 4 marks
In the diagram, $A B$ lies on circle with centre $O$ , and is produced to $C$ which lies on circle with centre $P$ . Similarly, $FE$ is produced to $D$ . $BE ∥ C D$ .
8.1 Prove that $A C D F$ is a cyclic quadrilateral.(4)
Show the memo (3 ticks)Memo
1. ✓
  Relate A to E2 $\hat{A} = \hat{E}_{2}$ (Exterior angle of cyclic quad $A BEF$ )
  1 mark · Use exterior angle of cyclic quadrilateral
2. ✓
  Relate E2 to D $\hat{D} + \hat{E}_{2} = 18 0^{\circ}$ (Co-interior angles, $BE ∥ C D$ )
  1 mark · Use angles on parallel lines
3. ✓
  Substitute and conclude $\hat{D} + \hat{A} = 18 0^{\circ}$
  Therefore $A C D F$ is a cyclic quad (Opposite angles supplementary)
  2 marks · Prove a quadrilateral is cyclic

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