Grade 11 Maths · Euclidean Geometry

Use exterior angle of cyclic quadrilateral

Use the fact that an outside angle of a quadrilateral in a circle equals the inside angle opposite it.

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Circle Geometry Grade 11 Cyclic Quadrilaterals: Exterior Angle Cyclic QuadMiss Martins Maths and Science

Exterior Angles in Cyclic QuadrilateralsMindset

Circle Geometry Grade 11 : Cyclic QuadrilateralKevinmathscience

Euclidean Geometry Theorem 6: Exterior Angle of a Cyclic QuadrilateralBRIGHT YOUNG BRAINS

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Question 8.1 · June 2024 · Paper 2 · Provincial · 4 marks

In the figure, two circles intersect at

E

and

D

D, A, B

and

C

are points on the circle such that

A BC D

is a parallelogram.

C D

is a tangent to the smaller circle. Chords

D C

and

BG

intersect at

F

A D G

and

A EB

are straight lines. Diagram for this question

8.1 Prove, with reasons, that

A D = D E

.(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: use exterior angle of cyclic quadrilateral

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3. Question 9.2.3 · November 2024 · Paper 2 · Provincial · 2 marks
9.2 In the diagram, O is the centre of the circle with a reflex angle of 230°. P, Q, R and S are points on the circumference of the circle with chord SR extended to point T. Q $\hat{R}$ T $= 85°$ and O $\hat{P}$ S $= 20°$ .

Calculate the magnitude of the following angles: $Diagram for question 9.2$
9.2.3 $\hat{P}_{1}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  S $\hat{P}_{1} + 20° = 85°$
  $\hat{P}_{1} = 65°$
  1 mark · Use exterior angle of cyclic quadrilateral
2. ✓
  R[ext. $∠$ of cyclic quad]
  1 mark · Use exterior angle of cyclic quadrilateral
4. Question 4.1.2(c) · March 2020 · Provincial · 2 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 $∠ B D A = 3 4^{\circ}$ .
4.1.2 Determine with reasons the sizes of the following angles:
4.1.2(c) $\hat{D}_{4}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Statement and reason $\hat{D}_{4} = \hat{B}_{1} + \hat{B}_{2} = 4 1^{\circ} + 6 1^{\circ}$ (ext $∠$ of cyclic quad ABCD)
  1 mark · Use exterior angle of cyclic quadrilateral
2. ✓
  Answer $\hat{D}_{4} = 10 2^{\circ}$
  1 mark · Use exterior angle of cyclic quadrilateral
5. Question 7.2 · March 2022 · Provincial · 4 marks
In the diagram, PQRS is a cyclic quadrilateral. PS and QR are produced to meet at T. RE is a tangent to the circle at R, with E on PT and RE || QS.
7.2 Prove that: $Δ RST ∣∣∣ Δ PQT$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Identify common angle $\hat{T}$ is common
  1 mark · Prove triangles similar
2. ✓
  Identify exterior cyclic quad angle $\hat{S}_{3} = P \hat{Q} R$ (ext angle of cyclic quad)
  1 mark · Use exterior angle of cyclic quadrilateral
3. ✓
  Identify third angle $S \hat{R} T = \hat{P}$ (ext angle of cyclic quad / sum of angles)
  1 mark · Use exterior angle of cyclic quadrilateral
4. ✓
  Conclusion $Δ RST ∣∣∣ Δ PQT$ (AAA)
  1 mark · Prove triangles similar
6. Question 6.2 · June 2024 · Paper 2 · Provincial · 5 marks
$PQ$ is a tangent to the circle at $C$ . $EBC D$ is a cyclic quadrilateral. $A EB$ and $A D C$ are straight lines with $A B ∣∣ QP$ . Prove with reasons that:
6.2 $\hat{D}_{3} = \hat{D}_{1}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  S $\hat{D}_{3} = \hat{B}_{1} + \hat{A}$
  1 mark · Use exterior angle of a triangle
2. ✓
  Rext $∠$ of $△$
  1 mark · Use exterior angle of a triangle
3. ✓
  S $\hat{D}_{3} = \hat{B}_{1} + \hat{B}_{2}$ ( $\hat{B}_{2} = \hat{A}$ , proven in 6.1)
  1 mark · Use the tan-chord theorem
4. ✓
  S $\hat{D}_{1} = \hat{B}_{1} + \hat{B}_{2}$
  1 mark · Use exterior angle of cyclic quadrilateral
5. ✓
  Rext $∠$ cyclic quad; $∴ \hat{D}_{3} = \hat{D}_{1}$ (both equal $\hat{B}_{1} + \hat{B}_{2}$ / both equal $\hat{C}_{3}$ )
  1 mark · Use exterior angle of cyclic quadrilateral

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