Grade 10 Maths · Euclidean Geometry

Use exterior angle of a triangle

Find an angle using the fact that an outside angle of a triangle equals the two opposite inside angles added together (for example 40° + 70° = 110°).

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Exterior Angle Theorem For Triangles, Practice Problems - GeometryThe Organic Chemistry Tutor

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Question 2.1 · June 2024 · Paper 2 · Provincial · 2 marks

In the diagram below,

A (- 2; 4)

B (- 2; - 1)

and

C (p; - 4)

are the vertices of

Δ A BC

Q

is the x-intercept of the line

A C

θ

is the angle of inclination of

A C

and

B \hat{A} C = 3 7^{\circ}

2.1 Determine the size of

θ

.(2)

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More past paper questions: use exterior angle of a triangle

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3. Question 10.2.4 · May/June 2025 · Paper 2 · National (DBE) · 3 marks
10.2 In the diagram, O is the centre of the circle. Points D and B lie on the circle. Points A and C lie outside the circle such that side AC of $Δ A D C$ passes through B. F is a point on BC such that $FO ∥ B D$ . $D R = RB$ and RO is drawn.
10.2.4 Calculate the size of $A \hat{B} D$ .(3)
Show the memo (1 tick)Memo
1. ✓
  METHOD 1: Using trigonometry in the right-angled triangle and exterior anglesIn right-angled $Δ D RO$ :
  $tan R \hat{D} O = \frac{RO}{R D} = \frac{4 , 5}{6} = \frac{3}{4}$
  $∴ R \hat{D} O = 36, 8 7^{\circ}$
  
  Given $F \hat{C} O = R \hat{D} O$ , so $F \hat{C} O = 36, 8 7^{\circ}$ (this is angle $\hat{C}$ )
  In $Δ B D C$ , $B D = BC$ , so $B \hat{D} C = \hat{C} = 36, 8 7^{\circ}$
  
  $A \hat{B} D$ is an exterior angle of $Δ B D C$ :
  $A \hat{B} D = B \hat{D} C + \hat{C}$ [ext $∠$ of $Δ$ ]
  $A \hat{B} D = 36, 8 7^{\circ} + 36, 8 7^{\circ}$
  $A \hat{B} D = 73, 7 4^{\circ}$
  3 marks · Use exterior angle of a triangle
4. Question 4.2.2 · June 2024 · Paper 2 · Provincial · 2 marks
4.2 In the given figure, $D \hat{C} O = 2 5^{\circ}$ , $O$ is the center of the circle. $A$ , $B$ , $E$ , $C$ and $D$ are points on the circumference. Calculate with reasons, the size of:
4.2.2 $\hat{O}_{1}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  S $\hat{O}_{1} = 2 5^{\circ} + 2 5^{\circ} = 5 0^{\circ}$
  1 mark · Use exterior angle of a triangle
2. ✓
  Rext $∠$ of $△$
  1 mark · Use exterior angle of a triangle
5. Question 4.1.1 · March 2021 · Provincial · 3 marks
4.1 In the diagram below, $\hat{E} = 2 x$ , $\hat{C}_{1} = 7 5^{\circ}$ , $\hat{C}_{2} = 2 x - 1 5^{\circ}$ and $\hat{B}_{2} = 13 5^{\circ}$ .
4.1.1 Determine the value of x.(3)
Show the memo (3 ticks)Memo
1. ✓
  Statement $\hat{B}_{2} = \hat{E} + \hat{C}_{1}$ , so $13 5^{\circ} = 2 x + 7 5^{\circ}$
  1 mark · Use exterior angle of a triangle
2. ✓
  Reason(ext $∠$ of $△$ )
  1 mark · Use exterior angle of a triangle
3. ✓
  Answer $2 x = 6 0^{\circ} ⟹ x = 3 0^{\circ}$
  1 mark · Use exterior angle of a triangle
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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