Grade 10 Maths · Euclidean Geometry

Use angles on a straight line

Find an angle using angles on a straight line adding to 180°, angles around a point adding to 360°, or vertically opposite angles being equal.

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Grade 8: Angles on a straight line (Geometry of straight lines)Mrs Le Roux - Maths

Angles in a Straight Line - Corbettmathscorbettmaths

Grade 8: Vertically opposite angles (Geometry of straight lines)Mrs Le Roux - Maths

Grade 8: Angles around a point/ Angles in a revolution (Geometry of straight lines)Mrs Le Roux - Maths

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Question 10.4 · November 2024 · Paper 2 · Provincial · 6 marks

The circle is given below with TQA as a tangent. PS = SR and PQ = PR.

PQ ∥ SR

.
SR is extended to A.

\hat{Q}_{1} = x

and

\hat{Q}_{3} = y

. $Diagram for this question$

10.4 Calculate the value of

x

and

y

.(6)

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

More past paper questions: use angles on a straight line

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3. Question 5.2.1(e) · June 2024 · Paper 2 · Provincial · 3 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 $\hat{B} = 7 2^{\circ}$ .
5.2.1 Calculate, giving reasons, the size of:
5.2.1(e) $\hat{D}_{1}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Angles on a straight line (S/R) $\hat{D}_{1} + (\hat{D}_{2} + \hat{D}_{3}) + \hat{D}_{4} = 18 0^{\circ}$ ( $∠$ s on a straight line $E D G$ )
  1 mark · Use angles on a straight line
2. ✓
  Angle in a semi-circle (S/R) $\hat{D}_{2} + \hat{D}_{3} = A \hat{D} C = 9 0^{\circ}$ ( $∠$ in semi-circle)
  $\hat{D}_{1} + 9 0^{\circ} + 3 6^{\circ} = 18 0^{\circ}$
  1 mark · Use angle in a semicircle
3. ✓
  Answer (S) $\hat{D}_{1} = 18 0^{\circ} - 12 6^{\circ} = 5 4^{\circ}$
  1 mark · Use angles on a straight line
4. Question 9.1.2 · November 2025 · Paper 2 · Provincial · 2 marks
9.1 In the diagram below, O is the centre of the circle. ACDE and BCDE are cyclic quadrilaterals. GCF is a tangent to the circle at point C. Chord CE and line OC are drawn. $\hat{E}_{2} = 15°$ .

Determine the size of the following angles below, giving reasons: $Diagram for question 9.1$
9.1.2 $\hat{A}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  S $\hat{O}_{2} = 150°$ [ $∠$ s on a str. line]
  1 mark · Use angles on a straight line
2. ✓
  S/R $\hat{A} = 75°$ [ $∠$ at centre $= 2∠$ at circumf.]
  1 mark · Use angle at the centre
5. Question 5.2.4 · March 2020 · Provincial · 2 marks
5.2 AE is a diameter of the circle. AE is produced to D. DB is a tangent to the circle at B. DC is perpendicular to AD. CBA is a straight line. BE and EC are drawn. Let $\hat{A} = x .$
5.2.4 Is CE a tangent to the circle through A, B and E? Give a reason for your answer.(2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate E2 $\hat{E}_{2} = 18 0^{\circ} - (\hat{E}_{1} + \hat{E}_{3})$ ( $∠$ s on str line)
  $= 18 0^{\circ} - (9 0^{\circ} - x + 9 0^{\circ} - x) = 2 x$
  1 mark · Use angles on a straight line
2. ✓
  Compare and Conclude $\hat{E}_{2} \neq = \hat{A}$ (since $2 x \neq = x$ )
  $∴$ CE is not a tangent (converse of tan-chord theorem does not apply)
  1 mark · Prove a line is a tangent
6. Question 4.1 · March 2022 · Provincial · 4 marks
4.1 In the diagram below, $Δ A BC$ is drawn with $A B ∥ ST$ , $\hat{C}_{1} = 10 x$ , $\hat{C}_{2} = y$ , $\hat{S}_{1} = 7 8^{\circ}$ , $\hat{T}_{1} = 4 x$ and $\hat{A} = z$ . Calculate, with reasons, $x, y$ and $z$ .(4)
Show the memo (3 ticks)Memo
1. ✓
  Calculate x $10 x = 7 8^{\circ} + 4 x$ (Ext $∠$ of $Δ$ CST)
  $6 x = 7 8^{\circ} ⟹ x = 1 3^{\circ}$
  2 marks · Use exterior angle of a triangle
2. ✓
  Calculate y $y + 10 x = 18 0^{\circ}$ (Angles on straight line)
  $y = 18 0^{\circ} - 13 0^{\circ} = 5 0^{\circ}$
  1 mark · Use angles on a straight line
3. ✓
  Calculate z $z = 7 8^{\circ}$ (Corresp $∠$ s, $ST ∥ A B$ )
  1 mark · Use angles on parallel lines

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