Home · IGCSE Maths 0580 · Geometry · Angles: lines, parallel lines and triangles · Practice
IGCSE · Practice questions · Angles: lines, parallel lines and triangles

Find it, then justify it.

Six original Cambridge-style questions on angle properties, with every answer backed by a named geometric reason.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

Find an angle, name the fact.

01
Calculation
[2 marks]

Four angles meet at a point. They are \(105^\circ\), \(87^\circ\), \(2x\) and \(x\). Find \(x\).

  • Angles at a point add to \(360^\circ\): \(105 + 87 + 3x = 360\). ✓
  • \(3x = 168\), so \(x = 56\). ✓
02
Calculation
[2 marks]

The angles \(4x\), \(5x\) and \(27^\circ\) lie on a straight line. Find \(x\).

  • Angles on a straight line add to \(180^\circ\): \(9x + 27 = 180\). ✓
  • \(9x = 153\), so \(x = 17\). ✓
03
Analysis
[3 marks]

Two parallel lines are crossed by a transversal. One of the angles formed is \(58^\circ\).

Find (a) the angle corresponding to it, (b) the angle co-interior with it, (c) the angle vertically opposite the co-interior angle. Give a reason for each.

  • (a) \(58^\circ\): corresponding angles are equal. ✓
  • (b) \(180^\circ - 58^\circ = 122^\circ\): co-interior angles add to \(180^\circ\). ✓
  • (c) \(122^\circ\): vertically opposite angles are equal. ✓
04
Calculation
[3 marks]

The angles of a quadrilateral are \(x\), \(x + 30^\circ\), \(2x\) and \(2x - 30^\circ\). Find each angle.

  • Angle sum of a quadrilateral is \(360^\circ\): \(6x = 360\), so \(x = 60\). ✓
  • The angles are \(60^\circ\), \(90^\circ\), \(120^\circ\) and \(90^\circ\). Check: total \(360^\circ\). ✓
05
Proof
[3 marks]

Triangle ABC has interior angles \(a\), \(b\) and \(c\) at A, B and C. Side BC is extended beyond C to D.

Prove that the exterior angle ACD is equal to \(a + b\).

  • Angles in a triangle add to \(180^\circ\): \(a + b + c = 180^\circ\), so \(a + b = 180^\circ - c\). ✓
  • BCD is a straight line, so angle ACD \(= 180^\circ - c\) (angles on a straight line). ✓
  • Therefore angle ACD \(= a + b\). ✓
06
Analysis
[4 marks]

PQ and RS are parallel lines, with Q to the right of P and S to the right of R. Point T lies between the lines, to the right of P and R. Angle QPT = \(38^\circ\) and angle SRT = \(47^\circ\).

Find angle PTR, giving reasons.

  • Draw a line through T parallel to PQ and RS, extending LEFT from T to a point U (towards the side of P and R). ✓
  • Angle PTU = angle QPT = \(38^\circ\) (alternate angles). ✓
  • Angle RTU = angle SRT = \(47^\circ\) (alternate angles). ✓
  • Angle PTR \(= 38^\circ + 47^\circ = 85^\circ\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.