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PSLE Mathematics · Primary 5 and 6

Angles and Geometric Figures

Geometry questions at PSLE ask you to find unknown angles in figures made of lines, triangles and four-sided figures. You need a small set of angle facts and the properties of special triangles and quadrilaterals, and then the skill of applying them one step at a time. This topic builds on the angle work from earlier years and leads into the geometry of secondary school. This guide lists the facts you must know, shows how to set out reasons, and works through a typical parallelogram question.

What the syllabus expects

  • Use the facts that angles on a straight line add up to 180°, angles at a point add up to 360°, and vertically opposite angles are equal
  • Use the angle sum of a triangle, which is 180°
  • Use the properties of isosceles, equilateral and right-angled triangles
  • Use the properties of the square, rectangle, parallelogram, rhombus and trapezium
  • Find unknown angles in composite figures made of these shapes, giving reasons

Angle facts on lines and at a point

A right angle is 90° and a straight line makes 180°. Angles on a straight line add up to 180°, so if one angle on a line is 115°, the angle next to it is 180° − 115° = 65°. Angles at a point, which go all the way round, add up to 360°. If three angles at a point are 125°, 98° and x, then x = 360° − 125° − 98° = 137°. When two straight lines cross, the angles opposite each other are equal. These are called vertically opposite angles, and each pair next to each other still adds up to 180°. Many PSLE questions start with one of these facts, so look for straight lines and crossing points first.

Triangles

The three angles of any triangle add up to 180°. An isosceles triangle has two equal sides, usually shown with small matching marks, and the two angles opposite those sides are equal. These are often called the base angles. If the angle between the equal sides is 40°, each base angle is (180° − 40°) ÷ 2 = 70°. An equilateral triangle has three equal sides and three angles of 60°. A right-angled triangle has one angle of 90°, so the other two angles add up to 90°. Only use a property if the question tells you the sides are equal or the angle is a right angle. A triangle that looks isosceles in the figure might not be.

Special four-sided figures

A parallelogram has two pairs of parallel sides. Its opposite sides are equal, its opposite angles are equal, and the two angles between a pair of parallel sides, along the same side, add up to 180°. So if one angle of a parallelogram is 70°, the angles are 70°, 110°, 70° and 110°. A rhombus is a parallelogram with all four sides equal, so it has all the parallelogram properties as well. A rectangle has four right angles and a square has four right angles and four equal sides. A trapezium has exactly one pair of parallel sides. The two angles along each non-parallel side, between the parallel lines, add up to 180°. Learn which angles pair up in each shape, because pairing the wrong angles is a common error.

Working through a composite figure

Most questions combine shapes, for example a triangle drawn inside a parallelogram. Start by marking every given angle and every pair of equal sides on the figure. Then look for an angle you can find in one step, write it on the figure, and use it to find the next one. Write a short reason for each step, such as 'angles on a straight line' or 'base angles of an isosceles triangle', because it helps you see the chain of reasoning. If the figure shows that a shape has been folded, the folded part and its original position are identical, so the matching angles are equal. Figures are usually marked as not drawn to scale, so never measure with a protractor.

Worked example

ABCD is a parallelogram with AB parallel to DC and ∠DAB = 70°. E is a point on DC such that AD = DE. Find (a) ∠DEA (b) ∠EAB (c) ∠AEC.

  1. ∠ADC = 180° − 70° = 110°, because angles between the parallel sides AB and DC along AD add up to 180°.
  2. Triangle ADE is isosceles with AD = DE, so its base angles ∠DAE and ∠DEA are equal.
  3. ∠DEA = (180° − 110°) ÷ 2 = 70° ÷ 2 = 35°, using the angle sum of a triangle.
  4. ∠DAE = 35° too, so ∠EAB = ∠DAB − ∠DAE = 70° − 35° = 35°.
  5. D, E and C lie on a straight line, so ∠AEC = 180° − ∠DEA = 180° − 35° = 145°.
  6. Check: in triangle ADE, 110° + 35° + 35° = 180°.

Answer: (a) ∠DEA = 35° (b) ∠EAB = 35° (c) ∠AEC = 145°

Common mistakes

  • Assuming a triangle is isosceles or an angle is a right angle because it looks that way. Use only what the question states or marks.
  • Putting the equal angles of an isosceles triangle in the wrong place. The equal angles are opposite the equal sides, not between them.
  • Mixing up 180° and 360°. Angles on a straight line add up to 180°, while angles at a point add up to 360°.
  • In a trapezium, adding the wrong pair of angles to make 180°. The pairs that add up to 180° lie along the same non-parallel side, between the two parallel sides.
  • Measuring with a protractor when the figure is not drawn to scale. Every angle must come from a reason.
  • Finding an angle correctly but labelling it with the wrong letters. Read angle names carefully: ∠AEC has its vertex at E, the middle letter.

Exam tips

  • Copy the given angles onto the figure first, then write each new angle on the figure as soon as you find it, so the next step becomes visible.
  • Write a short reason beside each step. It keeps your reasoning clear and lets you spot a wrong fact before it spoils the rest of the answer.
  • Look for hidden isosceles triangles, such as triangles with two sides that are radii of the same circle or two sides of a rhombus.

Practice questions

1. Easy: Three angles meet at a point. Two of them are 125° and 98°. Find the third angle.

Show answer

137°. Working: angles at a point add up to 360°, so 360° − 125° − 98° = 137°.

2. Medium: In triangle PQR, PQ = PR and ∠QPR = 46°. Find ∠PQR.

Show answer

67°. Working: the triangle is isosceles, so ∠PQR = ∠PRQ. Each is (180° − 46°) ÷ 2 = 134° ÷ 2 = 67°.

3. Harder: WXYZ is a trapezium with WX parallel to ZY. WY is a diagonal. ∠XWZ = 64° and ∠WYZ = 30°. Find ∠ZWY.

Show answer

34°. Working: ∠WZY = 180° − 64° = 116°, because angles between the parallel sides along WZ add up to 180°. In triangle WZY, ∠ZWY = 180° − 116° − 30° = 34°.

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