Edexcel International GCSE Mathematics A 4MA1: Geometry and trigonometry – Study Guide

This study guide covers Topic 4, Geometry and trigonometry, sections 4.1–4.8, of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on that specification. Sections 4.1–4.5 are the same on both tiers. Sections 4.6–4.8 have Foundation content plus extra statements marked below as Higher tier only. A calculator may be used on every paper. Mensuration, 3D shapes and similarity (4.9–4.11) are not part of this unit.

Afterwards, use the revision notes and the practice questions. The Edexcel IGCSE Mathematics course hub links the rest of the course, and the printable checklist lists every statement.

What this unit covers

Section	What you must be able to do	Tier
4.1	Angle types; angles with intersecting and parallel lines; triangle angle facts	Both
4.2	Name polygons; quadrilateral properties; interior and exterior angles; congruence	Both
4.3	Lines of symmetry and order of rotational symmetry	Both
4.4	Scales, time, estimates, bearings, speed, density, pressure	Both
4.5	Measure and construct; scale drawings; perpendicular and angle bisectors	Both
4.6	Circle vocabulary; chord and tangent properties	Both
4.6	Intersecting chords; cyclic quadrilaterals; circle theorems	Higher tier only
4.7	Give reasons for angle calculations	Both (Higher extends to circles)
4.8	Pythagoras and right-angled trigonometry in 2D, including bearings	Both
4.8	Obtuse angles; elevation and depression; sine and cosine rules; ½ab sin C; 3D problems	Higher tier only

4.1 Angles, lines and triangles

• Acute: less than 90°. Right: 90°. Obtuse: between 90° and 180°. Reflex: between 180° and 360°.
• Angles at a point add to 360°; angles on a straight line add to 180°; vertically opposite angles are equal.
• With parallel lines: alternate angles are equal (Z-shape), corresponding angles are equal (F-shape), allied (co-interior) angles add to 180° (C-shape).
• Angles in a triangle add to 180°. The exterior angle of a triangle equals the sum of the two interior opposite angles.
• Isosceles: two equal sides and two equal base angles. Equilateral: all sides equal, every angle 60°. Right-angled: one angle of 90°.

Worked example. A triangle has angles (2x + 5)°, (3x − 10)° and (x + 5)°. Find each angle.

2x + 5 + 3x − 10 + x + 5 = 180   (angles in a triangle add to 180°)
6x = 180, so x = 30
Angles: 65°, 80°, 35°

4.2 Polygons

Learn the names: triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), octagon (8), and the special quadrilaterals.

Quadrilateral	Key properties
Parallelogram	Opposite sides parallel and equal; opposite angles equal
Rectangle	Four right angles; diagonals equal and bisect each other
Square	Four equal sides, four right angles; diagonals equal and meet at 90°
Rhombus	Four equal sides; opposite angles equal; diagonals meet at 90°
Trapezium	Exactly one pair of parallel sides
Kite	Two pairs of equal adjacent sides; one pair of equal opposite angles; diagonals meet at 90°

• Angles in a quadrilateral add to 360°.
• The interior angles of an n-sided polygon add to (2n − 4) right angles, which is (2n − 4) × 90°. An octagon: (16 − 4) × 90° = 1080°.
• A regular polygon has equal sides and equal angles. Its exterior angles add to 360°, so each exterior angle is 360° ÷ n, and interior angle = 180° − exterior angle.
• Congruent shapes are the same shape and size.

Worked example. A regular polygon has interior angles of 156°. How many sides has it?

Exterior angle = 180° − 156° = 24°
n = 360° ÷ 24° = 15 sides

Worked example. A quadrilateral has angles 100°, 2x°, (3x − 10)° and (x + 30)°. Find x.

100 + 2x + 3x − 10 + x + 30 = 360
6x + 120 = 360, so x = 40
Angles: 100°, 80°, 110°, 70° (total 360°)

4.3 Symmetry

A line of symmetry splits a shape into two mirror-image halves. The order of rotational symmetry is the number of times the shape looks the same in one full turn. A regular hexagon has 6 lines and order 6. A parallelogram (not a rhombus or rectangle) has no lines of symmetry and rotational symmetry of order 2.

4.4 Measures

• Scales: work out what one small division is worth before reading a value.
• Time: 21:45 is 9:45 pm. From 21:45 to 06:20 the next day is 2 h 15 min to midnight plus 6 h 20 min, which is 8 h 35 min.
• Bearings are measured clockwise from north and written with three figures: 070°, not 70°. If B is on a bearing of 070° from A, then A is on a bearing of 070° + 180° = 250° from B.
• Speed = distance ÷ time: 150 km in 2 h 30 min is 150 ÷ 2.5 = 60 km/h.
• Density = mass ÷ volume: 540 g with volume 200 cm³ has density 2.7 g/cm³.
• Pressure = force ÷ area (the formula is given in the exam): 600 N on 0.25 m² gives 2400 N/m².
• To convert 72 km/h to m/s: 72 × 1000 ÷ 3600 = 20 m/s.

4.5 Construction

Measure lines to the nearest millimetre and angles to the nearest degree. To draw a triangle from three sides, draw one side, then set your compasses to each of the other lengths and draw arcs from each end; the arcs cross at the third vertex.

Perpendicular bisector of AB (straight edge and compasses only):

1. Open the compasses to more than half of AB.
2. Draw arcs above and below AB from A, then from B with the same radius.
3. Join the two crossing points.

Bisector of angle ABC:

1. From B, draw an arc crossing both arms.
2. From each crossing point, draw arcs of equal radius that meet inside the angle.
3. Join B to that meeting point.

Leave all construction arcs visible. For scale drawings, convert with the scale: on a 1 : 50 000 map, 7.4 cm represents 7.4 × 50 000 = 370 000 cm = 3.7 km.

4.6 Circle properties

Know the terms centre, radius, diameter, chord, circumference, tangent, arc, sector and segment.

Both tiers:

• A tangent is perpendicular to the radius at the point of contact.
• Two tangents from an external point are equal in length.
• The line from the centre perpendicular to a chord bisects the chord (and the converse).

Worked example. A chord of length 16 cm is in a circle of radius 10 cm. How far is the chord from the centre? The perpendicular from the centre halves the chord to 8 cm, so the distance is √(10² − 8²) = √36 = 6 cm.

Higher tier only

Formal proof is not required, but you must use these:

1. The angle at the centre is twice the angle at the circumference subtended by the same arc (124° at the centre gives 62° at the circumference).
2. The angle in a semicircle is 90°.
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral (all four vertices on the circle) add to 180°.
5. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Intersecting chords. If chords AB and CD cross at X inside the circle, AX × XB = CX × XD. If lines through an external point P cut the circle at A, B and at C, D, then PA × PB = PC × PD.

Worked example. From P, one line meets the circle at A and B with PA = 5 cm and AB = 7 cm. Another meets it at C and D with PC = 4 cm. Find CD.

PB = 5 + 7 = 12
5 × 12 = 4 × PD, so PD = 15
CD = 15 − 4 = 11 cm

4.7 Geometrical reasoning

When a question says “give reasons”, write the fact in standard words next to each step, for example “alternate angles are equal”, “angles in a triangle add up to 180°”, “base angles of an isosceles triangle are equal”, “the tangent is perpendicular to the radius”. On Foundation, reasons are only required for calculations based on lines (including chords and tangents), triangles or polygons. On Higher, they may be needed in any context involving lines, polygons and circles, so learn the circle theorems word for word.

4.8 Trigonometry and Pythagoras’ theorem

Both tiers. In a right-angled triangle with hypotenuse c: a² + b² = c². A 6.5 m ladder with its foot 2.5 m from a wall reaches √(6.5² − 2.5²) = 6 m up the wall.

For acute angles in a right-angled triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. If the opposite side is 7 cm and the adjacent side is 10 cm, θ = tan⁻¹(0.7) = 35.0° (1 d.p.).

For bearings, split a journey into north and east parts. A ship sailing 12 km on a bearing of 040° goes 12 cos 40° = 9.19 km north and 12 sin 40° = 7.71 km east.

Higher tier only

• Obtuse angles: sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ. So sin 150° = sin 30° = 0.5 and cos 120° = −0.5. When sin⁻¹ gives an acute answer, check whether the obtuse angle is also possible.
• Elevation and depression are measured from the horizontal. From 40 m away, a 32° angle of elevation gives a height of 40 tan 32° = 25.0 m.
• Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. Area = ½ab sin C. All three are on the Higher formulae sheet.

Worked example. In triangle ABC, AB = 7 cm, AC = 8 cm and angle A = 50°.

BC² = 7² + 8² − 2(7)(8)cos 50° = 41.01…,  BC = 6.40 cm
Area = ½(7)(8)sin 50° = 21.4 cm²

With A = 48°, B = 67° and a = 9 cm, the sine rule gives b = 9 sin 67° ÷ sin 48° = 11.1 cm. For sides 5, 7 and 9 cm, the largest angle is opposite 9: cos C = (25 + 49 − 81) ÷ 70 = −0.1, so C = 95.7°.

3D problems. Find a right-angled triangle inside the solid. In a 6 × 8 × 5 cm cuboid, the base diagonal is √(6² + 8²) = 10 cm, the space diagonal is √(10² + 5²) = 11.2 cm, and the angle between the space diagonal and the base is tan⁻¹(5/10) = 26.6°. The angle between a line and a plane is the angle between the line and its projection onto the plane. The angle between two planes is not required.

Common errors

• Giving a bearing as two figures, or measuring it anticlockwise.
• Using 360° ÷ n as the interior angle of a regular polygon.
• Calling allied angles equal: they add to 180°.
• Leaving your calculator in radians.
• Using the cosine rule for the included angle with the wrong side opposite it.
• Writing “Z angles” or “because of the circle” instead of a proper reason.

Next steps

• Condensed recall and a self-test: revision notes
• Twelve mark-scheme style questions: practice questions
• Find your weak spots quickly: free diagnostics

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 4, Geometry and trigonometry, sections 4.1–4.8 (Foundation and Higher tier content).