Angles at a point and on a line

The first job of geometry is to find missing angles using rules you can quote:

• Angles on a straight line add to 180°.
• Angles around a point add to 360°.
• Vertically opposite angles (where two lines cross) are equal.
• Angles in a triangle add to 180°; in a quadrilateral to 360°.

Reasoning matters (4.7): in the exam you often must state the rule you used — e.g. "angles on a straight line sum to 180°" — to earn full marks.

Your turn — missing angle

Angles in parallel lines

When a straight line (a transversal) crosses two parallel lines, special equal and supplementary pairs appear:

• Corresponding angles (in "F" shapes) are equal.
• Alternate angles (in "Z" shapes) are equal.
• Co-interior / allied angles (in "C/U" shapes) sum to 180°.

Name that angle pair

Naming triangles & quadrilaterals

You must know the properties of each special shape — sides, angles and symmetry:

• Equilateral triangle — 3 equal sides, all angles 60°.
• Isosceles triangle — 2 equal sides and 2 equal base angles.
• Scalene triangle — all sides and angles different.
• Square, rectangle, parallelogram, rhombus, trapezium, kite — each with its own side, angle and diagonal properties.

Watch out: in an isosceles triangle the two equal angles sit under the two equal sides — not next to the "odd" side.

Interior & exterior angles

A polygon with n sides can be split into n − 2 triangles, so its interior angles add up to:

Common trap: the exterior angles always sum to 360° (not the interior angles). Interior + exterior at each vertex = 180°.

Your turn — regular polygon

Interior angle of a hexagon

Line & rotational symmetry

Two kinds of symmetry are tested:

• Line symmetry — the number of mirror lines that fold a shape exactly onto itself. A square has 4; a rectangle has 2.
• Rotational symmetry — how many times a shape looks identical in one full turn. A square has order 4; an equilateral triangle has order 3.

Constructions and loci

Using only a ruler and a pair of compasses (leave your arcs showing — they earn marks), you must be able to construct:

• The perpendicular bisector of a line (the locus of points equidistant from its two ends).
• The bisector of an angle (the locus of points equidistant from its two arms).
• A perpendicular from a point to a line.

A locus is the set of all points obeying a rule. The locus of points a fixed distance r from a point is a circle of radius r; a fixed distance from a line is a "racetrack" of two parallels joined by semicircles.

Circle theorems

On the Higher tier you use these circle rules (with reasons):

• The angle at the centre is twice the angle at the circumference on the same arc.
• The angle in a semicircle is 90°.
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral sum to 180°.
• A tangent meets a radius at 90°; two tangents from a point are equal.
• Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
• A radius perpendicular to a chord bisects the chord.

Angle in a semicircle

Pythagoras' theorem

In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two:

Misconception: the hypotenuse is the side opposite the right angle, not just "the sloping one." To find a shorter side, rearrange to a² = c² − b² and subtract.

Your turn — Pythagoras

SOHCAHTOA

In a right-angled triangle, label the sides relative to the angle θ: the hypotenuse (opposite the right angle), the opposite (facing θ) and the adjacent (next to θ):

Misconception: you must pick the ratio that uses the two sides you know/want. Using sin when you only have opposite & adjacent (that's tan) is the classic slip.

Your turn — SOHCAHTOA

Which ratio or rule?

Tap the correct method for each triangle situation.

Bearings

A bearing gives a direction as an angle measured:

• clockwise,
• from north,
• always written with three figures (e.g. 060°, 135°, 285°).

The bearing of B from A and the bearing of A from B differ by 180° (a "back bearing"). Trigonometry and Pythagoras are often used to solve bearing problems.

The sine rule

For non-right-angled triangles (Higher tier), use the sine rule when you have a matching side–angle pair:

Use the sine rule when you know a side and its opposite angle. If instead you know two sides and the angle between them, you'll need the cosine rule (next).

The cosine rule & ½ab sin C

The cosine rule handles the two cases the sine rule can't: two sides and the included angle, or all three sides:

The area of any triangle (given two sides and the angle between them) is:

Your turn — area of a triangle

Trigonometry in 3D

Higher-tier problems apply Pythagoras and trigonometry in three dimensions — for example finding the length of the space diagonal of a cuboid, or the angle between a line and a plane.

Strategy: identify a right-angled triangle inside the solid, solve it, then use the answer in the next triangle. The angle between a line and a plane sits between the line and its "shadow" on the plane.

Perimeter, area, circles & sectors

You must know the standard 2D formulas:

• Rectangle area = base × height; triangle = ½ × base × height.
• Parallelogram = base × height; trapezium = ½(a + b) × h.
• Circle: circumference = πd = 2πr; area = πr².

Your turn — area of a circle

Surface area & volume of solids

For a prism (uniform cross-section), volume = area of cross-section × length. Key formulas:

• Cuboid: V = length × width × height.
• Cylinder: V = πr²h; curved surface area = 2πrh.
• Cone: V = ⅓πr²h; curved surface area = πrl.
• Sphere: V = 4⁄3 πr³; surface area = 4πr².
• Pyramid: V = ⅓ × base area × height.

Your turn — volume of a prism

Match each shape to its formula

Tap a shape on the left, then its matching formula on the right.

Similar shapes & scale factors

Congruent shapes are identical (same size & shape). Similar shapes are the same shape but enlarged by a linear scale factor k (equal ratios of corresponding sides).

Misconception: the area scale factor is k² (not k), and the volume scale factor is k³. Double the lengths ⇒ 4× the area, 8× the volume.

Your turn — area scale factor

Volume scale factor

The rules & formulas to know

You've covered all of Edexcel 4MA1 Section 4 — angles, polygons, symmetry, bearings, constructions & loci, circle theorems, Pythagoras, trigonometry, mensuration, 3D solids and similarity. Press Finish to see your score.