Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Triangles & Quadrilaterals

In one sentence. The special triangles and quadrilaterals, their angle totals, symmetries and diagonal properties — the shape vocabulary of every geometry question.

Geometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Euclid. Learn Triangles & Quadrilaterals, teach it, then defend it against Euclid.

Where this sits in the specification

Specification 4MA1, 4.1 Triangles · 4.3 Symmetry. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.1 — Triangles
  • 4.3 — Symmetry

1 · Understand it

What it is

Quadrilateral angles sum to 360°. The special quadrilaterals each carry a bundle of properties: a parallelogram has opposite sides parallel and equal, opposite angles equal, co-interior angles supplementary; a rhombus adds four equal sides and perpendicular bisecting diagonals; a kite has one line of symmetry and perpendicular diagonals; a trapezium has one pair of parallel sides.

Why it works

Any quadrilateral splits into two triangles along a diagonal — two 180°s make the 360° total. A parallelogram's angle facts are the parallel-lines facts in disguise: adjacent angles are co-interior (sum 180°). A rhombus's diagonals meet at right angles because its four equal sides make four congruent right triangles — which is also why Pythagoras computes its side from its diagonals.

When a question is asking for it

Angle chases in quadrilaterals; 'how many lines of symmetry?'; and any question naming a special shape — the name is a licence to use every property in its bundle, and questions rely on you cashing it.

How to hold it in your head

Learn the shapes as a family tree: square ⊂ rhombus ⊂ parallelogram, and square ⊂ rectangle ⊂ parallelogram. Every property of a parent is inherited by the child — a square has everything a rhombus has, plus right angles.

2 · The mistakes that cost marks

  • Quadrilateral angles summing to 180°. Two triangles' worth: 360°.
  • A parallelogram's adjacent angles equal. OPPOSITE angles are equal; adjacent ones sum to 180°.
  • A kite with two lines of symmetry. One only — along the diagonal joining the unequal-pair vertices. (Two would make it a rhombus.)
  • Assuming an isosceles triangle's given angle is the apex. If a triangle 'has an angle of 50°', the 50° could be apex OR base — two cases.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Three angles of a quadrilateral are 85°, 97° and 102°. Find the fourth.

  1. Angle sum of a quadrilateral: 360°.
  2. 85 + 97 + 102 = 284.
  3. Fourth angle: 360 − 284 = 76°.

Answer: 76°

A rhombus has diagonals of length 12 cm and 16 cm. Find its side length.

  1. Diagonals of a rhombus bisect each other at right angles: half-diagonals 6 and 8.
  2. Each side is the hypotenuse of a 6-8 right triangle.
  3. Side = √(6² + 8²) = √100 = 10 cm.

Answer: 10 cm

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Three angles of a quadrilateral are 85°, 97° and 102°. Work out the fourth angle. (answer in °)
  2. Grade 7 — How many lines of symmetry does a kite have?
  3. Grade 8 — One angle of a parallelogram is 65°. Work out the size of an ADJACENT angle. (answer in °)
  4. Grade 9 — An isosceles triangle has an angle of 50°. There are two possible triangles. Work out the LARGER possible size of a base angle. (answer in °)
  5. 9+ — A rhombus has sides of length 13 cm. One of its diagonals is 10 cm long. Work out the AREA of the rhombus. (answer in cm²)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Triangles & Quadrilaterals, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Triangles & Quadrilaterals works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Triangles & Quadrilaterals against Euclid — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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Carry on through the course