Why the Squares Add

The motion: “a² + b² = c² is true because it works for every right-angled triangle anyone has measured.”

Your opponent: Euclid · Course: Pearson Edexcel International GCSE Mathematics (4MA1) · Assessment objectives: AO1, AO2

That is verification, not proof — and it is Proposition 47 of his own first book.

What you are asked to argue

State the theorem, justify why it MUST hold, and then find the invalid step in the argument Euclid puts to you.

The statement of the theorem, including which side is the hypotenuse
At least one proof: the rearrangement of four congruent triangles inside a square, or the similar-triangles proof
The converse and how it is used to test whether a triangle is right-angled
Where the theorem fails — non-right-angled triangles, and what replaces it

Disagreeing with the figure is not an error. Any of these can score full marks if argued well:

Any valid proof, including one not taught on this course, provided every step is justified
The argument that measurement gives grounds for BELIEF while proof gives grounds for CERTAINTY — a precise and creditable position

Wrong: Checking several triangles proves the theorem.
The correction: Examples can only disprove. No finite number of confirming cases establishes a general claim; one counterexample would destroy it.
Wrong: c is always the longest side, so you can always use the formula.
The correction: c is the side opposite the right angle. In a triangle with no right angle the relation simply does not hold, and the longest side is irrelevant.
Wrong: It works because it is the rule.
The correction: A rule is a statement of what is true. The justification is what shows that it is.

A complete proof with every step justified, the proof/verification distinction stated precisely, and the invalid step located and explained.

Read the life before you argue with it — the case you are about to meet was built by a real person, over a real career: