Measure 3 units along one side of a corner, 4 units along the other, and the diagonal between those two marks should be exactly 5. If it is, the corner is a right angle. If it is not, it is not.
The method predates the theorem that explains it, and it is still the fastest way to check a corner on site.
Why those three numbers
Pythagoras' theorem says that in a right triangle the squares of the two short sides add to the square of the long one. 3 squared is 9, 4 squared is 16, and 9 plus 16 is 25, which is 5 squared. The relationship only holds when the angle between the short sides is exactly 90 degrees, so measuring the diagonal tests the angle.
What makes 3-4-5 special is not the right angle, which any number of triangles have, but that all three sides are whole numbers. Sets like this are called Pythagorean triples, and they are the reason the method works with a plain tape measure and no arithmetic on site.
Bigger is more accurate
Any multiple of a triple is also a triple, so 6-8-10 and 9-12-15 work identically. Use the largest that fits the space.
The reason is error, not geometry. A tape read one centimetre out over a 3 metre leg throws the angle much further than the same one centimetre over a 9 metre leg. Scaling the triangle up divides your reading error by the same factor it multiplies the sides.
Other primitive triples are available when the space suits them better. 5-12-13 is a long thin one that fits a narrow run where a 3-4-5 will not, and the Pythagorean triples calculator generates them.
Checking a whole rectangle at once
For a finished rectangle there is a quicker test: measure both diagonals. In any rectangle they are equal, and if they are not, the shape is a parallelogram no matter how square each corner looked individually.
This catches a failure the corner test can miss. Four corners each checked separately can still drift out of square as a set, and the diagonals reveal it in two measurements.
Where else the same triangle turns up
The diagonal of a rectangular opening, the true length of a rafter against its rise and run, the bracing across a frame and the depth of a ramp are all the same calculation with different names. Anything involving a right angle and two known sides is this triangle.
The Pythagoras calculator solves for the hypotenuse or either leg, so it works whether you are finding a diagonal or checking a measurement back against one. For a roof specifically, the rise and run relationship is covered in the piece on roof pitch, where the same triangle sets both the rafter length and the true roof area.