Cut a 13×5 'triangle' into four pieces, slide them around, reassemble the same 'triangle' — with one unit square missing. Nothing is hidden and no piece changes size. The con is in a word you just accepted without checking, and the reason the con works so well is a one-line identity about Fibonacci numbers.
Nobody checks the diagonal. You were handed the word triangle and you took it for free — and once the outline is a triangle its area is 13·5/2 = 32.5, so four pieces adding to 32 are a scandal. The outline was never a triangle. Its long edge has a kink in it, bending down in one arrangement and up in the other.
The pieces are the honest party. Not one of them is bent, stretched, flipped or trimmed — they only slide. What changes is the region they enclose: 32 cells one way, 33 the other. The pieces can only supply 32, so the second arrangement has to show one cell of nothing, and it does.
Why doesn't your eye object? Because the two slopes that meet at the kink, 3/8 = 0.375 and 2/5 = 0.400, are 1.25° apart — thinner than the pencil line you would draw them with. Close enough to lie to the eye; nowhere near close enough to lie to the grid.