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Answer: one triangle, with angles $100^\circ, 64^\circ, 16^\circ$


We want to find angles $x^\circ$, $y^\circ$ and $z^\circ$ so that $$x^2+y^2+z^2=180.$$
We know that the largest angle must be smaller than $180^\circ$ and bigger than $180^\circ/3 = 60^\circ$. So the largest angle must be $169^\circ, 144^\circ, 121^\circ, 100^\circ, 81^\circ$ or $64^\circ$.

largest angle medium angle smallest angle all squares?
$169$ $9$ $2$ no
  $4$ $7$ no
$144$ $36$ $0$ not a triangle
  $25$ $11$ no
  $16$ $20$ no
$121$ $49$ $10$ no
  $36$ $23$ no
$100$ $64$ $16$ yes!
  $49$ $31$ no
  $36$ $44$ no
$81$ $81$ $18$ no
  $64$ $35$ no
  $49$ $50$ no
$64$ $64$ $52$ no
  $49$ $67$ no

So there is exactly one triangle with all three angles perfect squares, viz a triangle with angles $10^2,8^2,4^2$

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.