Apply the Pythagorean Theorem to find a missing side of a right triangle, and use its converse to classify a triangle as acute, right, or obtuse.
The Pythagorean Theorem is probably the most famous fact in geometry: in a right triangle with legs a and b and hypotenuse c (the side opposite the right angle, always the longest side), a^2 + b^2 = c^2. It works ONLY for right triangles, but it works for every single one of them, and it lets you find any one side once you know the other two. A handful of "Pythagorean triples" -- whole-number solutions like 3-4-5, 5-12-13, and 8-15-17 (and any multiple of one, like 6-8-10) -- are worth recognizing on sight, since they save a lot of arithmetic.
The converse is just as useful: if the three side lengths of a triangle satisfy a^2 + b^2 = c^2 (with c the longest side), the triangle IS a right triangle -- a way to test for a right triangle using only lengths, no protractor needed. Push the comparison further and it classifies any triangle: if c^2 is less than a^2 + b^2, the triangle is acute; if c^2 is greater than a^2 + b^2, the triangle is obtuse; only exact equality gives a right triangle.
Mastery looks like: They apply the Pythagorean Theorem to find a missing side of a right triangle and use its converse, including the acute/obtuse extension, to classify a triangle from its three side lengths.
Common stumbles: Adding the legs and squaring the sum instead of squaring each leg first; using the wrong side as c when the longest side is not obviously labeled as the hypotenuse.