D(5, 7), E(4, 3), and F(8, 2) form the vertices of a triangle. What is m∠DEF?

We calculate side lengths by Pythagorean theorem.

\(DE = \sqrt{17}\\ EF = \sqrt{17}\\ DF = \sqrt{34}\)

Obviously, \(DE^2 + EF^2 = DF^2\)

So by converse of Pythagorean theorem, m∠DEF = 90^{o}.