the converse of the Pythagorean theorem

The Pythagorean theorem says 'right triangle, therefore a^2 + b^2 = c^2'. Its converse runs the implication backwards: if the three sides of a triangle happen to satisfy a^2 + b^2 = c^2 (with c the longest), then the triangle must be right-angled, with the right angle opposite the longest side. It lets you test for a right angle using only a tape measure — no protractor required.

Concretely: measure the three sides, identify the longest as c, and check whether the sum of the squares of the other two equals c^2. If it does, the angle opposite c is exactly 90 degrees. Sides 6, 8, 10 satisfy 36 + 64 = 100, so that triangle is right-angled. This is precisely how builders square a corner with the '3-4-5' trick: a triangle with those sides must contain a right angle, so a wall laid out that way is guaranteed perpendicular to the floor.

The same comparison classifies any triangle. If a^2 + b^2 is greater than c^2, the largest angle is acute and the triangle is acute; if a^2 + b^2 is less than c^2, the largest angle is obtuse. Only equality gives a right triangle. A logical caution worth remembering from foundations: a theorem and its converse are different statements — one being true does not automatically make the other true — but here both the Pythagorean theorem and its converse happen to hold, and each must be proved on its own.