Converse of Pythagoras: proving a right angle

If a²+b²=c² for the sides of a triangle, then the angle opposite c is 90°.

Key formulas

Pythagoras starts from a right angle. The converse starts from the side relation and concludes the right angle.

Sides 5, 12, 13. 25+144=169, so the angle between sides 5 and 12 is 90°.

5, 12, 14 does not work. Check the arithmetic before claiming a right triangle.

Problem. A triangle has sides 9, 12 and 15. Is it right-angled?

Solution. Check 9² + 12² = 81 + 144 = 225 = 15². Yes — the angle opposite the side of length 15 is 90°.

Always test the longest side as c.

Sides 7, 24, 25. The triangle is:

No — the side sum would fail for any labelling.