Triangle Inequalities

Not every set of three lengths makes a triangle. The two shorter ones have to reach across the longest, so:

any two sides added together must exceed the third.

For 3, 4 and 8: 3 + 4 = 7, which is less than 8. Those cannot form a triangle. The short sides simply do not reach.

Testing all three pairs works, but there is a shortcut: check only the two smallest against the largest. If that passes, the others automatically do.

There is also an ordering rule. The longest side is always opposite the largest angle, and the shortest side opposite the smallest. So in a triangle with angles 30°, 60° and 90°, the sides go in that same order of size.

The Triangle Inequality Theorem states that for any triangle, a + b > c for every labelling of the sides. Equality would collapse the triangle into a segment, so the inequality is strict.

Given two sides a and b, the third satisfies |a − b| < c < a + b, which is the standard way to state the range of possible third sides.

The side-angle ordering theorem states that in any triangle, the larger angle lies opposite the longer side, and conversely. Both directions are used: side lengths order the angles, and angle measures order the sides.

The Hinge Theorem (SAS Inequality) extends this to two triangles: given two pairs of congruent sides, the triangle with the larger included angle has the longer third side. Its converse orders the included angles from the third sides.

These results are typically proved indirectly, by assuming the negation and deriving a contradiction with the Exterior Angle Theorem — which is why triangle inequality is the standard setting for introducing proof by contradiction.