Inequalities in Triangles

In any triangle, sides and angles are paired by size. The longest side always lies opposite the largest angle, and the shortest side lies opposite the smallest angle. So in triangle ABC, if side AC is the longest, then angle B (the angle facing AC) is the largest. This pairing works both ways: it does not matter whether you start by comparing the sides or the angles, the order you get is the same.

Theorem 1 says: if two sides of a triangle are unequal, the greater side has the greater angle opposite to it. Its converse, Theorem 2, says: if two angles are unequal, the greater angle has the greater side opposite to it. These are proved formally using the exterior-angle property and the isosceles-triangle result. Together they let you arrange the sides of a triangle in order just by looking at the angles, and the other way round.

The sum of any two sides of a triangle is always greater than the third side. The reason is simple: a straight line is the shortest path between two points, so going from A to C directly (side AC) must be shorter than going from A to B and then B to C. From this we also get a difference rule: the third side is greater than the difference of the other two. So for a triangle to exist at all, every side must be smaller than the sum of the other two.

Of all the lines that can be drawn from a point to a given line, the perpendicular is the shortest. If P is a point and AB is a line, the perpendicular from P to AB beats every slanting line, because each slanting line is the hypotenuse of a right triangle and the hypotenuse is always the longest side. This is why height in a triangle is measured along the perpendicular.