Inequalities in One Triangle

In any triangle, the sides and the angles are linked: the longest side sits opposite the largest angle, and the shortest side sits opposite the smallest angle. The word that does all the work is opposite. A side and the angle it faces are across the triangle from each other — the angle opens up toward that side.

This relationship lets you order the sides of a triangle using only its angle measures, or order the angles using only its side lengths, without measuring anything. It also comes with a bonus fact about angles that poke outside the triangle: the exterior angle inequality.

Longest side, largest angle

To order sides from angles: rank the angles from smallest to largest, then replace each angle with the side opposite it. The side opposite the smallest angle is the shortest; the side opposite the largest angle is the longest. In a triangle with angles $60^\circ$ at $A$ , $80^\circ$ at $B$ , and $40^\circ$ at $C$ , the largest angle is at $B$ , so the longest side is $\overline{AC}$ — the side that does not touch $B$ .

To order angles from sides, run the same idea in reverse: rank the side lengths, then replace each side with the angle opposite it. The angle opposite the longest side is the largest.

The trap answer is always the side next to the big angle. A side opposite an angle shares no endpoint with that angle's vertex. If the sides you are comparing all touch the vertex you are looking at, you are pairing them wrong.

Find the missing angle first

Many problems give only two angle measures. Before ordering anything, find the third angle with the Triangle Angle-Sum: the three angles add to $180^\circ$ . The missing angle is often the largest one — the problem is written that way on purpose, to catch students who rank only the two printed angles.

The exterior angle inequality

Extend one side of a triangle past a vertex and you create an exterior angle. The exterior angle inequality says an exterior angle is greater than either remote interior angle — the two interior angles far away from it, not the one it sits next to.

Nothing is guaranteed about the comparison with the adjacent interior angle; the exterior angle and its neighbor simply add to $180^\circ$ . So if $\angle 1$ is exterior with remote interior angles $\angle 2$ and $\angle 3$ , the only safe conclusions are $m\angle 1 > m\angle 2$ and $m\angle 1 > m\angle 3$ . The figure below shows the exterior angle $1$ and its two remote interior angles, $2$ and $3$ .

Worked examples

Example 1: order the sides from the angles

In $\triangle ABC$ , $m\angle A = 60^\circ$ , $m\angle B = 80^\circ$ , and $m\angle C = 40^\circ$ . List the sides from shortest to longest.

Rank the angles from smallest to largest

40^\circ < 60^\circ < 80^\circ

Replace each angle with its opposite side

\angle C \to \overline{AB},\; \angle A \to \overline{BC},\; \angle B \to \overline{AC}

Write the sides in the same order

\overline{AB} < \overline{BC} < \overline{AC}

Answer: $\overline{AB}$ , $\overline{BC}$ , $\overline{AC}$

Example 2: two angles given, find the longest side

In $\triangle DEF$ , $m\angle D = 48^\circ$ and $m\angle E = 65^\circ$ . Which side is the longest?

Find the third angle

m\angle F = 180^\circ - 48^\circ - 65^\circ = 67^\circ

Identify the largest angle

67^\circ > 65^\circ > 48^\circ

The longest side is opposite the largest angle

\overline{DE}

Answer: $\overline{DE}$

Example 3: order the angles from the sides

In $\triangle TUV$ , $TU = 6$ , $UV = 11$ , and $TV = 9$ . List the angles from smallest to largest.

Rank the sides from shortest to longest

6 < 9 < 11

Replace each side with its opposite angle

\overline{TU} \to \angle V,\; \overline{TV} \to \angle U,\; \overline{UV} \to \angle T

Write the angles in the same order

\angle V < \angle U < \angle T

Answer: $\angle V$ , $\angle U$ , $\angle T$

Try one yourself

Common questions

Is the longest side next to the largest angle?

No — it is across from it. The largest angle opens up toward the longest side, so the longest side is the one that does not touch the largest angle's vertex.

What if two angles are equal?

Then the sides opposite them are equal too — the triangle is isosceles. The ordering still works; the two equal angles just tie, and so do their opposite sides.

Do I need both the side lengths and the angle measures?

No. Either set alone determines the order of the other. Angles order the sides, and sides order the angles — that is the whole point of the theorem.

Which angles does an exterior angle beat?

Only the two remote interior angles — the ones at the other two vertices. It tells you nothing certain about the interior angle right next to it; those two just add to $180^\circ$ .

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Inequalities in One Triangle

Longest side, largest angle

Find the missing angle first

The exterior angle inequality

Worked examples

Example 1: order the sides from the angles

Example 2: two angles given, find the longest side

Example 3: order the angles from the sides

Try one yourself

Common questions

Keep learning

Indirect Proof & the Triangle Inequality

Perpendicular Bisectors & the Circumcenter

Medians & the Centroid

Angles of Triangles