Triangle Inequality Theorem
State and apply the Triangle Inequality Theorem: the sum of any two sides must exceed the third side. Determine whether given side lengths form a valid triangle and find the range of possible lengths for a third side.
A lesson of Geometry, in Triangle Relationships, about 10 minutes.
By the end of this lesson, you will be able to apply the Triangle Inequality Theorem to determine if three lengths can form a triangle, use the inequality relationships between sides and angles, and find the range of possible lengths for a missing side.
The Big Idea
Not every three lengths can form a triangle. The sides must "reach each other" — if one side is too long, the triangle collapses. The Triangle Inequality Theorem gives the exact condition for triangle existence and has a natural geometric interpretation: the shortest path between two points is a straight line.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For sides $a$, $b$, $c$:
$$a + b > c, \quad a + c > b, \quad b + c > a$$
In practice, you only need to check that the sum of the two shorter sides exceeds the longest side.
Example 1 — Can a Triangle Be Formed?
(a) Sides: 5, 7, 9
$5 + 7 = 12 > 9$ ✓ → triangle possible
(b) Sides: 3, 4, 8
$3 + 4 = 7 < 8$ ✗ → no triangle possible
(c) Sides: 6, 6, 12
$6 + 6 = 12$, not $> 12$ ✗ → degenerate (a line, not a triangle)
Example 2 — Range for a Missing Side
Two sides of a triangle are 5 and 9. Find the range of possible lengths for the third side $x$.
Inequalities:
$$5 + 9 > x \implies x < 14$$ $$5 + x > 9 \implies x > 4$$ $$9 + x > 5 \implies x > -4 \text{ (always true for positive } x\text{)}$$
$$4 < x < 14$$
The third side must be strictly between 4 and 14.
Side-Angle Inequality
In a triangle, the larger angle is opposite the longer side.
$$AB > BC \iff \angle C > \angle A$$
Conversely, the larger side is opposite the larger angle.
Corollary: In a right triangle, the hypotenuse is the longest side (opposite the 90° angle, which is the largest angle).
Example 3 — Ordering Sides and Angles
$\triangle ABC$: $\angle A = 50°$, $\angle B = 80°$, $\angle C = 50°$.
Order the sides from shortest to longest.
Largest angle: $\angle B = 80°$ → opposite side $\overline{AC}$ is longest.
Equal angles: $\angle A = \angle C = 50°$ → opposite sides $\overline{BC} = \overline{AB}$.
Order: $BC = AB < AC$.
Example 4 — Ordering Angles from Sides
$\triangle PQR$: $PQ = 5$, $QR = 7$, $PR = 9$.