Triangle Inequality Rules

Not every combination of three lengths can form a triangle. There are rules — called triangle inequalities — that determine what side lengths are possible.

The Triangle Inequality Theorem

For any triangle with sides \(a\), \(b\), and \(c\):

Each side must be less than the sum of the other two, and greater than their difference.

In other words, for all three combinations to hold:

$$a < b + c \quad b < a + c \quad c < a + b$$

Triangle with sides a, b, and c labeled, showing the three triangle inequality relationships

This makes physical sense. If you try to build a triangle and one side is longer than the other two combined, those two sides can't bridge the gap — they don't reach each other.

Can These Be Triangle Sides?

Example: AB = 4, BC = 5, AC = 1. Do these form a valid triangle?

Check all three inequalities:

  • Is 4 < 5 + 1? Yes, 4 < 6. ✓
  • Is 1 < 4 + 5? Yes, 1 < 9. ✓
  • Is 5 < 1 + 4? No, 5 = 5. ✗

The third check fails. Sides of 4, 5, and 1 do not form a triangle. (If the two shorter sides add to exactly the length of the third, you get a flat line, not a triangle.)

Finding the Range of a Missing Side

Example: A triangle has sides of length 3, 9, and \(x\). How many integer values of \(x\) are possible?

From the inequality rules, \(x\) must satisfy:

$$x < 3 + 9 = 12 \quad \Rightarrow \quad x < 12$$ $$x > 9 - 3 = 6 \quad \Rightarrow \quad x > 6$$

So: \(6 < x < 12\)

The integers in that range (not including the endpoints) are: 7, 8, 9, 10, 11 — five values.

A Multiple-Choice Problem

City X to City Y is 100 miles. City Y to City Z is 450 miles. What is a possible distance from City X to City Z?

(a) 150 miles (b) 250 miles (c) 350 miles (d) 400 miles

Applying the triangle inequality to XZ:

$$XZ > 450 - 100 = 350 \quad \text{and} \quad XZ < 450 + 100 = 550$$

So XZ must be between 350 and 550 miles (not including 350). The only answer in that range is (d) 400 miles.

Angles and Sides Together

The triangle inequalities also describe the relationship between a triangle's angles and its sides:

  • The longest side is opposite the largest angle
  • The shortest side is opposite the smallest angle
  • If two sides are equal, the angles opposite them are also equal (isosceles triangles)

Example: In isosceles triangle ABC, ∠A = 30° and AB = AC. Find the shortest side.

Since AB = AC, angles B and C are equal. The three angles sum to 180°:

$$30 + \angle B + \angle C = 180$$ $$\angle B = \angle C = 75°$$

Angle A (30°) is the smallest, so side BC — the side opposite A — is the shortest side.

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