Triangle Inequality Theorem
The sum of any two sides of a triangle must be greater than the third side. This is the fundamental rule for when three lengths can form a triangle.
Introduction
Can you form a triangle with sides 3, 4, and 100? No — the two short sides can't “reach” each other. The Triangle Inequality Theorem tells us exactly when three lengths work.
Past Knowledge
Triangle basics (5.1). “Shortest path” intuition. Segment addition.
Today's Goal
State, prove, and apply the Triangle Inequality Theorem.
Future Success
Side-angle ordering (6.2.2), Hinge Theorem (6.2.3), indirect proof.
Key Concepts
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
For a triangle with sides : , ,
Shortcut Check
You only need to check: the sum of the two shorter sides > the longest side. If that passes, all three inequalities hold automatically.
Theorem & Proof
Two-Column Proof: Triangle Inequality Theorem
Prove: In ,
Strategy: Extend side to create an isosceles triangle, then use the angle-side relationship.
Extend CB past B, create isosceles △ABD, use angle-side relationship to show DC > AC.
| # | Statement | Reason |
|---|---|---|
| 1 | Extend past to point so that | Ruler Postulate (we can mark any length on a ray) |
| 2 | is isosceles with | Definition of isosceles (step 1) |
| 3 | Isosceles Triangle Theorem (base angles are equal) | |
| 4 | Angle Addition Postulate: includes as an extra part, so | |
| 5 | Steps 3 and 4 combined () | |
| 6 | In : | In a triangle, the side opposite the larger angle is longer |
| 7 | (step 1); so |
∎ The sum of any two sides exceeds the third. Repeat the argument for the other pairs.
Worked Examples
Can These Form a Triangle?
Can lengths 5, 8, 14 form a triangle?
Shortcut: check the two shorter sides: . ✗
No. 5 + 8 = 13, which is NOT greater than 14.
Finding the Range of a Third Side
Two sides of a triangle are 7 and 11. Find the possible lengths of the third side.
Let = third side. The inequalities give us:
→
Triple Check
For which values of do lengths form a triangle?
Three inequalities (all three pairs):
→ → ✓ (always true)
→ →
→ → ✓ (always true for positive )
Also: all sides positive → , →
Common Pitfalls
Allowing Equality
It must be strictly greater, not ≥. If , the three points are collinear (a degenerate “triangle” with no area).
Checking Only One Pair
There are three inequalities. The shortcut (two shortest > longest) works, but if you check a random pair and it passes, you haven't proven the triangle forms.
Real-Life Applications
Network Routing — Shortest Path
The triangle inequality is the foundation of shortest-path algorithms in computer science. It guarantees that the direct route between two nodes is never longer than any two-hop route — a fact used by GPS, internet routing, and logistics optimization.
Manufacturing — Beam Lengths
When building triangular trusses, manufacturers check that the three ordered beam lengths satisfy the triangle inequality before cutting materials, preventing waste from impossible configurations.
Practice Quiz
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