A constraint on what side lengths can actually form a triangle — not every set of three numbers works.
The Mnemonic
Not Every Trio of Lengths Makes a Triangle
The Triangle Inequality Theorem states: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This has to be true for ALL THREE possible pairings of the sides — not just one pairing.
This means you can't just pick any three numbers and assume they form a triangle. Three lengths of 2, 3, and 10, for example, can never close into a triangle — the two shorter sides (2 + 3 = 5) simply can't stretch far enough to meet the endpoints of a side that's 10 long.
Sides of 3 and 4 can never reach far enough to close a triangle with a third side of 10 — 3 + 4 = 7, which is less than 10
💡 Memory Trick
"A constraint on what side lengths can form a triangle." Picture two rigid sticks hinged at one end, trying to reach across to close a triangle with a third stick. If the two shorter sticks combined are too short to reach past the third stick's endpoints, the triangle physically can't close — that's exactly what the inequality is checking.
Why It Works
The Physical Reason Behind the Rule
Think of building a triangle out of three physical rods, hinged at their ends. If you lay two of the rods flat, end to end, in a straight line, together they span a length exactly equal to their sum. To form an actual triangle (a closed shape with a bend), those two rods have to angle inward slightly rather than lying perfectly flat — which makes the straight-line distance between their two free ends slightly LESS than their combined length.
So the third side — which is exactly that straight-line distance between the two free ends — must always be shorter than the sum of the other two. If the "third side" you're given is equal to or longer than that sum, there's no way to angle the first two rods enough to reach it; the shape simply won't close.
Using It In A Proof
Testing Whether Three Lengths Form a Triangle
Checking this theorem is mostly about knowing the shortcut: you don't actually need to check all three pairings individually if you're smart about which pair you pick.
1
Add the two SHORTEST sides
If the two shortest sides sum to more than the longest side, all three pairings will automatically work — you only need to check this one combination, not all three.
2
Compare to the longest side
If (shortest + middle) > longest, the three lengths form a valid triangle. If (shortest + middle) ≤ longest, they don't.
3
For finding a range instead of testing specific numbers
Given two sides, the third side must be greater than their difference and less than their sum — this creates a range rather than a single test.
Sides of 5 and 9: the third side must be between 4 (9−5) and 14 (9+5), not including the endpoints.
Full Worked Example
Finding the Range of a Missing Third Side
Given: A triangle has two sides of length 7 and 12. Find: the range of possible values for the third side, x.
1
Find the upper bound
The third side must be less than the sum of the other two: x < 7 + 12 = 19.
2
Find the lower bound
The third side must be greater than the difference of the other two: x > 12 − 7 = 5.
3
State the full range
5 < x < 19.
4
Verify with a specific value
Try x = 10 (inside the range): sides 7, 10, 12. Check all three pairings — 7+10=17>12 ✓, 7+12=19>10 ✓, 10+12=22>7 ✓. A valid triangle.
Try x = 5 (the excluded boundary): 7 + 5 = 12, which is NOT greater than 12 — confirming why the range uses strict inequality (< and >), not ≤ and ≥.
🎯 Quick Worked Example
Can side lengths 4, 6, and 11 form a triangle?
1
Add the two shortest sides. 4 + 6 = 10.
2
Compare to the longest side. 10 is NOT greater than 11.
3
Conclude. No — since the two shortest sides don't sum to more than the longest, these three lengths cannot form a triangle.
📌 Exam Application
You only ever need to check ONE pairing to test validity — always the two shortest sides against the longest. If that pairing passes, every other pairing automatically passes too, since the other two combinations involve the longest side plus something else, which is always safely larger.
⚠️ Most Common Triangle Inequality Theorem Mistakes
Trap 1 — Checking the wrong pairing: Only checking (shortest + longest) vs. middle, or a similarly mismatched pairing, can give a false pass — always specifically add the TWO SHORTEST sides and compare that sum to the longest side.
Trap 2 — Using ≤ instead of strict inequality: The sum of two sides must be STRICTLY greater than the third — if they're exactly equal (like 5, 7, 12, where 5+7=12), the three segments lie flat in a straight line instead of closing into a triangle, so this case fails, not passes.
✓ Quick Self-Test
1) State the Triangle Inequality Theorem. 2) Do side lengths 5, 5, and 11 form a triangle? Why or why not? 3) Given two sides of 6 and 15, what is the range of possible values for the third side? 4) Why do you only need to check one pairing (shortest two vs. longest) rather than all three? 5) Why does the theorem use strict inequality rather than 'greater than or equal to'?