Worksheets · Geometry

Triangle inequality theorem worksheet

Any two sides together have to beat the third, which pins the missing side between the difference and the sum of the other two.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Triangle Inequality

Date Period

Give the range for the third side.

  1. Give the range for the third side132x
  2. Give the range for the third side1220x
  3. Give the range for the third side207x
  4. Give the range for the third side1315x
  5. Give the range for the third side1020x
  6. Give the range for the third side1011x
  7. Give the range for the third side49x
  8. Give the range for the third side64x
  9. Give the range for the third side113x
  10. Give the range for the third side916x
  11. Give the range for the third side75x
  12. Give the range for the third side210x
  13. Give the range for the third side1317x
  14. Give the range for the third side1412x
  15. Give the range for the third side165x
  16. Give the range for the third side1510x
  17. Give the range for the third side1620x
  18. Give the range for the third side616x
  19. Give the range for the third side48x
  20. Give the range for the third side96x

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Triangle InequalityAnswer keyVersion 1

Date Period

Give the range for the third side.

  1. Give the range for the third side132x11<x<15
  2. Give the range for the third side1220x8<x<32
  3. Give the range for the third side207x13<x<27
  4. Give the range for the third side1315x2<x<28
  5. Give the range for the third side1020x10<x<30
  6. Give the range for the third side1011x1<x<21
  7. Give the range for the third side49x5<x<13
  8. Give the range for the third side64x2<x<10
  9. Give the range for the third side113x8<x<14
  10. Give the range for the third side916x7<x<25
  11. Give the range for the third side75x2<x<12
  12. Give the range for the third side210x8<x<12
  13. Give the range for the third side1317x4<x<30
  14. Give the range for the third side1412x2<x<26
  15. Give the range for the third side165x11<x<21
  16. Give the range for the third side1510x5<x<25
  17. Give the range for the third side1620x4<x<36
  18. Give the range for the third side616x10<x<22
  19. Give the range for the third side48x4<x<12
  20. Give the range for the third side96x3<x<15

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Two sides of a triangle are 7 and 10. Give the range for the third side

  1. It must be less than 7 plus 10.
  2. And more than 10 minus 7.

The answer is 3<x<17.

Where students go wrong

Giving only the upper limit. A side of 1 with sides 7 and 10 does not close up either, so the difference is just as real a limit as the sum.

Also called possible side lengths or third side of a triangle.

Questions about these worksheets

Yes. Every triangle inequality theorem sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the triangle inequality theorem questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Giving only the upper limit. A side of 1 with sides 7 and 10 does not close up either, so the difference is just as real a limit as the sum.