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Triangle Inequality Theorem Worksheets

Decide whether side lengths make a triangle and find the possible range of a third side, using whole numbers or tenths.

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Check each triple, then find the strict range for each missing third side.

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Compare the two shorter sides first

A nondegenerate triangle needs the sum of its two shortest sides to be greater than the longest. Equality makes a straight line.

Learning objective

What this activity builds

Apply the strict triangle inequality to side-length feasibility and third-side ranges.

Worked example

Two sides are 4.2 and 6.1. What can the third side x be?

1.9 < x < 10.3.

Subtract the known lengths for the lower bound and add them for the upper bound. Neither endpoint makes a nondegenerate triangle.

Teaching notes

Make the practice count

  • Compare the sum of the two shortest sides with the longest.
  • A sum equal to the longest side makes a straight line, not a triangle.
  • Keep third-side bounds strict, including when the lower bound is zero.

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The two sections stay on separate pages: triangle feasibility first, then strict third-side ranges.

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Math SheetNameScoreTeacherDateTriangle Inequality TheoremDecide whether each set of lengths can form a triangle. Then find the strict range for each missing third side.Section 1: Can these sides form a triangle?Circle Yes or No. Compare the two shortest sides with the longest side and explain your decision.1.Side lengths: 19 in, 9 in, 10 inTriangle? Yes / NoExplain:2.Side lengths: 27 ft, 8 ft, 11 ftTriangle? Yes / NoExplain:3.Side lengths: 10 yd, 8 yd, 5 ydTriangle? Yes / NoExplain:4.Side lengths: 13 mi, 12 mi, 6 miTriangle? Yes / NoExplain:5.Side lengths: 4 in, 3 in, 2 inTriangle? Yes / NoExplain:6.Side lengths: 8 ft, 10 ft, 4 ftTriangle? Yes / NoExplain:7.Side lengths: 2 yd, 23 yd, 2 ydTriangle? Yes / NoExplain:8.Side lengths: 3 mi, 12 mi, 12 miTriangle? Yes / NoExplain:9.Side lengths: 8 in, 3 in, 10 inTriangle? Yes / NoExplain:10.Side lengths: 8 ft, 8 ft, 3 ftTriangle? Yes / NoExplain:Triangle inequalities use strict bounds.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateTriangle Inequality TheoremDecide whether each set of lengths can form a triangle. Then find the strict range for each missing third side.Section 2: Find the strict range of the third sideUse the absolute difference and the sum of the given sides. Write both strict inequality signs.11. Two sides are 28 yd and 28 yd. Write the strict range for the third side x.Given sides: 28 yd, 28 ydStrict interval:Work:12. Two sides are 25 mi and 21 mi. Write the strict range for the third side x.Given sides: 25 mi, 21 miStrict interval:Work:13. Two sides are 18 in and 23 in. Write the strict range for the third side x.Given sides: 18 in, 23 inStrict interval:Work:14. Two sides are 8 ft and 12 ft. Write the strict range for the third side x.Given sides: 8 ft, 12 ftStrict interval:Work:15. Two sides are 1 yd and 6 yd. Write the strict range for the third side x.Given sides: 1 yd, 6 ydStrict interval:Work:16. Two sides are 28 mi and 7 mi. Write the strict range for the third side x.Given sides: 28 mi, 7 miStrict interval:Work:Triangle inequalities use strict bounds.math-sheet.comPage 2 of 2

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