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Grade 10 · lesson 4 of 4

Test Whether Three Lengths Can Form a Triangle

A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Two triangle sides have lengths 2 and 3. Which proposed third length makes a nondegenerate triangle? Candidate answers: (A) 1 (B) 4 (C) 5 (D) 6.
  • Independent check inputs: Two triangle sides have lengths 3 and 4. Which proposed third length makes a nondegenerate triangle? Candidate answers: (A) 1 (B) 6 (C) 7 (D) 8.

Learn the idea

Parallel-line and triangle theorems connect geometric conditions to numerical equations. Some side and angle values are impossible, so reasoning must check the defining configuration rather than rely only on arithmetic. A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides. Both strict bounds express a possible shape.

Here is the method on the supplied teaching case. The third side z must satisfy |3−2|<z<2+3, or 1<z<5. The value 4 is inside; equality would make a collapsed straight arrangement.

Equality creates a collapsed straight arrangement rather than a triangle with area. Test both the lower and upper bound; positive lengths by themselves do not guarantee a valid triangle.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Two triangle sides have lengths 2 and 3. Which proposed third length makes a nondegenerate triangle? Candidate answers: (A) 1 (B) 4 (C) 5 (D) 6. Solution: The third side z must satisfy |3−2|<z<2+3, or 1<z<5. The value 4 is inside; equality would make a collapsed straight arrangement. The final answer is 4. Equality creates a collapsed straight arrangement rather than a triangle with area. Test both the lower and upper bound; positive lengths by themselves do not guarantee a valid triangle.

Your turn

Check your reasoning

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Helpful tips
  • A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.
  • Equality creates a collapsed straight arrangement rather than a triangle with area.
Two triangle sides have lengths 3 and 4. Which proposed third length makes a nondegenerate triangle?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

6

The third side z must satisfy |4−3|<z<3+4, or 1<z<7. The value 6 is inside; equality would make a collapsed straight arrangement.

Take it further

Changed-input task: Two triangle sides have lengths 9 and 10. Which proposed third length makes a nondegenerate triangle? Candidate answers: (A) 1 (B) 18 (C) 19 (D) 20. Show the mathematical reason, not only the selected answer. Expected result: 18. The third side z must satisfy |10−9|<z<9+10, or 1<z<19. The value 18 is inside; equality would make a collapsed straight arrangement.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Two triangle sides have lengths 4 and 5. Which proposed third length makes a nondegenerate triangle?
Need a hint?
  • A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.
  • Equality creates a collapsed straight arrangement rather than a triangle with area.
2. Two triangle sides have lengths 5 and 6. Which proposed third length makes a nondegenerate triangle?
Need a hint?
  • A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.
  • Equality creates a collapsed straight arrangement rather than a triangle with area.
3. Two triangle sides have lengths 6 and 7. Which proposed third length makes a nondegenerate triangle?
Need a hint?
  • A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.
  • Equality creates a collapsed straight arrangement rather than a triangle with area.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.