The sum of any two side lengths exceeds the third; count integer possibilities for a third side or judge whether given lengths form a triangle. Added post-audit: surfaced by the bank (5 questions across 5 tests), a standard middle-school geometry fact not separately labeled in the 8 audited forms.
Free, timed SHSAT practice sets on triangle inequality— every question includes a full worked explanation, and there's no account required.
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Three lengths make a triangle only if any two sides add up to more than the third. In practice, check that the two shorter sides add to more than the longest one.
If you know two sides, \(a\) and \(b\), the third side \(c\) must be between their difference and their sum: \(|a - b| < c < a + b\).
Example: sides 4 and 9 give \(5 < c < 13\). The whole-number choices are 6 through 12, which is 7 values. The least is 6 and the greatest is 12.
To test a set like 3, 4, 8: \(3 + 4 = 7\), which is not more than 8, so it is not a triangle.
Common traps: including the endpoints (5 and 13 make a flat line, not a triangle), counting the integers wrongly (use \(\text{high} - \text{low} - 1\)), and checking only the two longest sides instead of the two shortest.
The third side \(c\) must satisfy \(|13 - 20| < c < 13 + 20\), so \(7 < c < 33\). There are \(33 - 7 - 1 = 25\) whole numbers from 8 to 32. (Counting both 7 and 33, which make a flat triangle, gives 27. Subtracting \(33 - 7\) counts one endpoint, giving 26. Using only \(c < 33\) and forgetting \(c > 7\) gives 32.)
The third side \(c\) must satisfy \(6 < c < 30\). The greatest whole number in that range is 29. (Using 30 itself, where the triangle lies flat, gives 30. Giving the least possible length instead gives 7. Thinking the third side cannot be longer than the longest given side gives 18.)
Three lengths form a triangle only if each pair adds to more than the third. For 5, 7, 8, all three checks pass. (For 5, 7, 12: \(5 + 7 = 12\) is not more than 12, so the sides lie flat. For 5, 7, 13: \(5 + 7 = 12\) is less than 13. For 5, 7, 1: checking only \(5 + 7 > 1\) misses that \(1 + 5 = 6\) is not more than 7.)