Triangle inequality and side/angle relationships
Triangle Inequality and Side–Angle Relationships
Triangle Inequality and Side–Angle Relationships is a topic in Triangle Theorems in the California Common Core State Standards. It is aligned to Standard G-CO.10, which requires students to prove theorems relating the side lengths and angle measures of a triangle.
The triangle inequality says any two sides sum to more than the third, and the largest angle lies opposite the longest side.
Theory
Two rules connect a triangle's sides and angles:
- Triangle Inequality: the sum of any two side lengths is greater than the third. Equivalently, the third side lies between the difference and the sum of the other two.
- Side-Angle Relationship: the larger angle is opposite the longer side (and vice versa).
The inequality and the side bound:
How to apply the theorems
- Existence: check the two smallest sides sum to more than the largest.
- Range of a side: \(|a-b|<x<a+b\).
- Order angles: match the order of their opposite sides.
Check that the two smaller sides sum to more than the largest.
| \(3+4\) | \(=\) | \(7>5\ \checkmark\) |
Yes — the triangle inequality holds.
Test the two smaller sides against the largest.
| \(2+3\) | \(=\) | \(5<6\) |
No — the sum is too small, so no triangle exists.
The third side lies between the difference and the sum of the other two.
| \(10-7\) | \(<\) | \(x<10+7\) |
| \(3\) | \(<\) | \(x<17\) |
The larger angle is opposite the longer side, so order the angles like their opposite sides \(5<6<8\).
| \(A\) | \(<\) | \(C<B\) |
Common pitfalls
Frequently asked questions
What is the triangle inequality theorem?
The sum of any two sides of a triangle is greater than the third side.
How do you check if three lengths form a triangle?
Add the two shortest lengths; if the sum exceeds the longest length, they form a triangle.
What is the range of the third side?
It lies strictly between the difference and the sum of the other two sides: \(|a-b|<x<a+b\).
How are a triangle's sides and angles related in size?
The larger angle is opposite the longer side, and the smaller angle is opposite the shorter side.