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 Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.5(D), which requires students to verify the Triangle Inequality theorem and the relationships between side lengths and angle measures.
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.