What are Triangle Inequalities?
In the previous topic, we looked at what happens when sides and angles are perfectly equal. But what happens when they are not equal? Inequalities deal with relationships of "greater than" (>) or "less than" (<).
There are two major laws that govern inequalities in triangles:
- The Side-Angle Inequality Theorem: If two sides of a triangle are unequal in length, the angle opposite to the longer side is always larger than the angle opposite to the shorter side. Think of it like a crane or a laptop lid: the wider you open the angle, the longer the distance between the tips becomes! Conversely, the side opposite to the larger angle is always the longer side.
- The Triangle Inequality Theorem: This is a fundamental rule of nature. It states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. If this rule is broken, the two sides will lie flat or won't reach each other, and you cannot form a closed triangle at all!
Let us see how the Triangle Inequality Theorem works with an example. Suppose a triangle has sides of length \(a\), \(b\), and \(c\). For this triangle to exist, all three of these conditions must be true:
- \(a + b > c\)
- \(b + c > a\)
- \(a + c > b\)
A helpful shortcut rule derived from this is: The difference between any two sides of a triangle is always less than the third side (\(|a - b| < c\)).