Triangle Inequality Theorem
Grade 7 Geometry — When Can Three Lengths Form a Triangle?
🔍 Introduction: Can Any Three Lengths Form a Triangle?
Imagine you have three sticks. Can you always arrange them to form a triangle? The answer is no!
🤔 Think about it: If you have sticks of lengths 2 cm, 3 cm, and 100 cm, could you ever connect them into a triangle? The two short sticks together (2 + 3 = 5 cm) aren't even close to reaching across the long stick!
There are restrictions on what combinations of side lengths can form a valid triangle. The Triangle Inequality Theorem tells us exactly when three lengths can — and cannot — form a triangle.
Interactive visualization — available in the full lesson
👆 Drag point C to explore different triangles!
📐 Theorem Statement
Triangle Inequality Theorem
Each side of a triangle is less than the sum of the other two sides.
For a triangle with sides $a$, $b$, and $c$, all three of the following inequalities must be true:
Inequality 1
$a < b + c$
Inequality 2
$b < a + c$
Inequality 3
$c < a + b$
Equivalent statement: The length of any side must be less than the sum of the other two sides.