Triangle Inequality Theorem

Geometry · Axiom Academy

LESSON Triangle Inequality Theorem Determining when three side lengths can actually close up into a triangle. 1. Two Sides Must Reach Across the Third Pin a base of length c and hinge the other two sides, of lengths a and b , at its ends. Swing them inward: they meet at an apex above the base only if together they are long enough to span the gap. That is the whole idea. The sum of any two sides exceeds the third 2. All Three Inequalities Must Hold Any side could be the "long one," so the rule has to hold for every pairing. For a triangle with sides a , b , and c , all three of these must be true at once: What happens right at the edge? Shrink the two sides until they only just reach. When the sum exactly equals the third side, the apex flattens onto the base — the three points fall in a straight line. This is the degenerate case , and it's why the inequality must be strict (>, not ≥). Run the full check on two sets of lengths and the pattern is clear: The two sides over-reach; the apex sits above the base and a genuine triangle closes. The two sides fall short (or just touch). The points are collinear — no area, no triangle. You've seen exactly when three lengths can close into a triangle — and watched the boundary case flatten before your eyes. Scroll up to revisit any step.

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