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Triangle Inequality Theorem
The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
Try this Adjust the triangle by dragging the points A,B or C. Notice how the longest side is always shorter than the sum of the other two.

In the figure above, drag the point C up towards the line AB. As it gets closer you can see that the line AB is always shorter than the sum of AC and BC. It gets close, but never quite makes it until C is actually on the line AB and the figure is no longer a triangle.

The shortest distance between two points is a straight line. The distance from A to B will always be longer if you have to 'detour' via C.

To illustrate this topic, we have picked one side in the figure above, but this property of triangles is always true no matter which side you initially pick. Reshape the triangle above and convince yourself that this is so.

The Converse

A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two.
For more on this see Triangle inequality theorem converse.
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Other triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises