Triangles
Rule

Triangle Inequality Theorem

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
A triangle with movable vertices. The side lengths are printed. Three inequalities.
Therefore, given a triangle three inequalities hold true.

Proof

Consider a general triangle and the three inequalities given by the theorem.

Triangle ABC

Here, it will be shown that The other two inequalities can be proved following the same procedure. Start by extending to the left of Then, consider a point on this line such that

Triangle ABC and point D

In the sides and are congruent. This means that by the Isosceles Triangle Theorem, the angles opposite them are congruent angles. Therefore, which means that

Triangle ABC and point D
Notice that is made of and Therefore, the measures of these three angles can be related thanks to the Angle Addition Postulate.
From this equation, is greater than
Now consider From the previous inequality and the Triangle Larger Angle Theorem, the side opposite is longer than the side opposite
The length of is equal to the sum of the lengths of and This is because of the Segment Addition Postulate. Substituting the corresponding sum into the left-hand side results in the following inequality.
Recall that was plotted so that Therefore, substitute for into the last inequality.
Notice that the last inequality is the desired one. Consequently, the proof is complete.
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