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Triangle Side and Angle Inequalities - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

In any triangle, the largest angle is opposite the largest side (the opposite side of an angle is the side that does not form the angle). The shortest angle is opposite the shortest side. Therefore, the angle measures can be used to list the size order of the sides. The converse is also true: the lengths of the sides can be used to order the relative size of the angles. Triangle side and angle inequalities are important when solving proofs.

There exists a special relationship between the length of a side and its angle in a triangle, so if we start by saying if the measure of angle a is bigger than measure of angle b is bigger than the measure of angle c, essentially all I'm saying is a is the biggest c is the smallest then we can say that the side opposite of angle a, so the side opposite of angle a is side bc, will be the largest. We can say that the side opposite of b, so I'm going opposite of b and that's side ac, so bc must be larger than ac and the side opposite of c will be your smallest so ab is your smallest. So the largest angle is opposite the largest side, the smallest angle is opposite the smallest side. Is the converse true? What if I said instead of the measure of angle a what if I just said, side what if we said ac is greater than side bc is greater than side ab. Then we could say what, which angle would be your largest? Well if ac is your largest angle excuse me if ac, remember this is a side is opposite of b then that means that measure of angle b is the largest. If bc is your next largest, what's opposite bc that's measure of angle a and last ab is opposite angle c so we'd say that measure of angle c would be the smallest. So it works both ways.

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