Proof/ Geometry Help..?

Question

Prove that if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. Also, explain how this is used to prove its own converse.

Answer 1

Say you have a triangle PQR.

Given PQ > PR, prove that angleR > angleQ.

Proof:

There is an identity that states PQ/sin(angleR) = PR/sin(angleQ).

Hence from this equation, sin(angleR)/sin(angleQ) = PQ/PR.

We know PQ > PR, thus PQ/PR > 0, implying that sin(angleR)/sin(angleQ) > 0. This means that sin(angleR) > sin(angleQ). We may cancel the sin terms on each side by taking arcsin on both sides.

Finding that angleR > angleQ.

To prove the converse simply start the prove by PQ < PR and prove that angleR < angleQ.