Relationship of third side of triangle to opposite angle

Question

In any (Euclidian) triangle with two sides of fixed length, when the angle between them is larger, the opposite side is always longer. Intuitively this seems right, and I found this statement in an old textbook, but I have not found a proof. Can anyone provide a proof?

Answer 1

The easiest way is to use the cosine rule: $c^2=a^2+b^2-2ab\cos \theta$, where $\theta$ is the angle opposite the side of length $c$. If $a,b$ are fixed and $\theta$ varies then increasing $\theta$ (in the range $(0,\pi)$) decreases $\cos\theta$ so increases $c^2$ and therefore $c$.

Answer 2

A purely geometrical proof is contained in Euclid's elements, Book 1, proposition 24:

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI24.html