Is there a combinatorial interpretation of the triangular numbers? The triangular numbers count the number of items in a triangle with $n$ items on a side, like this:

This can be calculated exactly by the formula $T_n = \sum_{k=1}^n k = \frac{n(n+1)}{2} = {n+1 \choose 2} = {n+1 \choose n-1}$.
Is there any combinatorial interpretation to that formula, as in some way to interpret arranging objects in a triangle with $n$ on a side as the number of ways to choose 2 or $n-1$ objects out of a collection of $n+1$ objects?

Imagine a row of $n{+}1$ buttons underneath the triangle (as an extra row). Then for any two of those buttons you select they will designate a point of the triangle, and every point of the triangle can be identified with a pair of buttons:
 


Edit: David K notes in comments that a route distance triangle is a practical application of this idea. Pick two locations, read off the distance at the intersecting point of the triangle. Adapted slightly from the image given:

