vertical angles

Cross two straight lines, like the blades of an open pair of scissors, and you make four angles around the crossing point. The two angles that sit directly across from each other — not side by side, but back to back through the point — are a pair of vertical angles. The X shape made by the scissors has a top wedge and a bottom wedge that match, and a left wedge and a right wedge that match.

Precisely: when lines AB and CD cross at a point P, the angle APC and the angle BPD lie on opposite sides of P and form one pair of vertical angles; angle APD and angle BPC form the other pair. The key fact, the vertical-angles theorem, is that vertical angles are always equal in measure. The reason is short and beautiful. Angle APC and angle APD together make a straight line, so m(angle APC) + m(angle APD) = 180. Angle BPD and angle APD also make a straight line, so m(angle BPD) + m(angle APD) = 180. Both sums equal 180, so m(angle APC) = m(angle BPD). Each pair shares the same supplement, so each pair is equal.

This is often a learner's very first genuine theorem — a fact that is not obvious until you prove it, and is proven in two clean lines. It also flags a common slip: vertical angles are the across-the-point pairs, not the next-to-each-other pairs. The side-by-side angles are adjacent and form a linear pair; they add to 180 rather than being equal.