Vertically Opposite Angles

When two straight lines cross, they make four angles at the meeting point. The two angles directly across from each other — sharing only the crossing point, opening in opposite directions — are called vertically opposite angles (or just "vertical angles"). The surprise is that they are always exactly equal.

\text{vertically opposite angles are equal}

It follows from one fact you already know: angles on a straight line add up to 180^\circ.

When two straight lines intersect:

the two pairs of opposite angles are equal — a = c and b = d;
each angle and the one next to it lie on a straight line, so a + b = 180^\circ;
so the four angles come in just two sizes, x and 180^\circ - x.

Why it works

No measuring needed — two straight lines do all the work. Step through the reason.

The same argument works for the other pair, giving b = d. So whenever lines cross, the angles opposite each other match.

Practise: chase the angles

Two lines cross. Fill in every angle you can — using vertically opposite angles and angles on a straight line — ending with the highlighted one. Refresh for a new figure; Check explains each step.