Vertical Angles: Thales, Pyramids & Pool Shots

Picture two straight lines crossing like an X. You might notice four angles at the intersection—but did you know that the angles sitting opposite each other (called vertical angles) are always equal? It’s a simple fact that feels almost magical: no matter how slanted or skewed those lines are, opposite angles match exactly. Let’s dive into how a Greek philosopher first spotted this, why it matters in real life, and a few traps students often fall into.

Where did this come from?

Around 600 BC, Thales of Miletus—often called the first true mathematician—noticed that when two lines cross, the angles directly across from each other are always the same size. He used this insight not only to build the foundations of geometry but also, legend has it, to measure the height of Egyptian pyramids. By comparing the pyramid’s shadow with the shadow of a pole at the same time of day, he applied angle relationships (and equal vertical angles) to work out the pyramid’s height without climbing it!

Where you’ll see this in real life

1. Mirrors and optics: The law of reflection—angle in equals angle out—relies on the idea of vertical angles. When light bounces, the incoming and outgoing rays form equal opposite angles relative to the normal (an imaginary line perpendicular to the surface). 2. Periscopes and periscopic devices: Submarines and playground periscopes use mirrors set at particular angles so you see around obstacles. Vertical angles ensure your line of sight stays accurate. 3. Billiards or pool: Pros use “diamond systems” on the table’s rails. By visualising equal opposing angles, they predict exactly how the cue ball will rebound for trick shots. 4. Surveying and construction: Modern laser levels and surveying equipment still rest on the fact that opposite angles in crosshairs are identical, keeping measurements precise when laying foundations or mapping land.

A common misconception

Lots of students think vertical angles are the ones “on top and bottom,” but really it’s about position across the intersection—across from each other, not necessarily up versus down. Another trap is mixing them up with adjacent angles (two angles that share a side) or supplementary angles (which add up to 180°). Remember: vertical angles are non-adjacent and always equal, whereas supplementary angles sit side by side and total 180°.