AI Tutor
Open a pair of scissors slightly, and watch the sharp blades and plastic handles. As the tips spread apart, the finger loops widen by the exact same amount.
Whenever two straight lines cross at an intersection point, they create two pairs of twin angles that mirror each other perfectly.

The Crossing Lines

When two straight lines cross, the angles sitting directly across from each other at the vertex are called vertically opposite angles (or simply vertical angles).
Why must these opposite angles always match? The secret hides in a basic geometric rule: a straight flat line forms an angle of 180∘180^\circ.

Proving the Theorem

Around 600 BCE, the Greek philosopher Thales of Miletus became the first person to formally prove that vertical angles are always congruent.
Let's follow his logic using three adjacent angles around an intersection: ∠1\angle 1, ∠2\angle 2, and ∠3\angle 3.
Because ∠1\angle 1 and ∠2\angle 2 sit on a straight line, they are a linear pair and sum to 180∘180^\circ (m∠1+m∠2=180∘m\angle 1 + m\angle 2 = 180^\circ).
The angles ∠3\angle 3 and ∠2\angle 2 also sit along a straight line, so m∠3+m∠2=180∘m\angle 3 + m\angle 2 = 180^\circ.
Subtracting m∠2m\angle 2 from both equal expressions leaves m∠1=m∠3m\angle 1 = m\angle 3. This proves the Vertical Angles Theorem: vertical angles are always congruent.