AI Tutor
Grab a pair of scissors and open the blades. As the metal tips spread farther apart, the plastic handles spread apart by the exact same amount.
Whenever two straight lines cross, they form an "X" shape that creates four separate angles sharing one central corner, called a vertex.
The angles sitting directly across from each other are called vertically opposite angles, and they are always congruent, which means they have the exact same angle measure.
Around 600 BCE, the ancient Greek philosopher Thales of Miletus became the first person to formally prove that these opposite angles must always be equal.
How could Thales know this was true for every intersection in the universe without measuring each one?

The Proof

Two angles that sit side-by-side on a straight line form a linear pair and are supplementary, meaning their measures add up to 180∘180^\circ.
Look at angle aa and its neighbor bb on the top straight line: a+b=180∘  ⟹  a=180∘−ba + b = 180^\circ \implies a = 180^\circ - b
Now look at angle cc sitting opposite to aa. Because angle cc also sits on a straight line next to bb: c+b=180∘  ⟹  c=180∘−bc + b = 180^\circ \implies c = 180^\circ - b
Since both aa and cc equal 180∘−b180^\circ - b, angle aa must equal angle cc!
Knowing that vertical angles are equal and adjacent angles sum to 180∘180^\circ gives you a superpower: what if you only know ONE single angle in an intersection?