“Things which are equal to the same thing are also equal to one another.”Euclid · Elements
From simple, self-evident axioms, entire geometry is built through logical steps.
Euclid started with rules so simple nobody could disagree, like you can draw a straight line between any two points. From these tiny rules, through clear logical steps, he proved hundreds of truths about triangles, circles, and angles. Each step rests only on the ones that came before it.
Euclid showed that truth can grow from almost nothing. For instance, if you say two friends are the same age as you, then they are the same age as each other. This is a simple rule, but from it you can deduce complex relationships. The next step is to see how the same logic applies in geometry.
Why it matters This step-by-step way of proving things underlies all the mathematics taught in schools today.