(Mathematician, not historian) Mathematics is generally built of axioms: things that are just obviously true and can't be proved. The Peano axioms are about the natural numbers (and includes "obvious" things like x = x for all natural numbers x).

Now Euclid's Elements goes through the axioms of geometry and uses them to prove a lot of other geometry. Essentially all your high school geometry would come from 5-ish simple axioms:

1. Given two points, you can draw a line between them (called a line segment)
2. Given a line segment, you can extend it out indefinitely to be a straight line
3. Given a line segment you can draw a circle where the radius is the line segment and the centre is one of the line points
4. All right angles are the same (congruent)
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (this is equivalent to saying that if you have parallel lines, draw a line intersection both, than the interior angles must add up to 180 degrees).

Until at least 1820 all geometry would be taught from these axioms, making Euclid's axioms part of an equivalent high school education.

Now I said 5-ish because that last axiom is really not obvious like the others. Mathematicians were obsessed about it for years, convinced that it couldn't be a real axiom and could be derived from the other 4. It wasn't until around 1820 that an Hungarian mathematician János Bolyai worked it out. (fun fact: he became so obsessed with it that his father warned him "For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life").

What Bolyai showed was that the 5th axiom only holds in what we call Euclidean spaces: think flat surfaces. If you try to draw parallel lines on a sphere, draw a line across them you wills see that the in the interior angles don't add up to 180. This is important as it then means that a triangle of a sphere might not have an angle sum of 180 (you can draw triangles will angle sums over 1000 degrees). This is important to consider when we are doing any sort of triangulation over the earth's surfaces (e.g. gps).

tl;dr; Euclid's elements would have been part of a classical education, but they weren't completely correct and interesting mathematics comes out if it.