Euclidean geometry is a system in mathematics. People think Euclid was the first person who described it; therefore, it bears his name. He first described it in his textbook Elements. The book was the first systematic discussion of geometry as it was known at the time. In the book, Euclid first assumes a few axioms. These form the base for later work. They are intuitively clear. Starting from those axioms, other theorems can be proven.
Any straight line segment can be made longer (extended) to infinity, so it becomes a straight line.
With a straight line segment it is possible to draw a circle, so that one endpoint of the segment is the center of the circle, and the other endpoint lies on the circle. The line segment becomes the radius of the circle.
Parallel postulate. If two lines 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.
Status
Euclidean geometry is a first-order theory. With it, statements like For all triangles... can be made, and be proven. Statements like For all sets of triangles... are outside the scope of the theory.