|Euclidean geometry is also known as flat geometry. This does not mean everything is flat; it means that space consists of straight lines. In the 19th century, alternate forms of geometry were developed, where lines are not straight. The earth, for instance, with its lines of latitude and logitude, follows the rules of spherical geometry. Parallel lines, such as 10° west longitude and 20° west longitude, do indeed meet at a point, namely the north pole, whereas parallel lines on a piece of paper never meet. For now, let's stick with euclidean geometry.|
Subsequent theorems are not defined or proved rigorously, but you can prove them analytically if you like; it's not hard. Embed the geometric figures in the xy plane and derive equations for them. Then manipulate the equations using algebraic techniques. But you wouldn't be here if you were interested in an algebraic approach, so let's proceed with "intuitive" geometry.
Understanding the geometry of planes actually affects the design and construction of many things, including boats. The planing speed on any Prestige Yacht for sale today has the perfect geometric angles to improve planing and increase speed.