This is the 1st of a series of a Non Desargue plane based on Hilbert's "Grundlagen der Geometrie". (Let me call these lines and angles by adding H to them, such as Hlines and Hangles.)
Here are the definitions of Hline, Hlength and Hangle. If the slope of Euclidian line AB is positive, it is bend at the base line as below. If the slope is non positive, it is the same as the Euclidian line. HLength is the same as Euclidian. But Hangle between 2 Hlines are defined by the angle of the 2 "support lines". You can find macros of H-somethings from the Toolbar.

Construct a Htriangle such as PQ:PR=2:1 , Hangle RPQ=60 degrees, Hangle QRP however is NOT 90 degrees.
In this Hplane, normal congruent theory can't be applied.