On the non-Paschian ordered planes

We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle and angle measure, respectively. With respect to our definitions the plane has an elliptic character, meaning that the sum of the angles of a triangle is greater than $\pi$.