DIRECTIONS: For each of the following questions, you may only cite propositions 2.1, plane sepa- ration, line separation, and crossbar theorem; the incidence axioms, the betweenness axioms, and any definitions from the incidence and betweenness sections of Greenberg's textbook. You may also use exercises from the textbook, but only if you (or we as a class) have proved them already, and their number comes before whatever exercise you are trying to use it to prove. 2. Consider the following model on Z2: Points are A = (a1, 12) Z2, and lines are given by ar+by+c=0 where a,b,c eZ, where you cannot have both a = 0 and b=0. Recall that Z is the set of all integers. Betweenness is defined on points A * B * C where aj bi > c or a2 b2 > 02. Note, all three of the incidence axioms (11) - (13) are satisfied in this model. (a) Show that the betweenness axioms (B1) - (B3) hold in this model. (Hint: look at the Canvas quiz on Betweenness] (b) Can you find a triangle AABC and a line l in this model, where l intesects one of the sides of the triangle neither of other 2 sides of the triangle. [hint: the key here is to recall what constitutes as a point in this model.] a DIRECTIONS: For each of the following questions, you may only cite propositions 2.1, plane sepa- ration, line separation, and crossbar theorem; the incidence axioms, the betweenness axioms, and any definitions from the incidence and betweenness sections of Greenberg's textbook. You may also use exercises from the textbook, but only if you (or we as a class) have proved them already, and their number comes before whatever exercise you are trying to use it to prove. 2. Consider the following model on Z2: Points are A = (a1, 12) Z2, and lines are given by ar+by+c=0 where a,b,c eZ, where you cannot have both a = 0 and b=0. Recall that Z is the set of all integers. Betweenness is defined on points A * B * C where aj bi > c or a2 b2 > 02. Note, all three of the incidence axioms (11) - (13) are satisfied in this model. (a) Show that the betweenness axioms (B1) - (B3) hold in this model. (Hint: look at the Canvas quiz on Betweenness] (b) Can you find a triangle AABC and a line l in this model, where l intesects one of the sides of the triangle neither of other 2 sides of the triangle. [hint: the key here is to recall what constitutes as a point in this model.] a