5. Consider the following axiomatic system. Undefined Terms: Point, Line, Plane, Incident Defined Terms: Collinear Axioms: 1. There exist at least 2 points. 2. Given any 2 distinct points, there is a unique line incident with both. 3. Every line is incident with exactly 2 points. 4. Given any line, there exists a point that is not incident with the line. 5. Given any 3 distinct noncollinear points, there exists a unique plane incident with all 3 points. 6. Given any plane, there exists a point that is not incident with the plane. Construct a model for this system. How many points, lines, and planes must this system contain? Prove your results from the axioms (don't rely on a diagram). 6. Consider only the points and lines of this system in #5 above. Is this a model of an incidence geometry? Why or why not? Which parallel property does it have? Is it isomorphic to another system we have studied? 7. Consider only the points and lines of this system in #5 above. Construct a model for the dual system. Is the dual an incidence geometry? 8. Define a geometry whose points are all points in the interior of a unit circle (exclude points on the boundary) and whose lines are open chords of the circle (straight segments with endpoints on the circle, excluding the endpoints). Using the typical notion of incidence for a point and a line. is this an incidence geometry? Which parallel property does it have? 5. Consider the following axiomatic system. Undefined Terms: Point, Line, Plane, Incident Defined Terms: Collinear Axioms: 1. There exist at least 2 points. 2. Given any 2 distinct points, there is a unique line incident with both. 3. Every line is incident with exactly 2 points. 4. Given any line, there exists a point that is not incident with the line. 5. Given any 3 distinct noncollinear points, there exists a unique plane incident with all 3 points. 6. Given any plane, there exists a point that is not incident with the plane. Construct a model for this system. How many points, lines, and planes must this system contain? Prove your results from the axioms (don't rely on a diagram). 6. Consider only the points and lines of this system in #5 above. Is this a model of an incidence geometry? Why or why not? Which parallel property does it have? Is it isomorphic to another system we have studied? 7. Consider only the points and lines of this system in #5 above. Construct a model for the dual system. Is the dual an incidence geometry? 8. Define a geometry whose points are all points in the interior of a unit circle (exclude points on the boundary) and whose lines are open chords of the circle (straight segments with endpoints on the circle, excluding the endpoints). Using the typical notion of incidence for a point and a line. is this an incidence geometry? Which parallel property does it have