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Transcript
ASSIGNMENT 5 – Euclid’s Axiomatic Geometry Definitions State the definitions of the following terms: Point Line Extremities of a line Straight line Right- angled triangle Trapezia Surface Extremities of a surface Plane surface Rectilinear Right angle/ Perpendicular Obtuse angle Boundary Figure Diameter Semicircle Circle Plane angle Acute angle Obtuseangled triangle Parallel Acute-angled triangle Center of circle Square Rectilinear figures Trilateral Multilateral Equilateral triangle Isosceles triangle Scalene triangle Oblong Rhomboid Postulates 1. Given two points, one can construct a line connecting these points ( it does not say that the line is unique). 2. Finite portions of lines (i.e segments) can be extended continuously in a straight line. 3. Given a point and a distance from that point, we can construct a circle with the point as center and the distance as radius (the principle of continuity of circles is assumed) 4. All right angles are equal to each other 5. If a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which are the angles less than two right angles. Common Notions 1. 2. 3. 4. 5. Things that are equal to the same thing are also equal to one another. If equals be added to equals, the wholes are equal. If equals be subtracted from equals, the remainders are equal. Things that coincide with one another are equal to one another The whole is greater than the part. Exercises: 1. Comment on the consistency and independence of the five postulates. 2. State “Playfair’s Postulate” and show that the postulate is logically equivalent to Euclid’s fifth postulate. 3. Think of another independent axiom that could be added to Euclid’s axioms to show that Euclid’s axiomatic system is not complete. 4. [Definition of a great circle: A great circle is a circle on the sphere cut by a plane passing through the center of the sphere]. Suppose we re-interpreted the term ‘point’ in Euclid’s postulates to mean a point on a sphere and a line to be a part of a great circle. How would you define circle on a sphere? How would you define angle on a sphere? Show that Euclid’s first four postulates hold in this new ‘spherical geometry’, but that the fifth postulate (or equivalently Playfair’s postulate) does not hold. 5. In the following exercises, experiment with spherical geometry to determine if each statement is most likely true or false in spherical geometry. Give short explanations for your answers: The sum of the angles of a triangle is 180 degrees. Given a line and a point not on the line, there is a perpendicular to the line through the point. A four-sided figure with three right angles must be a rectangle.