Download ASSIGNMENT 4

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.