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Euclid’s Postulates
Name ___________________________
Date _____________________
Geometry and much of modern mathematics is based on the work of Euclid, a Greek
mathematician of the school of Pythagoras. Euclid composed a series of propositions or theorems
based on five postulates:
ONE:
A straight line segment can be drawn joining any two points.
TWO:
Any straight line segment can be extended indefinitely in a straight line.
THREE: Given any straight line segment, a circle can be drawn having the segment as radius and
one endpoint as center.
FOUR:
All right angles are congruent.
FIVE:
If two lines are drawn which intersect a third in such a way that the sum of the inner
angles on one side is less than two right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
1. The first four postulates are simple and elegant; the fifth is not. The Fifth Postulate is far more
frequently referred to in the form of an equivalent postulate. What is it? State it.
Because the first four postulates are intuitive and the fifth is not, for centuries mathematicians
attempted to prove the Fifth Postulate as a theorem based on the first four postulates. Instead
they discovered situations in which the Fifth Postulate does not hold true. In fact, there are two
alternatives to the equivalent postulate given in Problem 1.
2. The equivalent proposition given in Problem 1 describes what is usually known as Euclidean
geometry or parabolic geometry. Do some research and rephrase your answer to Problem 1
slightly to define elliptic geometry and hyperbolic geometry.
3. Match each definition to the correct figure: parabola, hyperbola or ellipse. Draw an example.
a. The set of all points in the plane equidistant from a fixed point (the focus) and a fixed line
(the directrix).
b. The set of all points in the plane the sum of whose distance from two fixed points (the
foci) is a constant.
c. The set of all points in the plane the difference of whose distance from two fixed points
(the foci) is a constant.
4. Match each definition to the correct geometry: parabolic, hyperbolic or elliptical.
a. The plane extends infinitely far in any direction but is not continuous.
b. The plane extends infinitely far in any direction and is continuous.
c. The plane extends only a finite distance in any direction before “curving back around” on
itself. The plane is continuous.
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Homeschool Learning Network
Euclid’s Postulates
Name ___________________________
Date _____________________
Euclidean or parabolic geometry is the geometry we are all most familiar with. It is the geometry of
the “flat” piece of paper you are reading, extended infinitely in all directions. Parallel lines behave
themselves.
Elliptical geometry is the geometry of the surface of a sphere. (More accurately, it’s the geometry
of the surface of an ellipsoid.) On the surface of a sphere, parallel lines cannot exist. Let’s see
why. You’ll need something round like an orange and a black marker to draw on it with. It would
be a good idea to have several oranges.
5. First of all, any line drawn on the surface of a sphere will not extend infinitely. Instead, what
form must it take?
6. Draw two points on your orange. Given your answer to (5), how many different lines can you
draw through those two points?
Remember the definition of a line: a line is defined by two POINTS, and two POINTS define one
and only one line. This definition never changes, no matter what geometry you are looking at. So
what is wrong with our answer to (6)?… What’s wrong is our definition of a POINT. The definition
of a line never changes, but the definition of a POINT does change.
Let’s use “point” to refer to our ordinary, everyday definition of a point and POINT to refer to our
new, specialized meaning. In elliptical geometry, a POINT is defined as consisting of two antipodal
points.
7. What are antipodes?
8. Draw two POINTS on another orange. How many distinct lines can you draw through those
two POINTS?
9. How many distinct lines could you draw through a single POINT?
10. Draw another two POINTS on the same orange and connect them with a line. What do you
notice about the two lines?
11. In general, any two lines on the surface of a sphere will intersect in exactly _______ POINT(S).
12. By our new definition of a POINT, many of the “lines” we drew in answer to (5) are not really
lines at all. In elliptical geometry, a line is a _________ whose _________ is the same as that
of the sphere. Anything else is not a line but only a __________.
13. One of the important things to notice about the surface of a sphere is that triangles work
differently. How many right angles is it possible for a triangle to have? Try it out! What does
this imply about the sum of the interior angles of a triangle in elliptical geometry?
© 2002 The Homeschool Learning Network, all rights reserved. The Homeschool Learning Network permits teachers and parents to reproduce this
page for non-profit and educational purposes only. http://www.homeschoollearning.com
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Euclid’s Postulates
Name ___________________________
Date _____________________
ANSWERS TO QUESTIONS
1. The parallel postulate: “For any given line in the plane and for any point not on that line, there
will be one and only one line through that point that does not intersect the first line, if they are
extended far enough.”
2. The equivalent postulate for elliptical geometry is: “For any given line in the plane and for any
POINT not on that line, there will be zero lines through that POINT that do not intersect the first
line.”
The equivalent postulate for hyperbolic geometry is: “For any given line in the plane and for any
POINT not on that line, there will be at least two (and possibly infinite) lines through that POINT
that do not intersect the first line.”
(We use POINT here to indicate that our definition of “point” has changed somewhat from the
usual one.)
3. a. parabola b. ellipse c. hyperbola
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Euclid’s Postulates
Name ___________________________
Date _____________________
4. a. hyperbolic b. parabolic c. elliptical
5. a circle
6. infinitely many (try it out!)
7. Antipodes are diametrically opposite points on the surface of a sphere. The North Pole and the
South Pole are therefore antipodes.
8. one and only one
9. infinitely many, just as you would expect
10. They intersect.
11. one point
12. circle, radius (or diameter), circle
13. Two, therefore the sum of the interior angles will be greater than 180°.
© 2002 The Homeschool Learning Network, all rights reserved. The Homeschool Learning Network permits teachers and parents to reproduce this
page for non-profit and educational purposes only. http://www.homeschoollearning.com