Axioms and Postulates

1. Mathematical System and
Axiomatic Structure
UndefinedTerms, DefinedTerms,Theorems, and Postulates/Axioms

2. Axioms and Postulates
▪Euclid, starting from his definitions, assumed
certain properties, which were not to be
proved.These assumptions are actually
‘obvious universal truths’, in which he divided
into two: axioms and postulates.

3. Axioms and Postulates
▪Assumptions that were specific to Geometry
are called POSTULATES
▪Common notions or AXIOMS, on the other
hand, were assumptions used throughout
Mathematics and not specifically linked to
Geometry

4. Axioms
1. Things which are equal to the same thing
are equal to one another
2. If equals are added to equals, the wholes
are equal.
3. If equals are subtracted from equals, the
remainders are equals.

5. Axioms
4. Things which coincide with one another are
equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are
equal to one another.
7. Things which are Half of the same things are
equal to one another.

6. Postulates
1. A straight line segment can be drawn joining
any two points
2. Any straight line can be extended indefinitely in
a straight line.
3. Given any straight line segment, a circle can be
drawn having the segment as radius and one
endpoint as center.

7. Postulates
1. A straight line segment can be drawn joining
any two points
Axiom 5.1
Given two distinct points, there is a unique line that
passes through them.

8. Postulates
4. All right angles are congruent.
5. 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.

9. Theorem
5.1 Two distinct lines cannot have more than
one point in common.

10. Exercise
1. Which of the following statements are true and
which are false? Give reasons for your answers.
a. Only one line can pass through a single point.
b. There are an infinite number of lines which pass
through two distinct points.
c. Any straight line can be extended indefinitely in a
straight line.
d. If two circles are equal, then their radius are equal.

11. Exercise
2. Give a definition for each of the following terms.
a. parallel lines d. radius of a circle
b. perpendicular lines e. square
c. line segment

12. Exercise
3. Consider two postulates given below:
a. Given any two distinct points A and B, there
exists a third point C which is in between A and B.
b.There exist at least three points that are not
on the same line.
Are these postulates consistent? Do they
follow Euclid’s postulates? Explain.