Axioms and postulates (Euclidean geometry)

ppt on axioms

1. During Euclid's period, the
notions of points, line, plane
(or surface), and so on were
derived from what was seen
around them.

2. —Euclid (325 to 265 B.C.) is known as the Father of
Geometry. Heis credited for organizing all the works
of mathematics, then known, into a logical
representation in Elements, which is a collection of
thirteen books

3. —Some definitions given in his book I of the Elements
are as follows.
—A point is that which has no part.
—A line is breadth-less length.
—A straight line is a line which lies evenly with the
points on itself.
—A surface is that which has length and breadth only.
—Theedges of a surface are lines.
—A plane surface is a surface which lies evenly with the
straight lines on itself.

4. —Axioms and postulates are the assumptions that are
obvious universal truths, but are not proved. Euclid
used the term “postulate” for the assumptions that
were specific to geometry whereas axioms are used
throughout mathematics and are not specifically
linked to geometry.

5. —Things that are equal to the same things are equal to one
another.
—If equals are added to equals, then the wholes are also
equal.
—If equals are subtracted from equals, then the remainders
are equal.
—Things that coincide with one another are equal to one
another.
—The whole is greater than the part.
—Things that are double of the same things are equal to one
another.
—Things that are halves of the same things are equal to one
another.

7. —A straight line may be drawn from any one point
to any other point

9. —The above result can be stated in the form of an axiom
as follows.
—Axiom: Given two distinct points, there is a
unique line that passes through them.

10. —A terminated line can be produced indefinitely

12. A circle can be drawn with any centre and any
radius

14. All right angles are equal to one another.

15. 90°
90°90°
90°

16. —If a straight line falling on two straight lines forms
the interior angles that together measure less
than two right angles on the same side of it, then
the two straight lines, if produced indefinitely,
meet on that side on which the sum of the angles
is less than two right angles.

19. Euclid’s 5
th
postulate indirectly confirms the
existence of parallel lines

20. Two equivalent versions of Euclid‟sfifth postulate
are as follows.
—For every line l and for every point P not lying on l,
there exists a unique line „m‟passing through P and
parallel to l.
—Two distinct intersecting lines cannot be parallel to
the same line.

21. —THANK YOU………………………………………………
—BINITHA ANN JOHN
—GRADE : IX D
—ROLL NO.02