Axioms and postulates (Euclidean geometry)
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During Euclid's period, the notions of points, line, plane (or surface), and so on were derived from what was seen around them.
Transcript of Axioms and postulates (Euclidean geometry)
During Euclid's period, the notions of points, line, plane (or surface), and so on were derived from what was seen
around them.
� Euclid (325 to 265 B.C.) is known as the Father of Geometry. He is credited for organizing all the works of mathematics, then known, into a logical representation in Elements, which is a collection of thirteen books
� Some definitions given in his book I of the Elementsare 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. � The edges of a surface are lines. � A plane surface is a surface which lies evenly with the
straight lines on itself.
� 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.
� 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.
� A straight line may be drawn from any one point to any other point
� 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.
� A terminated line can be produced indefinitely
A circle can be drawn with any centre and anyradius
All right angles are equal to one another.
� 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.
Euclid’s 5th postulate indirectly confirms the existence of parallel lines
Two equivalent versions of Euclid‟s fifth 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.
� THANK YOU………………………………………………
� BINITHA ANN JOHN� GRADE : IX D� ROLL NO.02
