chapter
-5, euclid’s geometry
Important points:
1.
Axioms or postulates are the assumptions
which are obvious universal truths. They are not proved.
2.
Theorems are statements which are proved,
using definitions, axioms, previously proved statements and deductive
reasoning.
3.
Some of Euclid’s axioms were:
(i)
Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the
wholes are equal.
(iii) If equals are subtracted from
equals, the remainders are equal.
(iv) Things which coincide with one
another are equal to one another
(v) The whole is greater than the part.
(vi) Things which are double of the same
things are equal to one another
(vii) Things which are halves of the same
things are equal to one another
4.
Euclid’s postulates were:
Postulate
1:
A straight
line may be drawn from any one point to any other point.
Postulate
2:
A terminated line can be produced indefinitely
Postulate
3:
A circle can be drawn with any centre and any
radius.
Postulate
4:
All
right angles are equal to one another
Postulate 5 :
If a straight
line falling on two straight lines makes the interior angles on the same side
of it taken together less than two right angles, then the two straight lines,
if produced indefinitely, meet on that side on which the sum of angles is less
than two right angles.
5.
Two equivalent versions of Euclid’s fifth
postulate are:
(i)‘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’.
(ii) Two distinct intersecting lines
cannot be parallel to the same line
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