The Axioms of Euclidean Geometry
(1) A straight line may be draw between any two points.
(2) Any terminated straight line may be extended indefinitely
(3) A circle may be drawn with any given point as center and any given radius.
(4) All right angles are equal.
(5) If α + β < 180 ° then L1 and L2 will meet at some point to the left of L3 or at the side of angles.
Axiom: The basic facts which are taken for
granted without proof are called axioms.
Ex: (i) things which
are equal to the same thing are equal to one another;
(ii) Halves of
equals are equal;
(iii) A line contains infinitely many points;
(iv) The whole is greater than each of its parts.
The following axioms, corresponding to the first four rules of
arithmetic, are among those most commonly used in geometry:
(i) Addition: If equals are added to equals, the sums are equal;
(ii) Subtraction: If equals are taken from equals, the remainders
(iii) Multiplication: Things which are the same multiples
of equals are equal to one another;
(iv) Division: Things which are
the same parts of equals are equal to one another.
Statement: A sentence which can be judged to be true or false
is called a statement.
Ex: (i) The sum of the angles of a quadrilateral
is 360°, is a true statement;
(ii) The sum of the angles of a triangle is
360°, is a false statement;
(iii) a + 2 < 10 is a sentence but not a statement.
Proof of a statement: Providing a geometrical result by using
logical reasonings on previously proved and known results is called
a proof of a statement.
Theorem: A statement that requires
a proof is called a theorem. Establishing the truth of a
theorem is known as proving the theorem.
Ex: (i) The
sum of all the angles of a quadrilateral is 360°;
If one side of a triangle is produced, then the exterior
angle so formed is equal to the sum of the interior opposite angles.
Corollary: A statement whose truth can easily be deduced
from a theorem is called its corollary.
(i) If two angles of one triangle are equal to two
angles of another triangle, then the third angles
of both the triangles are also equal. This is the
corollary for the theorem: the sum of all the angles
of a triangle is 180°;
(ii) An exterior angle of a triangle is greater than
either of the two interior opposite angles. This is the corollary for the theorem:
if one side of a triangle is produced, then the exterior angle so formed is equal
to the sum of the interior opposite angles.