Short Notes - Introduction to Euclid’s Geometry

# Short Notes - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

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Euclid

What are Axioms?

The unproved universal truths are called axioms or postulates.

• Theorems are statements that can be proved using definitions and axioms.
• Euclid’s postulates are:
(i) A straight line may be drawn from anyone point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and radius
(iv) All right angles are equal to one another.
(v) 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 the side on which the sum of angles is less than two right angles.
• Playfair’s Axiom: For every line ‘ℓ’ and for every point, ‘P’ not lying on ℓ’, there exists a unique line passing through ‘P’ and parallel to ‘ℓ’’.

Euclid's Definitions, Axioms and Postulates

• Euclid was a Greek mathematician, who introduced the method of proving a geometrical result by using logical reasonings on previously proved and known results.
• The statement is a sentence that can be judged to be true or false.
• A Theorem is a statement that requires proof.

Note: Proving a theorem means establishing the truth of that theorem.

• The corollary is a statement whose truth can easily be deduced from a theorem.
• Axioms are the basic facts that are taken for granted without proof.
• Postulates are the basic facts that are taken for granted specific to geometry, without proof.

Some of Euclid’s axioms are:

• The whole is greater than the part.
• Things which are double of the same things are equal to one another.
• Things which are halves of the same things are equal to one another.
• Things which are equal to the same thing are equal to one another.
• Things which coincide with one another are equal to one another.
• If equals are added to equals, the wholes are equal.
• If equals are subtracted from equals, the remainders are equal.

Note:

• Euclidean Geometry is the study of flat surfaces.
• Nowadays ‘postulates’ and ‘axioms’ are terms that are used interchangeably and in the same sense.

Euclid's Five Postulates

• Postulate 1: A straight line may be drawn from any 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 is less than two right angles, 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.

Example: The straight line PQ falls on AB and CD such that the sum of the interior angles 1 and 2 is less than 180º on the left side of PQ. Thus, the lines AB and CD will intersect (on producing) on the left side of PQ.

Equivalent Versions of Euclid's Fifth Postulate

In the history of mathematics, Euclid’s fifth postulate is very significant. Though there are several equivalent versions of this postulate, we state the following axiom (called the Playfair's axiom) here: “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.

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