5. Introduction to Euclid's Geometry

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5. Introduction to Euclid’s Geometry

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Important Note :

1. Euclid’s axioms :
(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.
2. Euclid’s postulates were :
(i) A straight line may be drawn from any one point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and any 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 that side on which the sum of angles is less than two right angles

EXERCISE 5.1

1. Which of the following statements are true and which are false ? Given reasons for your answers .

(i) Only one line can pass through a single point .

(ii) There are an infinite number of lines which pass through two distinct points .

(iii) A terminated line can be produced indefinitely on both the sides .

(iv) If two circles are equal , then their radii are equal .

(v) In fig. 5.9 , if and , then .

2. Give a definition for each of the following term . Are there other terms that need to be defined first ? What are they and how might you define them ?

(i) parallel lines (ii) perpendicular lines (iii) line segment (iv) radius of a circle (v) square .

3. Consider two ‘ postulates’ given below :

(i) Given any two distinct points A and B , there exists a third point C which is in between A and B .

(ii) There exist at least three points that are not on the same line .

Do these postulates contain any undefined terms ? Are these postulates consistent ?

Do they follow from Euclid’s postulates ? Explain .

4. If a point C lies between two points A and B such that AC , then proves that . Explain by drawing the figure .

5. In Question 4 , point C is called a mid-point of line segment AB . Prove that every line segment has one and only one mid-point .

6. In Fig. 5.10 , if , then prove that .

7. Why is Axiom 5 , in the list of Euclid’s axioms , considered a universal truth ? ( Note that the question is not about the fifth postulate.)

EXERCISE – 5.2

1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand ?

2. Does Euclid’s postulate imply existence of parallel lines ? Explain.

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5. Introduction to Euclid's Geometry

NCERT