IX A Math Ch-5 Introduction to Euclid's Geometry
04/06/21 (Friday)
Diagnostic test for Euclid's Geometry.
Theory notes:
09/06/21 (Wednesday)
Euclid’s axioms
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Things which are halves of the same things are equal to one another.
Euclid’s five postulates
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 center 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.
Theorem 5.1 : Two distinct lines cannot have more than one point in common.
Equivalent Versions of Euclid’s Fifth Postulate :
(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.
Video Links:
Introduction to Euclid's Geometry
Introduction to Euclid Geometry part 2 (Elements by Euclid)
Introduction to Euclid Geometry part 3 (Terms)
Introduction to Euclid Geometry part 4 (Definition)
Introduction to Euclid Geometry_part 5(Axioms and Postulates)
Introduction to Euclid Geometry part 6 (Theorems)
Introduction to Euclid Geometry part 7 (Numerical)
Introduction to Euclid Geometry part 8 (5th Postulate Equivalent version)
The five postulates of Euclidean Geometry
10/06/21 (Thursday)
Exercise 5.1
1. Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
False
There can be infinite number of lines that can be drawn through a single point. Hence, the statement mentioned is False
(ii) There are an infinite number of lines which pass through two distinct points.
False
Through two distinct points there can be only one line that can be drawn. Hence, the statement mentioned is False
(iii) A terminated line can be produced indefinitely on both the sides.
True
A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely. Hence, the statement mentioned is True.
(iv) If two circles are equal, then their radii are equal.
True
The radii of two circles are equal when the two circles are equal. The circumference and the centre of both the circles coincide; and thus, the radius of the two circles should be equal. Hence, the statement mentioned is True.
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
True
According to Euclid’s 1st axiom- “Things which are equal to the same thing are also equal to one another”. Hence, the statement mentioned is True.
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.
Solution:
Yes, these postulates contain undefined terms. Undefined terms in the postulates are:
(i) There are many points that lie in a plane. But, in the postulates given here, the position of the point C is not given, as of whether it lies on the line segment joining AB or not.
(ii) Also there is no information about whether the points are in same plane or not.
And
Yes, these postulates are consistent as
(i) Point C is lying on the line segment AB in between A and B.
(ii) Point C does not lie on the line segment AB.
No, they don’t follow from Euclid’s postulates. They follow the axioms.
