NCERT Solutions for Chapter 5 Introduction to Euclid's Geometry Class 9 Maths
Chapter Name | NCERT Solutions for Chapter 5 Introduction to Euclid's Geometry |
Class | Class 9 |
Topics Covered |
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Related Study Materials |
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Short Revision for Ch 5 Introduction to Euclid's Geometry Class 9 Maths
- The terms point, line and plane are taken as undefined.
- Axioms (or postulates) are taken as universal truth without proof.
- Theorems are statements which are proved.
- Euclid's Axioms:
• Things which coincide with one another are equal to one another.
• Things which are equal to the same thing arc equal to one another.
• The whole is greater than the part.
• If equals are added to equals, the wholes are equal.
• If equals are subtracted from equals, the remainders are equal.
• Things which are double of the same things are equal to one another.
• Things which are half of the same things are equal to one another. - A point has no part.
- The ends of a line are points.
- The edges of a surface are lines.
- A line has no length.
- A surface has breadth and length only.
- Euclid's 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 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 sum of two interior - angles on the same side of it les than two right angles, then the two straight lines meet on that side.
Exercise 5.1
(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 AB = PQ and PQ = XY, then AB = XY.
2. Give a definition for each of the following terms. 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 = BC, then prove that AC = ½ AB. 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 AC = BD, then prove that AB = CD.
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.)
Solution
As statement is true in all the situations. Hence, it is considered a 'universal truth'.
Exercise 5.2
1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
2. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Solution
If a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then the lines, if produced indefinitely, will not meet on this side of l. In the same way, the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are, therefore, parallel.