RS Aggarwal Solutions Class 9 Chapter 3 Introduction to Euclid’s Geometry
Exercise 3A
Question 1:
A theorem is a statement that requires a proof. Whereas, a basic fact which is taken for granted, without proof, is called an axiom.
Example of Theorem: Pythagoras Theorem
Example of axiom: A unique line can be drawn through any two points.
Question 2:
(i) Line segment: The straight path between two points is called a line segment.
(ii) Ray: A line segment when extended indefinitely in one direction is called a ray.
(iii) Intersecting Lines: Two lines meeting at a common point are called intersecting lines, i.e., they have a common point.
(iv) Parallel Lines: Two lines in a plane are said to be parallel, if they have no common point, i.e., they do not meet at all.
(v) Half-line: A ray without its initial point is called a half-line.
(vi) Concurrent lines: Three or more lines are said to be concurrent, if they intersect at the same point.
(vii) Collinear points: Three or more than three points are said to be collinear, if they lie on the same line.
(viii) Plane: A plane is a surface such that every point of the line joining any two points on it, lies on it.
Question 3:
(i) Six points: A,B,C,D,E,F
(ii) Five line segments: \(\overline { EG } \), \(\overline { FH } \), \(\overline { EF } \), \(\overline { GH } \), \(\overline { MN } \)
(iii) Four rays: \(\overrightarrow { EP } \), \(\overrightarrow { GR } \), \(\overrightarrow { GB } \), \(\overrightarrow { HD } \)
(iv) Four lines: \(\overleftrightarrow { AB } \), \(\overleftrightarrow { CD } \), \(\overleftrightarrow { PQ } \), \(\overleftrightarrow { RS } \)
(vi) Four collinear points: M,E,G,B
Question 4:
(i) \((\overleftrightarrow { EF } \quad \overleftrightarrow { GH } ) \) and their corresponding point of intersection is R.
\((\overleftrightarrow { AB } \quad \overleftrightarrow { CD } ) \) and their corresponding point of intersection is P.
(ii) \(\overleftrightarrow { AB } \), \(\overleftrightarrow { EF } \), \(\overleftrightarrow { GH } \) and their point of intersection is R.
(iii) Three rays are: \(\overrightarrow { RB } \), \(\overrightarrow { RH } \), \(\overrightarrow { RG } \)
(iv) Two line segments are: \(\overline { RQ } \), \(\overline { RP } \)
Question 5:
(i) An infinite number of lines can be drawn to pass through a given point.
(ii) One and only one line can pass through two given points.
(iii) Two given lines can at the most intersect at one and only one point.
(iv) \(\overline { AB } \), \(\overline { BC } \), \(\overline { AC } \)
Question 6:
(i) False
(ii) False
(iii) False
(iv) True
(v) False
(vi) True
(vii) True
(viii) True
(ix) True
(x) False
(xi) False
(xii) True