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Q2(i):
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
Solution :
Step 1: Definition of Parallel Lines
Parallel lines are defined as two lines in a plane that do not intersect each other at any point, even when extended infinitely in both directions. In Euclidean geometry, two lines $l$ and $m$ are parallel if and only if the perpendicular distance between them remains constant at all points.
Step 2: Identification of Undefined or Primitive Terms
To define "parallel lines" with mathematical rigor, one must first possess clear definitions for several fundamental geometric concepts. These terms are considered "primitive" or require prior definition to avoid circular logic:
- Point: A mark of position that has no magnitude (length, breadth, or depth).
- Line: A breadthless length that extends indefinitely in two opposite directions.
- Plane: A flat surface that extends indefinitely in all directions within two dimensions.
- Intersection: The condition where two distinct lines share at least one common point.
- Distance: The shortest path between two points or between a point and a line.
Step 3: Formalizing the Definitions
To establish the definition of parallel lines, we define the constituent terms as follows:
| Term | Definition |
|---|---|
| Line | An object defined by a set of infinite points that possess length but no width. [Euclid’s Postulate/Definition 2] |
| Plane | A perfectly flat, two-dimensional surface that extends infinitely in all directions. |
| Intersection | The state where two lines occupy the same coordinates in a Cartesian plane $(x, y)$. |
Step 4: Geometric Representation
[By Euclidean definition, lines $l$ and $m$ are parallel if the perpendicular distance $d$ is constant and $l \cap m = \emptyset$.]
Final Solution: Parallel lines are defined as coplanar lines that never intersect, regardless of their length. This definition requires prior understanding of the terms "point," "line," "plane," and "intersection."
More Questions from Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1
- Q1(i): 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.
- Q1(ii): Which of the following statements are true and which are false? Give reasons for your answers. (ii) There are an infinite number of lines which pass through two distinct points.
- Q1(iii): Which of the following statements are true and which are false? Give reasons for your answers. (iii) A terminated line can be produced indefinitely on both the sides.
- Q1(iv): Which of the following statements are true and which are false? Give reasons for your answers. (iv) If two circles are equal, then their radii are equal.
- Q1(v): Which of the following statements are true and which are false? Give reasons for your answers. (v) In Fig. 5.9, if $AB = PQ$ and $PQ = XY$, then $AB = X
- Q2(ii): 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? (ii) perpendicular lines
- Q2(iii): 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? (iii) line segment
- Q2(iv): 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? (iv) radius of a circle
- Q2(v): 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? (v) square
- Q3: 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.
- Q4: If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.
- Q5: 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.
- Q6: In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
- Q7: 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.)
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