Chapter 5. Introduction to Euclid’s Geometry

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

Class 9 Chapter 5. Introduction to Euclid’s Geometry 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 .

 

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

 False . Because an infinite number of lines can pass through a single point.

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

False . This is based on Euclid's first axiom, which states that a straight line can be drawn between any two points. The straight line passing through these two points is unique and well-defined.

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

True. Because a terminated line can be produced indefinitely .

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

True. Two circles are equal if they have the same radius. If two circles have the same radius, they will have the same size and shape.

(v) In fig. 5.9 , if and  , then .

 

(v) True  .

If two things are equal to the same thing, then they are equal to each other. In this case, AB is equal to PQ, and PQ is equal to XY. Therefore, AB is also equal to XY.

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 .

Solution: (i) Parallel lines: Definition: Parallel lines are lines in a two-dimensional plane that never intersect, no matter how far they are extended.

Other terms that need to be defined first:

Lines: A line is a straight one-dimensional figure that extends infinitely in both directions.

(ii) Perpendicular lines: Definition: Perpendicular lines are two lines that intersect at a right angle (90 degrees).

Other terms that need to be defined first:

Lines: A line is breadthless lenght .

Intersection: The point where two or more lines meet or cross each other.

(iii) Line segment: Definition: A line segment is a part of a line that is bounded by two distinct endpoints. It is a finite portion of a straight line.

Other terms that need to be defined first:

Lines: Lines: A line is breadthless lenght .

Endpoints: The points at the beginning and end of a line segment that mark its boundaries.

(iv) Radius of a circle: Definition: The radius of a circle is the distance from the center of the circle to any point on the circumference (outer edge) of the circle.

Other terms that need to be defined first:

Circle: A circle is a two-dimensional geometric shape that consists of all points in a plane that are equidistant from a fixed center point.

(v) Square: Definition: A square is a four-sided polygon with equal sides and right angles at each corner. It is a special type of rectangle with all sides having the same length.

Other terms that need to be defined first:

Polygon: A polygon is a closed two-dimensional shape with straight sides.

Rectangle: A rectangle is a four-sided polygon with opposite sides being equal and all angles being right angles.

Right Angle: An angle that measures exactly 90 degrees.

Side (of a polygon): One of the line segments that form the edges of a polygon.

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.

The undefined terms are "points," "between" (in postulate i), and "line" (in postulate ii).

These postulates are consistent because they do not contradict any known geometrical principles.

No, these postulates do not directly follow from Euclid's postulates. Euclid's postulates are a set of five fundamental assumptions upon which Euclid's geometry is built, and these two postulates introduce new assumptions about the existence of points and non-collinearity. However, they are consistent with Euclid's postulates.

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

Solution:  In given figure:

Given that AC = BC,

 we can represent the segment AB as the combination of segments AC and CB.

AB = AC + CB

Since AC = BC, we can substitute BC with AC in the above equation:

AB = AC + AC

Therefore, we have proven that if a point C lies between two points A and B such that AC = BC, then

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 .

Solution:  Suppose there are two distinct midpoints, C and D, on the same line segment AB.

In figure :

Since C is a midpoint, we have:

AC = BC

Similarly, since D is a midpoint, we have:

AD = BD

Since C is the midpoint of AB, we have

AB = AC + CB

Similarly, since D is the midpoint of AB, we have

AB = AD + DB

But AC = BC and AD = BD (since C and D are midpoints).

AB = AC + AC = 2 × AC .......... (i)

AB = AD + AD = 2 × AD ...........(ii)

From (i) and (ii) , we get

2 × AC = 2 × AD

AC = AD

But this means that point C and point D coincide, which contradicts our assumption that they were distinct midpoints.

 Therefore, every line segment has one and only one midpoint.

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

 

Solution: In given figure :

Since, the points B lies between A and C

AC = AB + BC ............ (i)

Again , the point C lies between B and D

BD = BC + CD ............. (ii)

Given, AC = BD

AB + BC = BC + CD  [ From (i) and (ii) ]

AB = CD   Proved.

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:  Axiom 5 : The whole is greater than the part .

Example: Let us consider a pizza. The entire pizza is the "whole," and each slice of pizza is a "part."

Axiom 5 states that the entire pizza (the whole) will always be greater in size than any individual slice (the part) of the pizza.

 Class 9 Chapter 5. Introduction to Euclid’s Geometry Exercise 5.2

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

Solution: Given a straight line and a point not on that line, there is only one other straight line that does not intersect the first line and passes through the given point.

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

Solution:  Yes, Euclid's postulate implies the existence of parallel lines. The postulate states that if a line and a point not on that line are given, there is only one other line that does not intersect the first line and passes through the given point. When we have two such lines, they are parallel to each other and will never meet or intersect. Therefore, the existence of parallel lines is implied by Euclid's postulate.