NCERT Exemplar Class 9 Maths Solutions Chapter 5 Introduction to Euclids Geometry

NCERT Exemplar Class 9 Maths Solutions Chapter 5-Exercise 5.1

Question:1

The three steps from solids to points are:
(A) Solids - surfaces - lines - points
(B) Solids - lines - surfaces - points
(C) Lines - points - surfaces - solids
(D) Lines - surfaces - points - solids

Answer:

(A) Solids - surfaces - lines - points
Solution:
Solid: A three-dimensional object having a definite shape, size and volume is called a solid.
Surface: A surface is the outermost layer of a physical object (solid) or we can say that surface is a kind of a plane.
Lines: The boundaries of different surfaces are called curves. These curves are lines when they are straight. Lines join two or more points together.
Points: A point is an exact location. It has no size, it has only position. Different points join together to form a line.
Hence the three steps from solids to points are solids – surfaces – lines – points.
Therefore option (A) is correct.

Question:2

The number of dimensions, a solid has:
(A) 1
(B) 2
(C) 3
(D) 0

Answer:

(C) 3
Solution:
We know that solid is a three-dimensional object having definite shape, size and volume.
Three-dimensional means that to represent the shape, we need three axes (x, y, z).
The three dimensions of solids are generally known as length, width and height.
For example, a cuboid is a solid with three dimensions as shown:

In case of a cube, we know that length = width = height.
Hence all are called side length.
Solid has three dimensions therefore option (C) is correct.

Question:3

The number of dimensions, a surface has:
(A) 1
(B) 2
(C) 3
(D) 0

Answer:

(B) 2
Solution:
As we know that a surface is the outermost layer of any physical object or solids. In other words, we can say that the surface is a generalization of a plane. Therefore the surface has only length and breadth. Hence the surface has two dimensions
Option B is correct

Question:4

The number of dimension, a point has:
(A) 0
(B) 1
(C) 2
(D) 3

Answer:

(A) 0
Solution:
We know that the dimensions of a point are zero because we cannot define the size of a point. In other words, we can say that a point is an exact location. It has no size, but has only position. A point is a place, not a thing. Therefore it has no dimension.
Hence option (A) is correct.

Question:5

Euclid divided his famous treatise “The Elements” into:
(A) 13 chapters
(B) 12 chapters
(C) 11 chapters
(D) 9 chapters

Answer:

(A) 13 chapters
Solution:
Euclid divided his famous treatise “The Elements” into 13 chapters. It is a collection of definitions, postulates, theorems and mathematical proofs of these theorems.
The books cover elementary number theory, plane and solid Euclidean geometry. Euclid's Elements is considered as the most influential and successful textbook ever written.
Hence option (A) is correct

Question:6

The total number of propositions in the Elements are:
(A) 465
(B) 460
(C) 13
(D) 55

Answer:

Answer:(A) 465
Solution:
Euclid divided his famous treatise “The Elements” into 13 chapters. It is a collection of definitions, postulates, theorems and mathematical proofs of these theorems.
By propositions, we mean theorems and constructions. Euclid’s total propositions are 465.
Hence Euclid deduced 465 propositions in the elements using his axioms, postulates, definitions and theorems.

Question:7

Boundaries of solids are:
(A) surfaces
(B) curves
(C) lines
(D) points

Answer:

(A) surfaces
Solution:
Solid: A three-dimensional object having a definite shape, size and volume is called a solid.
Surface: A surface is the outermost layer of a physical object (solid) or we can say that surface is a kind of a plane.
Hence, the surfaces are the boundaries of solids.
Hence option (A) is correct.

Question:8

Boundaries of surfaces are:
(A) Surfaces
(B) curves
(C) lines
(D) points

Answer:

(B) curves
Solution:
Solid is a three-dimensional object having a definite shape, size and volume is called a solid. A surface is the outermost layer of a physical object (solid). The boundaries of different surfaces are called curves. These curves are called lines when they are straight. Lines join two or more points together.
Hence, option B is correct.

Question:9

In Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio
(A) 1 : 3 : 4
(B) 4 : 2 : 1
(C) 4 : 4 : 1
(D) 4 : 3 : 2

Answer:

(B) 4 : 2 : 1
Solution:
In Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio-
4 : 2 : 1 = length : breadth : thickness.
Hence option (B) is correct.

Question:10

A pyramid is a solid figure, the base of which is
(A) only a triangle
(B) only a square
(C) only a rectangle
(D) any polygon

Answer:

Answer:
(D) any polygon
Solution:
As we know that a pyramid is formed by connecting a polygonal base and a point called the apex.
The base of the pyramid is in the form of the polygon that can be a triangle or square or some other polygon.

From left to right: Cone or circular pyramid, Triangular pyramid, Rectangular pyramid, Pentagonal pyramid
Hence option (D) is correct.

Question:11

The side faces of a pyramid are:
(A) Triangles
(B) Squares
(C) Polygons
(D) Trapeziums

Answer:

(A) Triangles
Solution:
Pyramid:
We know that a pyramid is formed by connecting a polygonal base and a point called the apex.It is a conic solid with a polygonal base and triangular sides.
Hence the side faces of a pyramid are always triangles.
For example,

From left to right: Cone or circular pyramid, Triangular pyramid, Rectangular pyramid, Pentagonal pyramid
Option A is correct.

Question:12

It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is:

(A) First Axiom

(B) Second Axiom

(C) Third Axiom
(D) Fourth Axiom

Answer:

(B) Second Axiom
Solution:
Euclid’s second axiom illustrate this statement
Euclid’s second axiom: To produce (extend) a finite straight line continuously in a straight line: If equals are added to equals, then the wholes are equal (Addition property of equality). Here we can see that
x + y = 10
then we are adding an equal quantity, i.e., z to both.
x + y + z = 10 + z
Hence, option (B) is correct.

Question:13

In ancient India, the shapes of altars used for household rituals were:
(A) Squares and circles
(B) Triangles and rectangles
(C) Trapeziums and pyramids
(D) Rectangles and squares

Answer:

(A) Squares and circles
Solution:
In India, the shapes of altars used for household rituals were squares and circles. Other shapes like rectangles, triangles, trapeziums, pyramids and rectangles were used in public worship. In ancient times, a Hindu home was required to have three fires burning at three different altars. The three altars were to be of the same area but different shapes. Hence, in ancient India, square and circles shapes of altars were used for household rituals.

Question:14

The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is:
(A) Seven
(B) Eight
(C) Nine
(D) Eleven

Answer:

Answer:(C) Nine
Solution:
The Sriyantra is a mystical diagram used in Shri Vidya school of Hinduism. It consists of nine interlocking triangles. These triangles surround a central point known as a Bindu. These triangles represent the cosmos and the human body.
Hence there are nine interwoven isosceles triangles in Sriyantra (in the Atharvaveda).
Therefore, option C is correct

Question:15

Greeks emphasized on:
(A) Inductive reasoning
(B) Deductive reasoning
(C) Both A and B
(D) Practical use of geometry

Answer:

Answer:(B) Deductive reasoning
Solution:
Inductive reasoning is a technique for thinking where the premises are seen as providing some proof, however not full affirmation, of the reality of the end. It is depicted as a technique where one's encounters and perceptions, including what is found out from others, come up as an overall truth. Deductive reasoning is the way toward thinking from at least one proclamations (statements) to arrive at an obvious end result. Greeks emphasized the study of Greek philosophy and Greek history of mathematics and logic. Hence Greeks emphasized on deductive reasoning.
Hence option (B) is correct.

Question:16

In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for:
(A) Public worship
(B) Household rituals
(C) Both A and B
(D) None of A, B, C

Answer:

(A) Public worship
Solution:
An altar is a structure whereupon contributions, for example, sacrifices are made for religious purposes.
Public worship: In ancient India for public worship, combined shapes of rectangles, triangles and trapeziums were used.
Household rituals: In ancient India for Household rituals, square and circle shapes were used.
Hence option (A) is correct.

Question:17

Euclid belongs to the country:
(A) Babylonia
(B) Egypt
(C) Greece
(D) India

Answer:

(C) Greece
Solution:
Euclid was a Greek mathematician also known as the “founder of geometry” or “father of geometry” He was born in Greece. Hence, we can say that “Euclid belongs to Greece”.
Hence option (C) is correct.

Question:18

Thales belongs to the country:
(A) Babylonia
(B) Egypt
(C) Greece
(D) Rome

Answer:

(C) Greece
Solution:
Thales was born in Greece in 624 – 620 BC. Hence we can say that “Thales belongs to the country Greece”
Hence option (C) is correct.

Question:19

Pythagoras was a student of:
(A) Thales
(B) Euclid
(C) Both A and B
(D) Archimedes

Answer:

(A) Thales
Solution:
Pythagoras was an Ionian Greek philosopher and the originator of Pythagoreanism. He was a student of Thales.
Hence, option (A) is correct.

Question:20

Which of the following needs a proof?
(A) Theorem
(B) Axiom
(C) Definition
(D) Postulate

Answer:

(A) Theorem
Solution:
Theorem: A statement or proposition not plainly obvious but rather demonstrated by a chain of reasons; a fact set up by methods for acknowledged certainties
Axiom: A statement or proposition which is regarded as being self-obviously evident, established or accepted.
Definition: a statement that gives the exact meaning of the given word.
Postulate: to accept or guarantee as true, whether it is existent, or necessary.
Hence, Axioms, Definitions and postulates do not need a proof.
Hence only theorems need a proof.
Option (A) is correct.

Question:21

Euclid stated that all right angles are equal to each other in the form of
(A) an axiom
(B) a definition
(C) a postulate
(D) a proof

Answer:

(C) a postulate
Solution:
Euclid gave us five postulates.
Postulate: to accept or guarantee as true, whether it is existent, or necessary.
According to Euclid’s fourth postulate, all right angles are equal to each other.
Hence, Euclid’s statement “all right angles are equal to each other” is in the form of a postulate.
Option (C) is correct.

Question:22

‘Lines are parallel if they do not intersect’ is stated in the form of
(A) an axiom
(B) a definition
(C) a postulate
(D) a proof

Answer:

Answer:
(B) a definition
Solution:
Parallel lines: According to the definition of parallel lines “lines are parallel if they do not intersect”.

They are not intersecting.
Definition: a statement that gives the exact meaning of the given word.
Hence, ‘lines are parallel if they do not intersect’ is stated in the form of definition.
Hence, Option (B) is correct.

NCERT Exemplar Class 9 Maths Solutions Chapter 5-Exercise 5.2

Question:1

Write whether the following statements are True or False? Justify your answer: Euclidean geometry is valid only for curved surfaces.

Answer:

False
Solution:
Euclidean geometry is based on his axioms and postulates. These are valid only for plane surfaces. Hence we know that Euclidean geometry is valid only for figures in the plane. Thus, Euclidean geometry is not valid of curved surfaces.
Therefore, the given statement is false

Question:2

Write whether the following statements are True or False? Justify your answer: The boundaries of the solids are curves

Answer:

False
Solution: Solid: solid is an object that has length, width and height, it is not a flat shape. Hence if we notice a solid then we can see that the boundaries of solids are not curves but surface. Therefore, the given statement is false.

Question:3

Write whether the following statements are True or False? Justify your answer: The edges of a surface are curves.

Answer:

False
Solution:
The boundaries of different surfaces are called curves.
The edges of a surface are called lines.
Lines join two or more points together.
Let us consider a solid as shown:

In the given figure if we notice edge ab, we can see that it is not a curve it is a line.
Hence the given statement is false.

Question:4

Write whether the following statements are True or False? Justify your answer: The things which are double of the same thing are equal to one another.

Answer:

True
Solution:
Yes, it is true that the things which are double of the same thing are equal because it is the conclusion of one of Euclid’s axioms. Euclid’s first axiom states that things which are equal to the same thing are also equal to one another. Hence, the things which are double of the same thing are equal to one another.
Hence the given statement is true.

Question:5

Write whether the following statements are True or False? Justify your answer: If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.

Answer:

True
Solution:
Euclid’ fourth axiom states that things that coincide with one another are equal to one another (Reflexive Property). Now, it is given that a quantity B is a part of another quantity A. We can assume that another part of A can be equal to a quantity C. So, A = B + C. Yes, it is true “If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C” because it is the conclusion of one of Euclid’s axioms.
Hence the given statement is true.

Question:6

Write whether the following statements are True or False? Justify your answer: The statements that are proved are called axioms.

Answer:

False
Solution:
Axiom: A statement or proposition which is regarded as being self-obviously evident, established or accepted. Hence, an axiom is a statement that is well established. It is accepted without controversy or question.
Theorem: A statement or proposition not plainly obvious but rather demonstrated by a chain of reasons; a fact set up by methods for acknowledged certainties
The given statement “The statements that are proved are called axioms” is not true
Because the statements that are proved are called theorems, not axioms.
Hence the given statement is False.

Question:7

Write whether the following statements are True or False? Justify your answer:
“For every line l and for every point P not lying on a given line, there exists a unique line m passing through P and parallel to l is known as Playfair’s axiom.”

Answer:

True
Solution:
Euclid’s fifth postulate: according to Euclid’s fifth postulate if two lines are drawn which intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other. So according to Euclid’s fifth postulate, it is true that a line m can pass from point p with parallel to line l.
Playfair's axiom: Given a line in a plane, and a point not on this line. At most one line parallel to the given line can be drawn through the point. So Playfair's axiom is an axiom which can be used instead of the fifth postulate of Euclid.
Hence the given statement is true.

Question:8

Write whether the following statements are True or False? Justify your answer: Two distinct intersecting lines cannot be parallel to the same line.

Answer:

True
Solution:
Euclid’s fifth postulate: if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Using the above, we can see that two distinct intersecting lines cannot be parallel to the same line.
Here we can take an example: line AB and CD are intersecting. If we take a line EF parallel to AB then EF is not parallel to CD.

Hence the given statements is true.

Question:9

Write whether the following statements are True or False? Justify your answer:
Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

Answer:

True
Solution:
Euclid’s fifth postulate: if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
It means that if the sum of angles ÐA and ÐB in the figure is less than the sum of two right angles then A, B meet on the same side of angle A and B continued indefinitely.

All attempts to prove the fifth postulate as a theorem led to great achievement in the creation of several other geometries. These geometries are different from Euclidean geometry and called non-Euclidean geometry.
Hence the given statement is true.

NCERT Exemplar Class 9 Maths Solutions Chapter 5-Exercise 5.3

Question:1

Solve each of the following question using appropriate Euclid’s axiom: Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.

Answer:

Sales in September are also equal.
Solution:
Let us suppose that the sales of two salesmen in august is A. As per the question, in September, each salesman doubles his sale as compared to sale in august.
Thus, the sale of first salesman is 2A and sale of second salesman is also 2A.
Hence the sales of both the salesman are equal in September.
According to Euclid’s axiom, things which are double of the same things are equal to one another.
Hence, the sales in September are also equal.

Question:2

Solve each of the following question using appropriate Euclid’s axiom: It is known that x + y = 10 and that x = z. Show that z + y = 10?

Answer:

Euclid’s axiom According to Euclid’s axiom, if equals are added to equals, the wholes are equal.
To prove: z + y = 10
In this question given that x + y = 10, x = z.
Hence add y to both sides in x = z.
x + y = z + y
z + y = 10 ( x + y = 10)
Hence proved

Question:3

Solve each of the following question using appropriate Euclid’s axiom: Look at the Fig. 5.3. Show that length AH > sum of lengths of AB + BC + CD.

Answer:

To prove: AH > Sum of lengths of AB + BC + CD
Euclid axiom says that whole is greater than the part.
Here the given figure is

AB + BC + CD =

Here we notice that AB + BC + CD is a part of the whole line AH. According to Euclid axiom whole is always greater than the part. Hence, AH > sum of lengths of AB + BC + CD.

Question:4

Solve each of the following question using appropriate Euclid’s axiom: In the Fig.5.4, we have AB = BC, BX = BY. Show that AX = CY.

Answer:

To prove: AX = CY
Here it is given that
AB = BC, BX = BY
Euclid’s axiom says that if equals are subtracted from equals, the remainders are equal.
Subtract BX from AB and BY from BC.
AB – BX = BC – BY
AX = CY
Hence proved.

Question:5

Solve each of the following question using appropriate Euclid’s axiom: In the Fig.5.5, we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC.

Answer:

Solution:
To prove: AC = BC
Here given that
AX = CY … (1)
Here X is mid-point of AC

2AX = 2CX = AC … (2)
Also, Y is mid-point of BC

2BY = 2CY = BC … (3)
Euclid’s axiom says that things which are double of the same things are equal to one another.
Hence from equation (1, 2, 3)
2AX = 2CY
Implies AC = BC
Hence proved

Question:6

Solve each of the following question using appropriate Euclid’s axiom: In the Fig.5.6, we have

and AB = BC. Show that BX = BY.

Answer:

Solution:
To prove: BX = BY
Here, given that

;
2BX = AB ; 2BY = BC
AB = BC (Given)
According to Euclid’s axiom, things which are double of the same thing are equal to one another.
Hence,
2BX = 2BY
BX = BY
Hence Proved.

Question:7

Solve each of the following question using appropriate Euclid’s axiom: In the Fig.5.7, we have 1 = 2, 2 = 3. Show that 1 = 3.

Answer:

To Prove: 1 = 3
Here, it is given that 1 = 2, 2 = 3
We observe that 1 and 3 both are equal to 2. In other words, both are equal to the same thing. Hence from Euclid’s axiom, things which are equal to the same thing are equal to each other.
So, 1 = 3
Hence proved.

Question:8

Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.8, we have 1 = 3 and 2 = 4. Show that A = C.

Answer:

To prove: A = C
Here given that 1 = 3, 2 = 4
According to Euclid’s axiom, if equals are added to equals then wholes are also equal.
So here if we add 1, 2 then it is equal to the sum of 3, 4.
1 + 2 = 3 + 4
( 1 + 2 = A ; 3 + 4 = C)
A = C
Hence proved

Question:9

Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.9, we have ABC = ACB, 3 = 4. Show that 1 = 2.

Answer:

Solution:
Given that: ABC = ACB, 3 =4
To prove: 1 = 2
Euclid’s axiom says that if equals are subtracted from equals, then remainders are also equal.
So, as ABC = ACB, 3=4
We can write,
ABC - 4 =ACB –3
Now, ( ABC = 1 + 4 ;ACB = 2 + 3)
1 = 2
Hence proved

Question:10

Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.10, we have AC = DC, CB = CE. Show that AB = DE.

Answer:

Given:
AC = DC, CB = CE
To prove: AB = DE
Euclid axiom says that if equals added to equals, then wholes are also equal. Now, as
AC = DC, CB = CE
We can write:
AC + CB = DC + CE
{ AB= AC + CB; DE = DC + CE}
AB = DE
Hence proved.

Question:11

Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.11, if OX=0.5XY, PX=0.5XZ and OX = PX, show that XY = XZ.

Answer:

Given that:
OX = 0.5 XY, PX = 0.5 XZ, OX = PX
To prove: XY = XZ
OX = 0.5XY ; PX = 0.5 XZ
2OX = XY … (1) ; 2PX = XZ … (2)
OX = PX … (3) (Given)
Euclid axiom says that things which are double of the same things are equal to one another.
Hence from equation (1), (2), (3)
2OX = 2PX
XY = XZ
Hence proved.

Question:12

i) Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.12:

AB = BC, M is the mid-point of AB and N is the mid- point of BC. Show that AM = NC.
ii) Solve each of the following question using appropriate Euclid’s axiom: In the Fig. 5.12:

BM = BN, M is the mid-point of AB and N is the mid-point of BC. Show that AB = BC.

Answer:

i) Here AB = BC
If M is the midpoint of AB then
AM = MB = 0.5AB
If N is mid point of BC then
BN = NC = 0.5 BC
According to Euclid’s axiom, things which are halves of the same thing are equal to one another.
We have, AB = BC
Multiply both sides by 0.5
0.5 AB = 0.5 BC
AM = NC
Hence proved.
ii) Here, BM = BN
If M is mid-point of AB then
AM = MB
2AM = 2BM = AB
If N is mid-point of BC then
BN = NC
2BN = 2NC = BC
According to Euclid’s axiom, things which are double of the same thing are equal to one another.
Now, BM = BN
Multiply both sides by 2
2BM = 2BN
Hence,
AB = BC
Hence proved.

NCERT Exemplar Class 9 Maths Solutions Chapter 5-Exercise 5.4

Question:1

Read the following statement:
An equilateral triangle is a polygon made up of three-line segments out of which two-line segments are equal to the third one and all its angles are 60° each.
Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

Answer:

Defined terms are:
(1) Equilateral triangle: An equilateral triangle is a triangle with all three sides of equal length and all angles equal.
(2) Polygon: a plane figure with at least three straight sides and angles.
(3) Line segment: A line segment is a part of a line having two end points.
(4) Angle: The figure formed by two rays meeting at a common end point is called an angle.
(5) Acute angles: The angle whose measures are less than 90° are called acute angles.
Undefined terms are:
(1) Line
(2) Point
According to Euclid’s axiom, things which are equal to the same thing are equal to one another.
Here, angles of equilateral triangle are 60° and two line segments are equal to third one.
Therefore, by using Euclid’s axiom, all three sides and angles of an equilateral triangle are equal.

Question:2

Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate.

Answer:

Euclid’s fifth postulate: if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
It means that if the sum of angles A and B in the figure is less than the sum of two right angles then A, B meet on the same side of angle A and B continued indefinitely.

Now the given statement is “Two intersecting lines cannot be perpendicular to the same line”
Hence we can see that the lines A1 and B1 are intersecting and so, are not perpendicular to the same line (D).
The given statement “Two intersecting lines cannot be perpendicular to the same line” is an equivalent version to the Euclid’s fifth postulate. Because it is a conclusion from the fifth postulate.

Question:3

Read the following statements which are taken as axioms:

1. If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
2. If a transversal intersects two parallel lines, then alternate interior angles are equal.

Is this system of axioms consistent? Justify your answer.

Answer:

Given the system of axioms is not consistent.
Solution:
Axiom: A statement or proposition which is regarded as being self-obviously evident, established or accepted. Hence, it is accepted without controversy or question.
Statement (i) is False
We know that if a transversal intersects two parallel lines, then each pair of corresponding angles are equal. It is a theorem.
Theorem: A statement or proposition not plainly obvious but rather demonstrated by a chain of reasons; a fact set up by methods for acknowledged certainties
Statement (ii) is true
If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. It is also a theorem.

In the given figure, 3 = 6, 4 = 5 (alternate interior)
1 = 5, 2 = 6, (corresponding angles)
3 = 7, 4 = 8 (corresponding angles)
Hence statement (i) is not an axiom.
Statement (ii) is true and an axiom
Hence, the given system of axioms is not consistent.

Question:4

Read the following two statements which are taken as axioms:
(i) If two lines intersect each other, then the vertically opposite angles are not equal.
(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°.
Is this system of axioms consistent? Justify your answer.

Answer:

Given system of axioms is not consistent.
Solution:
Axiom: A statement or proposition which is regarded as being self-obviously evident, established or accepted. Hence, it is accepted without controversy or question.
Statement (i) is False
We know that if two lines intersect each other, then the vertically opposite angles are equal. This is a theorem.
Theorem: A statement or proposition not plainly obvious but rather demonstrated by a chain of reasons; a fact set up by methods for acknowledged certainties
Statement (ii) is true
We know that, if a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. It is an axiom.
Thus (i) statement is not an axiom and (ii) statement is an axiom.
Hence given system of axioms is not consistent.

Question:5

Read the following 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. Things which are double of the same thing are equal to one another.

Check whether the given system of axioms is consistent or inconsistent.

Answer:

Given the system of axioms is consistent.
Solution:
Euclid’s axioms:

1. Things that are equal to the same thing are also equal to one another
2. If equals are added to equals, then the wholes are equal
3. If equals are subtracted from equals, then the differences are equal
4. Things that coincide with one another are equal to one another
5. The whole is greater than the part.

Hence we can see (i) is an axiom. (same as axiom 1)
Hence we can see (ii) is also an axiom. (same as axiom 2)
(iii) Yes, it is true that the things which are double of the same thing are equal because it is the conclusion of one of Euclid’s axioms.
Euclid’s first axiom states that things which are equal to the same thing are also equal to one another.
Hence, the things which are double of the same thing are equal to one another.
Hence the given statement is true.
Therefore, the given system of axioms is consistent.