Euclid's Definitions, Axioms and Postulates

Introduction to Euclid's Geometry

Euclid's Definitions, Axioms, and Postulates

Euclid's Geometry, also known as Euclidean Geometry, is considered the study of plane and solid shapes based on different axioms and theorems. The word Geometry comes from the Greek words 'geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Euclid's Geometry was introduced by the Greek mathematician Euclid, where Euclid defined a basic set of rules and theorems for a proper study of geometry. In this section, we are going to learn more about the concept of Euclid's Geometry, the axioms and solve a few examples.

What is Euclid's Geometry?

Euclid's Geometry was introduced by the Father of Geometry i.e. Euclid and is also called Euclidean Geometry. Geometry was originated from the need for measuring land and was studied in various forms in every ancient civilization such as Egypt, Babylonia, India, etc. Euclid's geometry came into play when Euclid accumulated all the concepts and fundamentals of geometry into a book called 'Elements'. This book spoke about the definitions, the axioms, the theorems, and the proof of various shapes. Euclid specifically spoke about the shape, size, and position of solid shapes and various terms associated with them such as the surface, straight or curved lines, points, etc. Some of his fundamentals about solid shapes are :

  • A point has no parts.
  • A line is a breadthless length.
  • The ends of a line are points.
  • A straight line is a line that lies evenly with the points on itself.
  • A surface has a length and breadth only.
  • The edges of a surface are lines.
  • A plane surface is a surface that lies evenly with the straight lines on itself.

Definition of Euclid's Geometry

Euclid's geometry or the euclidean geometry is the study of Geometry based on the undefined terms such as points, lines, and planes of flat spaces. In other words, it is the study of geometrical shapes both plane shapes and solid shapes and the relationship between these shapes in terms of lines, points, and surfaces. Euclid introduced axioms and postulates for these solid shapes in his book elements that help in defining geometric shapes. Euclid's geometry deals with two main aspects - plane geometry and solid geometry. The table below mentions the theorems that were proved by Euclid.

Plane Geometry

Theorem Proved

Congruence of Triangles

Two triangles are congruent if they are similar in shape and size.

Similarity of Triangles

Two triangles are similar in shape but differ in size.

Areas

Area of a plane shape can be measured by comparing it with a unit square.

Pythagorean Theorem

Pythagorean theorem helps in calculating the distance in different situations for Geometric shapes.

Circles

Equal chord determines equal angles and vice versa in a circle.

Regular Polygons

Regular Polygons are equal in sides and angles.

Conic Section

Conic sections include Ellipse, Parabola, and Hyperbola.

 

Solid Geometry

Theorem Proved

Volume

Volume of a shape can be calculated.

Regular Solids

The existence of Platonic Solids.

 

 

 

 

 

 

 

Euclid's Axioms

Euclid's axioms or common notions are the assumptions of the obvious universal truths that have not been proven. But in his book, Elements, Euclid wrote a few axioms or common notions related to geometric shapes. Let us take a look:

Axiom 1: Things that are equal to the same thing are equal to one another.

Suppose the area of a rectangle is equal to the area of a triangle and the area of that triangle is equal to the area of a square. After applying the first axiom, we can say that that the area of the triangle and the square are equal. For example, if p = q and q = r, then we can say p = r.

Axiom 2: If equals are added to equals, the wholes are equal.

Let us look at the line segment AB, where AP = QB. When PQ is added to both sides, then according to axiom 2, AP + PQ = QB + PQ i.e AQ = PB.

Axiom 3: If equals are subtracted from equals, the remainders are equal.

Consider rectangles ABCD and PQRS, where the areas are equal. If the triangle XYZ is removed from both the rectangles then according to axiom 3, the areas of the remaining portions of the two triangles are equal.

Axiom 4: Things that coincide with one another are equal to one another.

Consider line segment AB with C in the center. AC + CB coincides with the line segment AB. Thus by axiom 4, we can say that AC + CB = AB.

Axiom 5: The whole is greater than the part.

Using the same figure as above, AC is a part of AB. Thus according to axiom 5, we can say that AB > AC.

Axiom 6 and Axiom 7: Things that are double of the same things are equal to one another. Things that are halves of the same things are equal to one another.

Axiom 6 and 7 are interrelated. Consider two identical circles with radii (r)1(r)1 and (r)2(r)2 with diameters as (d)1(d)1 and (d)2(d)2 respectively. Since the circles are identical, using both axioms 6 and 7, we can say that

(r)1(r)1 = (r)2(r)2 and (d)1(d)1 = (d)2(d)2.

Euclid's Postulate

For discussing Euclid's postulate, there are a few terms that we need to get familiarized with. Euclid talks about a three-step process from solids to points which is solids-surface-lines-points. At each step, one dimension is lost from the shape. Therefore, a solid is a 3D shape, a surface is a 2D shape, a line is a one dimension shape, and points are dimensions. The term surface means something that has length and breadth only. Whereas a point has no part, has a long length, etc. These terms will help in understanding the postulate better. There is 5 Euclid's postulate, let us take a look:

Postulate 1: A straight line segment can be drawn for any two given points.

This postulate shows us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. Look at the line below, only one line passes through P and Q which is PQ that passes through both Q and P respectively.

Postulate 2: A line segment can be extended in either direction to form a line.

A line segment can be extended in either direction to form a line is the second postulate.

Postulate 3: To describe a circle with any center and radius.

A circle is considered as a plane figure that consists of a set of points that are equidistant from a reference point and can be drawn with its center and radius. According to the third postulate, the shape of a circle does not change when the radius is different. What changes is the size of the circle?

Postulate 4: All right angles are equal to one another.

A right-angle measures at exactly 90° irrespective of the lengths of their arms. Hence according to postulate 4, all right angles are equal to each other. This holds good only for right-angled triangles and not acute angle triangles or obtuse angle triangles.

Postulate 5: If two lines are intersected by a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will intersect each other on that side if produced indefinitely.

When there are two lines cut by a third line, if the sum of the interior angles is less than 180°, then the two lines will meet when extended on that side.

In the image given below, 1+2<1801+2<180. Therefore, Line mm and nn will meet when extended on the side of 1 and 2

Non-Euclidean Geometry

There is a branch of geometry known as Non-Euclidean geometry. Basically, it is everything that does not fall under Euclidean geometry. However, it is commonly used to describe spherical geometry and hyperbolic geometry. Since spherical geometry comes under non-euclidean geometry, to convert it to euclidean or Euclid's geometry or basic geometry we need to change actual distances, location of points, area of the regions, and actual angles.

Examples of Euclid's Geometry

  • Example 1: Bella marked three points A, B, and C on a line such that, B lies between A and C. Help Bella prove that AB + BC = AC.

Solution

AC coincides with AB + BC.

Euclid’s Axiom (4) says that things that coincide with one another are equal to one another. So, it can be deduced that AB + BC = AC

It has been assumed that there is a unique line passing through two points.

  • Example 2: Prove that an equilateral triangle can be constructed on any given line segment.

Solution:

A line segment of any length is given, say AB. Using Euclid’s postulate 3, first, draw an arc with point A as the center and AB as the radius. Similarly, draw another arc with point B as the center and BA as the radius. Mark the meeting point of the arcs as C. Now, draw the line segments AC and BC to form ABCABC.

AB = AC; Arcs of same length. AB = BC; Arcs of same length.

Euclid’s axiom says that things which are equal to the same things are equal to one another. Hence, AB = BC = AC. Therefore, ABCABC is an equilateral triangle.

Example 3: Prove that things that are equal to the same thing are equal to one another.

Solution: According to Euclid's axiom 1, if the area of a triangle is equal to the area of a rectangle and the area of the rectangle is equal to the area of the square, we can say that the area of the triangle is also equal to the area of a square. Hence it is proved that things that are equal to the same thing are equal to one another.

Euclid's Definitions, Axioms and Postulates

CHAPTER -5

Introduction to Euclid's Geometry

Euclid's Definitions, Axioms and Postulates

Euclid's Geometry, also known as Euclidean Geometry, is considered the study of plane and solid shapes based on different axioms and theorems. The word Geometry comes from the Greek words 'geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Euclid's Geometry was introduced by the Greek mathematician Euclid, where Euclid defined a basic set of rules and theorems for a proper study of geometry. In this section, we are going to learn more about the concept of Euclid's Geometry, the axioms and solve a few examples.

What is Euclid's Geometry?

Euclid's Geometry was introduced by the Father of Geometry i.e. Euclid and is also called Euclidean Geometry. Geometry was originated from the need for measuring land and was studied in various forms in every ancient civilization such as Egypt, Babylonia, India, etc. Euclid's geometry came into play when Euclid accumulated all the concepts and fundamentals of geometry into a book called 'Elements'. This book spoke about the definitions, the axioms, the theorems, and the proof of various shapes. Euclid specifically spoke about the shape, size, and position of solid shapes and various terms associated with them such as the surface, straight or curved lines, points, etc. Some of his fundamentals about solid shapes are :

  • A point has no parts.
  • A line is a breadthless length.
  • The ends of a line are points.
  • A straight line is a line that lies evenly with the points on itself.
  • A surface has a length and breadth only.
  • The edges of a surface are lines.
  • A plane surface is a surface that lies evenly with the straight lines on itself.

Definition of Euclid's Geometry

Euclid's geometry or the euclidean geometry is the study of Geometry based on the undefined terms such as points, lines, and planes of flat spaces. In other words, it is the study of geometrical shapes both plane shapes and solid shapes and the relationship between these shapes in terms of lines, points, and surfaces. Euclid introduced axioms and postulates for these solid shapes in his book elements that help in defining geometric shapes. Euclid's geometry deals with two main aspects - plane geometry and solid geometry. The table below mentions the theorems that were proved by Euclid.

Plane Geometry

Theorem Proved

Congruence of Triangles

Two triangles are congruent if they are similar in shape and size.

Similarity of Triangles

Two triangles are similar in shape but differ in size.

Areas

Area of a plane shape can be measured by comparing it with a unit square.

Pythagorean Theorem

Pythagorean theorem helps in calculating the distance in different situations for Geometric shapes.

Circles

Equal chord determines equal angles and vice versa in a circle.

Regular Polygons

Regular Polygons are equal in sides and angles.

Conic Section

Conic sections include EllipseParabola, and Hyperbola.

 

Solid Geometry

Theorem Proved

Volume

Volume of a shape can be calculated.

Regular Solids

The existence of Platonic Solids.

 

 

 

 

 

 

Euclid's Axioms

Euclid's axioms or common notions are the assumptions of the obvious universal truths that have not been proven. But in his book, Elements, Euclid wrote a few axioms or common notions related to geometric shapes. Let us take a look:

Axiom 1: Things that are equal to the same thing are equal to one another.

Suppose the area of a rectangle is equal to the area of a triangle and the area of that triangle is equal to the area of a square. After applying the first axiom, we can say that that the area of the triangle and the square are equal. For example, if p = q and q = r, then we can say p = r.

Axiom 2: If equals are added to equals, the wholes are equal.

Let us look at the line segment AB, where AP = QB. When PQ is added to both sides, then according to axiom 2, AP + PQ = QB + PQ i.e AQ = PB.

Axiom 3: If equals are subtracted from equals, the remainders are equal.

Consider rectangles ABCD and PQRS, where the areas are equal. If the triangle XYZ is removed from both the rectangles then according to axiom 3, the areas of the remaining portions of the two triangles are equal.

Axiom 4: Things that coincide with one another are equal to one another.

Consider line segment AB with C in the center. AC + CB coincides with the line segment AB. Thus by axiom 4, we can say that AC + CB = AB.

Axiom 5: The whole is greater than the part.

Using the same figure as above, AC is a part of AB. Thus according to axiom 5, we can say that AB > AC.

Axiom 6 and Axiom 7: Things that are double of the same things are equal to one another. Things that are halves of the same things are equal to one another.

Axiom 6 and 7 are interrelated. Consider two identical circles with radii (r)1(r)1 and (r)2(r)2 with diameters as (d)1(d)1 and (d)2(d)2 respectively. Since the circles are identical, using both axioms 6 and 7, we can say that

(r)1(r)1 = (r)2(r)2 and (d)1(d)1 = (d)2(d)2.

Euclid's Postulate

For discussing Euclid's postulate, there are a few terms that we need to get familiarized with. Euclid talks about a three-step process from solids to points which is solids-surface-lines-points. At each step, one dimension is lost from the shape. Therefore, a solid is a 3D shape, a surface is a 2D shape, a line is a one dimension shape, and points are dimensions. The term surface means something that has length and breadth only. Whereas a point has no part, has a long length, etc. These terms will help in understanding the postulate better. There is 5 Euclid's postulate, let us take a look:

Postulate 1: A straight line segment can be drawn for any two given points.

This postulate shows us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. Look at the line below, only one line passes through P and Q which is PQ that passes through both Q and P respectively.

Postulate 2: A line segment can be extended in either direction to form a line.

A line segment can be extended in either direction to form a line is the second postulate.

Postulate 3: To describe a circle with any center and radius.

A circle is considered as a plane figure that consists of a set of points that are equidistant from a reference point and can be drawn with its center and radius. According to the third postulate, the shape of a circle does not change when the radius is different. What changes is the size of the circle.

Postulate 4: All right angles are equal to one another.

A right-angle measures at exactly 90° irrespective of the lengths of their arms. Hence according to postulate 4, all right angles are equal to each other. This holds good only for right-angled triangles and not acute angle triangles or obtuse angle triangles.

Postulate 5: If two lines are intersected by a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will intersect each other on that side if produced indefinitely.

When there are two lines cut by a third line, if the sum of the interior angles is less than 180°, then the two lines will meet when extended on that side.

In the image given below, 1+2<1801+2<180. Therefore, Line mm and nn will meet when extended on the side of 1 and 2

Non-Euclidean Geometry

There is a branch of geometry known as Non-Euclidean geometry. Basically, it is everything that does not fall under Euclidean geometry. However, it is commonly used to describe spherical geometry and hyperbolic geometry. Since spherical geometry comes under non-euclidean geometry, to convert it to euclidean or Euclid's geometry or basic geometry we need to change actual distances, location of points, area of the regions, and actual angles.

Examples on Euclid's Geometry

Example 1: Bella marked three points A, B, and C on a line such that, B lies between A and C. Help Bella prove that AB + BC = AC.

Solution

AC coincides with AB + BC.

Euclid’s Axiom (4) says that things that coincide with one another are equal to one another. So, it can be deduced that AB + BC = AC

It has been assumed that there is a unique line passing through two points.

Example 2: Prove that an equilateral triangle can be constructed on any given line segment.

Solution:

A line segment of any length is given, say AB. Using Euclid’s postulate 3, first, draw an arc with point A as the center and AB as the radius. Similarly, draw another arc with point B as the center and BA as the radius. Mark the meeting point of the arcs as C. Now, draw the line segments AC and BC to form ABCABC.

AB = AC; Arcs of same length. AB = BC; Arcs of same length.

Euclid’s axiom says that things which are equal to the same things are equal to one another. Hence, AB = BC = AC. Therefore, ABCABC is an equilateral triangle.

Example 3: Prove that things that are equal to the same thing are equal to one another.

Solution: According to Euclid's axiom 1, if the area of a triangle is equal to the area of a rectangle and the area of the rectangle is equal to the area of the square, we can say that the area of the triangle is also equal to the area of a square. Hence it is proved that things that are equal to the same thing are equal to one another.

Equivalent Versions of Euclid's Fifth Postulate

Equivalent versions of Euclid’s Fifth Postulate: Diagrams, Example

Equivalent versions of Euclid’s Fifth Postulate is very significant in the history of mathematics. We see that, by implication, no intersection of lines will occur when the sum of the measures of the interior angles on the same side of the falling line is exactly ({180^ circ }). There are several equivalent versions of this postulate. One of them is ‘Playfair’s Axiom’ (given by a Scottish mathematician John Playfair in (1729)), as stated below:

“For every line (l) and every point (P) not lying on (l,) there exists a unique line (m) passing through (P) and parallel to ( l.”)

Of all the lines passing through the point (P,) only line (m) is parallel to line (l,) as shown in the diagram below.

What is the Equivalent Version?

Geometry has been taken from a variety of civilisations. Almost every significant civilisation has studied and used geometry in its prime. The Egyptian and the Indian civilisations were more focused on using geometry as a tool. Then Euclid came and changed the way people used to think of geometry. So instead of making it the tool, he thought of geometry as an abstract model of the world in which Euclid lived.

Five Basic Postulates of Euclidean Geometry

Below you can see Euclid’s five postulates:

Postulate 1: A straight line can be drawn from any point to any other point.

This postulate tells you that at least one straight line crosses two distinct points, but it does not say that there cannot be more than one line. However, without mentioning it, Euclid frequently assumed that there is a unique line connecting two distinct points in his work. You can state this result in the form of an axiom as follows:

Axiom: Given two distinct points, there is a unique line that passes through them.

How many lines passing through (P) also pass through (Q) on the given diagram. Only one, that is, the line (PQ.) How many lines passing through (Q) also pass through (P?) Only one, that is, the line (PQ.) Hence, the statement that is given above is self-evident, and it is considered as an axiom.

Postulate 2: A terminated line can be produced indefinitely.

Note that what you call the line segment nowadays is what Euclid called a terminated line. So, according to the present-day terms, the second postulate says that a line segment can be extended on either side to form a line in the given diagram.

Postulate 3: A circle can be drawn with any centre and radius.

Postulate 4: All the right angles are similar (equal) to one another.

Postulate 5: If the straight line that is falling on two straight lines makes the interior angles on the same side of it is taken together less than two right angles, then the two straight lines, if it is produced indefinitely, they meet on the side on which the sum of the angles is less than the two right angles.

Example: The line (PQ) in the given diagram falls on lines (AB) and (CD) such that the sum of the interior angles (1) and (2) is less than ({180^ circ }) on the left side of (PQ.) Therefore, the lines (AB) and (CD) will eventually intersect on the left side of (PQ.)

Difference Between an Axiom and a Postulate

An axiom is a statement that is usually considered self-evident and assumed to be true without proof. It is used as the starting point in math proof for reducing other facts.

The axioms were considered different from the postulates. An axiom would mention a self-evident assumption common to many areas of the inquiry. In contrast, the postulate mentioned a hypothesis specific to a certain line of the inquiry that was approved without proof. 

Example: In Euclid’s Elements, we can compare “common notions” (axioms) with the postulates.

There is generally no difference between what was classically referred to as the “axioms” and the “postulates” in modern mathematics. Modern mathematics distinguishes between the logical axioms and the non-logical axioms, with the latter sometimes referred to as the postulates.

What Does Euclid’s Fifth Postulate Imply?

Euclid’s fifth postulate states that 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 right angles, then both straight lines, if produced indefinitely, will meet on the other side on which the sum of angles is less than two right angles. First, you have to use this postulate and then prove it by using graphs and angles.

Complete step by step answer:

We know that Euclid’s fifth postulate states that if (angle 1 + angle 2 < {180^ circ },), the line (a) and (b) meet on the right side of line (c.)

Now, since you know that (a) and (b) are parallel, then (angle 1 + angle 2 = {180^ circ }), you get to see the below diagram:

Hence, by Euclid’s fifth postulate, lines (a) and (b) will not meet on the right side of (c) as the sum is not less than ({180^ circ })

In the same way, you can see (angle 3 + angle 4 = {180^ circ })

Therefore, by Euclid’s fifth postulate, lines (a) and (b) will not meet on the left side of (c) as the sum is not less than ({180^ circ })

Thus, Euclid’s postulate implies the existence of parallel lines.

Equivalent Versions of Euclid's Fifth Postulate

Equivalent versions of Euclid’s Fifth Postulate: Diagrams, Example

Equivalent versions of Euclid’s Fifth Postulate is very significant in the history of mathematics. We see that, by implication, no intersection of lines will occur when the sum of the measures of the interior angles on the same side of the falling line is exactly \({180^ \circ }\). There are several equivalent versions of this postulate. One of them is ‘Playfair’s Axiom’ (given by a Scottish mathematician John Playfair in \(1729)\), as stated below:

“For every line \(l\) and every point \(P\) not lying on \(l,\) there exists a unique line \(m\) passing through \(P\) and parallel to \( l.”\)

Of all the lines passing through the point \(P,\) only line \(m\) is parallel to line \(l,\) as shown in the diagram below.

What is the Equivalent Version?

Geometry has been taken from a variety of civilisations. Almost every significant civilisation has studied and used geometry in its prime. The Egyptian and the Indian civilisations were more focused on using geometry as a tool. Then Euclid came and changed the way people used to think of geometry. So instead of making it the tool, he thought of geometry as an abstract model of the world in which Euclid lived.

Five Basic Postulates of Euclidean Geometry

Below you can see Euclid’s five postulates:

Postulate 1: A straight line can be drawn from any point to any other point.

This postulate tells you that at least one straight line crosses two distinct points, but it does not say that there cannot be more than one line. However, without mentioning it, Euclid frequently assumed that there is a unique line connecting two distinct points in his work. You can state this result in the form of an axiom as follows:

Axiom: Given two distinct points, there is a unique line that passes through them.

How many lines passing through \(P\) also pass through \(Q\) on the given diagram. Only one, that is, the line \(PQ.\) How many lines passing through \(Q\) also pass through \(P?\) Only one, that is, the line \(PQ.\) Hence, the statement that is given above is self-evident, and it is considered as an axiom.

Postulate 2: A terminated line can be produced indefinitely.

Note that what you call the line segment nowadays is what Euclid called a terminated line. So, according to the present-day terms, the second postulate says that a line segment can be extended on either side to form a line in the given diagram.

Postulate 3: A circle can be drawn with any centre and radius.

Postulate 4: All the right angles are similar (equal) to one another.

Postulate 5: If the straight line that is falling on two straight lines makes the interior angles on the same side of it is taken together less than two right angles, then the two straight lines, if it is produced indefinitely, they meet on the side on which the sum of the angles is less than the two right angles.

Example: The line \(PQ\) in the given diagram falls on lines \(AB\) and \(CD\) such that the sum of the interior angles \(1\) and \(2\) is less than \({180^ \circ }\) on the left side of \(PQ.\) Therefore, the lines \(AB\) and \(CD\) will eventually intersect on the left side of \(PQ.\)

Difference Between an Axiom and a Postulate

An axiom is a statement that is usually considered self-evident and assumed to be true without proof. It is used as the starting point in math proof for reducing other facts.

The axioms were considered different from the postulates. An axiom would mention a self-evident assumption common to many areas of the inquiry. In contrast, the postulate mentioned a hypothesis specific to a certain line of the inquiry that was approved without proof. 

Example: In Euclid’s Elements, we can compare “common notions” (axioms) with the postulates.

There is generally no difference between what was classically referred to as the “axioms” and the “postulates” in modern mathematics. Modern mathematics distinguishes between the logical axioms and the non-logical axioms, with the latter sometimes referred to as the postulates.

What Does Euclid’s Fifth Postulate Imply?

Euclid’s fifth postulate states that 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 right angles, then both straight lines, if produced indefinitely, will meet on the other side on which the sum of angles is less than two right angles. First, you have to use this postulate and then prove it by using graphs and angles.

Complete step by step answer:

We know that Euclid’s fifth postulate states that if \(\angle 1 + \angle 2 < {180^ \circ },\), the line \(a\) and \(b\) meet on the right side of line \(c.\)

Now, since you know that \(a\) and \(b\) are parallel, then \(\angle 1 + \angle 2 = {180^ \circ }\), you get to see the below diagram:

Hence, by Euclid’s fifth postulate, lines \(a\) and \(b\) will not meet on the right side of \(c\) as the sum is not less than \({180^ \circ }\)

In the same way, you can see \(\angle 3 + \angle 4 = {180^ \circ }\)

Therefore, by Euclid’s fifth postulate, lines \(a\) and \(b\) will not meet on the left side of \(c\) as the sum is not less than \({180^ \circ }\)

Thus, Euclid’s postulate implies the existence of parallel lines.

Solved Examples

Q.1. If \(A, B\) and \(C\) are three points on a line, and \(B\) lies between \(A\) and \(C\) in the given diagram, then prove that \(AB+BC=AC.\)

Ans: In the given diagram above, \(AC\) coincides with \(AB+BC\)
Also, Euclid’s Axiom \((4)\) says that things that coincide with one another are equal. So, it can be deduced that
\(AB+BC=AC\)
Note that in the given solution, it has been considered that there is a unique line that is passing through two points.

Q.2. Prove that an equilateral triangle may be constructed on any of the given line segments.
Ans: In the segment above, a line segment of any length is given, say AB in the given diagram.

Here, you have to do some construction. Using Euclid’s postulate \(3,\) you can draw a circle with point \(A\) as the centre and \(AB\) as the radius in the given diagram. Similarly, you draw another circle with point \(B\) as the centre and \(BA\) as the radius. The two circles that meet at a point say \(C.\)
Now, draw the line segments \(AC\) and \(BC\) to form \(\Delta A B C\) in the given diagram.
So, you have to show that this triangle is the equilateral triangle, i.e., \(AB=AC=BC.\)
Now, \(AB=AC\), since they are the radii of the same circle          (1)
Similarly, \(AB=BC\)   (Radii of the same circle)                                (2)
From the given two facts, and Euclid’s axiom that things that are equal to the same thing are equal, you can conclude that \(AB=BC=AC\)
So, \(\Delta A B C\) is an equilateral triangle.

Q.3. Prove that the two lines that are both parallel to the same line are parallel to each other.
Ans:
 Three lines, \(l, m\) and \(n\) in a plane such that \(m||l\) and \(n||l.\)
Prove: \(m||n\)
If possible, let \(m\) be not parallel to \(n\). Then, \(m\) and \(n\) intersect in a unique point, say \(P.\)

Thus, through a point \(P\) outside \(l,\) two lines \(m\) and \(n,\) both parallel to \(l.\) This is a  contradiction to the parallel axiom. So, the supposition is wrong.
Hence, \(m||n.\)

Q.4. Prove that two distinct lines cannot have more than one point in common.
Ans: 
Here, we are given two lines \(l\) and \(m.\) You need to prove that they have only a single point in common. Let us assume that the two lines intersect in two distinct points, say \(P\) and \(Q.\) So, you have two lines that are passing through two distinct points \(P\) and \(Q.\)

But this assumption clashes with the axiom that only one line can pass through two distinct points. So, the assumption that we started with that two lines can pass through two distinct points is wrong.
From this, we understand that we are forced to conclude that two distinct lines cannot have more than one point in common.

Q.5. Prove that if lines \(AB, AC, AD\) and \(AE\) are parallel to line \(l,\) then points \(A, B, C, D\) and \(E\) are collinear.
Ans:
 Lines \(AB, AC, AD\) and \(AE\) are parallel to a line \(l.\)
Prove: \(A,B,C, D\) and \(E\) are collinear.
Since lines \(AB, AC, AD\) and \(AE\) are parallel to a line \(l.\) 
Therefore, point \(A\) lies outside \(l\) and through \(A\) lines \(AB, AC, AD\) and \(AE\) are drawn parallel to \(l.\) But, by parallel axiom, only one line can be drawn parallel to \(l, \) through a point outside it. 
Therefore, the points \(A, B, C, D, E\) lie on the same line.
Hence, points \(A, B, C, D\) and \(E\) are collinear points.

Q.1. What is the equivalent version?
Ans:
 Geometry has taken from a variety of civilisations. Almost every significant civilisation has studied and used geometry in its prime. The Egyptian and the Indian civilisations were more focused on using geometry as a tool. Then Euclid came and changed the way people use to think in geometry. So instead of making it the tool, he thought of geometry as an abstract model of the world in which Euclid lived.

Q.2. Which of the five postulates is equivalent to Playfair’s postulate?
Ans: 
The definition of the fifth postulate is taken so that the parallel lines are the lines that do not intersect or have some line that is intersecting them in the same angles. Playfair’s axiom is contextually equivalent to Euclid’s fifth postulate and is thus logically independent of the first four postulates.

Q.3. How would you rewrite Euclid’s fifth postulate?
Ans:
 This postulate can be rewritten in reference to the given below diagram: You can consider a straight line \(PQ\) that falls on both straight lines \(AB\) and \(CD\) in such a way that the total sum of the interior angles \(\angle 1\) and \(\angle 2\) is less than \(180\) degrees on the left side of the straight line \(PQ.\) In this case, \(AB\) and \(CD,\) if produced, will meet at the left side of \(PQ.\)

Q 4. Who proved Euclid’s fifth postulate?
Ans:
 The Persian mathematician, astronomer, philosopher, and poet Omar Khayyam \((1050-1153)\) attempted to prove the fifth postulate.

Q.5. What is the equivalent of the fifth postulate?
Ans:
 If the straight line that is falling on two straight lines makes the interior angles on the same side of it is taken together less than two right angles, then the two straight lines, if it is produced indefinitely, they meet on the side on which the sum of the angles is less than the two right angles.

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