Chapter-5, Introduction To Euclid's Geometry

Explanation: 

Option (A)Solids-surfaces-lines-points.

1. Solid have form, size, and position. Boundaries are referred to as surfaces.

2. The boundaries of surfaces are curves or straight lines. The points are the endings of these lines.

3. The surface is a plane generalization that is not necessarily flat, that is, the curvature which is, not necessarily null.

4. This is also analogous to a straight-line generalizing curve.

5. There are many more accurate definitions, depending on the context and the mathematical instruments which are many more accurate definitions.

Hence, the three steps from solids to points are solids-surfaces-lines-points.

The explanation for the incorrect options are:

Option(B):1.The option Solids-lines-surfaces-points, does not match with the other three steps from the solids to points that are solids-surfaces-lines-points. Therefore option B is incorrect.

Option(C): This option Lines-points-surfaces-solids which does not match with the result of the three steps from solids to the points are solids-surfaces-lines-points therefore option C is also incorrect.

Option( D):1 The option Lines-surfaces-points-solids will not match with the result of three steps from solids to the points that are solids-surfaces-lines-points, therefore option D is also incorrect.

Explanation: 

 ( C) 3

The three dimensions of a cube or a cuboid are described by its length, breadth, and height. Therefore, a solid has three dimensions.

Explanation: 

(B) 2 

The plane surface will have two dimensions because there are two coordinates that are possible for each point on a surface.

Explanation: 

 (A) 

The number of dimensions in a point is zero because a point has no length, no width, and no height. A point is always zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.

Explanation: 

(A) 13 chapters

Euclid’s Elements is a mathematical treatise consisting of 13 chapters which are attributed to the ancient  Greek mathematician named Euclid in Alexandria, Ptolemaic Egypt in 300 BC. He presented a set of formal and logical arguments based on the basic terms and axioms, which provided a systematic method of rational exploration that helped mathematicians, philosophers, and scientists well into the 19th century.

Explanation: 

 ( A)465

1. The statements which are proved are called propositions or theorems, which are a part of geometry and were exercised by the famous Greek mathematician named Euclid, who also explained it in his book, Elements.

2. The elements were also divided into thirteen books that popularized geometry all over the world.

3. The book Elements also has a collection of definitions, axioms(postulates), propositions theorems and constructions), and also the mathematical proofs of the propositions.

4. Theorems or the propositions the statements that are proved by Euclid, All 465 propositions are deduced by Euclid in a logical chain by using his postulates, axioms, definitions, and by his theorems.

Explanation: 

(A) surfaces 

The boundaries of the solids are called surfaces as they have two dimensions. A solid is a three-dimensional object which has a shape, volume, and size, whereas a surface is a physical object's outermost layer, which is why the boundaries of solids are called surfaces.

Explanation: 

(B) curves

The boundaries of surfaces are called curves or straight lines. These lines will end in its points which state that the boundaries of the solids are surfaces. A surface always has length and breadth.

Explanation: 

(B) 4:2:1

The Indus Valley Civilization often denoted by the city Harappa spanned from3200 from 1300 BC.In Indus Valley Civilization the dimensions of the bricks used for the constructions were in the ratio of 4:2:1 that is length:breadth: thickness.

Explanation: 

(D) Any polygon

A pyramid is a 3D polyhedron with the base of a polygon along with three or more triangle-shaped faces which meet at the point that is above the base. The triangular sides are referred to as faces whereas the point above the base is referred to as the apex. A pyramid is made by the connection of the base to the apex.

Explanation: 

(A) Triangles

A pyramid is a polyhedron that has a base and three or more triangular faces which meet at the top called the apex. There will be at least three triangular faces and one base in a pyramid. The point where two faces meet is called the vertices, and a vertex at the top is where all the triangular faces meet. Hence, option (A) triangles, which are the side faces of a pyramid, is the correct answer.

Explanation: 

(B) Second axiom

The equation x+y=10

Now we have to add an equal value z to both the sides, x+y+z=10+z

As in Euclid’s second axiom, whenever the equals are added together, it will result in the wholes as equals. Thats why, the above statements illustrate that Euclid’s second axiom is the correct option.

Explanation: 

(A) Squares and circles

The geometrical shapes were used in ancient times to build altars and fireplaces which held the holy Agni and around which many other rituals used to take places such as yagya, and weddings. This is one of the classic examples that, in ancient times the knowledge of mathematics was present in India, and our Indian mathematicians made huge contributions to the world’s mathematical heritage. The concept of geometry came from the construction of these altars and fireplaces. For the household rituals, the altars and fireplaces used were in the shape of squares and circles.

Explanation: 

(c) Nine

The central portion of Sri Yantra has nine interwoven isosceles triangles. The Sri Yantra is a form of mystical diagram, known as a Yantra and the nine interlocking triangles radiate out from its central point. These triangles are arranged in such a way that they produce 43 subsidiary triangles.

Explanation:

(B) Deductive reasoning

The Greeks were interested in establishing the truth of the statements which they discover while using deductive reasoning. Thales, a Greek mathematician, was credited with giving the first known proof.

Explanation:

(A)Public rituals

In the Vedic period, people used squares and circular-shaped altars for their household rituals where as the altars in a combination with triangles, rectangles, and trapeziums were used for public rituals. The geometry of the Vedic period originated with the construction and fireplaces of the altar which is used for performing the rites.

Explanation:

(C) Greece

Euclid was one of the famous mathematicians, best known for his treatise on geometry called The Elements. There are thirteen books of Euclid’s Elements. He presented a set of formal logical arguments based on a few rems and axioms and provided a systematic method of rational exploration that helped many philosophers, mathematicians, and scientists.

Explanation:

(C ) Greece

Thales appears to be the first known Greek philosopher, scientist ant great mathematician. Thales was the first natural philosopher in Ionian School and is thought to have been Anaximander’s tutor. He invented the formula for calculating the distance between a ship and the shore. Thales is also credited for the invention of plane geometry.

Explanation:

(A) Thales

Pythagoras was the student of the famous mathematician, and philosopher Thales. Pythagoras was best known for the proof of the important Pythagorean theorem, which is about right-angle triangles. This theorem explains that for any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Explanation:

(A)Theorem

Axioms, definitions, and postulates are self-evident therefore no proof is required. A theorem is a mathematical statement that can and must be proven to be true. This proof may also tell us why the statement is true, as well as what ideas that statement connects or requires by virtue of being true or in order to be true.

Explanation:

(c) Postulate

Euclid's fourth postulate states that all the right angles are equal to each other. A right angle is an angle measuring 90 degrees. So, irrespective of the length of the right angle or its orientation all right angles are identical in form and they coincide exactly when placed one on top of the other.

Explanation:

(B) a definition

A postulate or an axiom is a statement that can be taken to be true without any proof. But a statement of the exact meaning of a word is a definition.” Lines are parallel if they do not intersect” is a definition of a parallel line as it gives the meaning of a parallel or what is meant by two lines being parallel. Hence, this statement is in the form of a definition.

Explanation:

A. Pyramid is a solid figure, the base of which is a triangle or square or some other polygon and its side faces are equilateral triangles that converge to a point at the top.

Answer: True. The structure of a pyramid whose outer surface is triangular and it will converge to a single point at the top. Whereas the base of the pyramid can be trilateral, quadrilateral, or polygon shape.

B. In the Vedic period, squares and circular-shaped altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles, and trapeziums were used for public rituals.

Answer: True.

In ancient India people were using square and circular-shaped altars in their households in order to perform their rituals, whereas the combination of rectangles, triangles, and trapeziums was used for public rituals. The geometry of the Vedic period originated with the construction of fireplaces in order to perform Vedic rites.

C. In geometry, we take a point line and a plane as undefined terms.

Answer: True

In geometry, the point will be defined as the location which does not have size. The line will have one dimension and the plane will extend indefinitely in two dimensions. That is why mathematicians leave these terms undefined.

D. If the area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.

Answer: True 

Area of the triangle=Area of the rectangle

Area of the rectangle=Area of the square

Therefore the area of the triangle=area of the square.

According to Euclid’s first axiom, “Things equal to the same thing are equal.

E.Euclid’s fourth axiom says that everything equals to itself

Answer: True.

Euclid’s fourth axiom states that “things which coincide with one another are equal to one another.

F. Euclidean geometry is valid only for figures in the plane.

Answer: True

Euclidean geometry is only valid for curved surfaces. Euclidian geometry is used in understanding the geometry of flat or two-dimensional spaces. The above statement is true because, by Einstein’s theory of general relativity, physical space itself is not Euclidean. Euclidean space is a good approximation for it where the gravitational field is weak. So, in space or multidimensional space, the Euclidean axioms are not applicable.

Explanation:

1. Euclidean geometry is valid only for curved surfaces.

Answer: False

Euclidean geometry is based on axioms and postulates that are valid only for the plane surface.Physical space itself is not Euclidean and curved surfaces require physical space so Euclidean geometry is not valid for a curved surface.

Explanation:

False

The boundaries of solids are not curves, the boundaries of the solids are surfaces.

Explanation:

False

The above statement is false because the edges of surfaces are not curves, the edges of the surfaces are lines

Explanation:

True

Let a=2x and b=2x

When a,b, and x are arbitrary numbers or things. The first axiom of Euclid states that “the things which are equal to the same are also equal to each other”

So a=b, hence the above statement is true.

Explanation:

True

Now consider quantity A having two parts B and C. When A coincides with B and C, from the fourth axiom of Euclid, the things which coincide with one another are equal to one another

A=B+C

Explanation:

False

The axioms are statements that are self-evident and also they are accepted without any kind of proof. There Are a few statements that require proof and experimental verifications in order to establish themselves. Such types of statements are called as theory.

Explanation:

True

Let line l be parallel to line m through a point P outside line l.

Let n be another line that passes through P.

A line PQ is drawn from P to point Q on l.

Let PQ make an angle with l and the alternate angle with m.

Since l and m are parallel

∠c=∠a.........(i)

Let another line n, which is parallel to l, pass through P and the angle, alternate to c, made by n with PQ be b.

Since l and n are parallel 

∠c=∠a.........(i)

Then, by Euclid’s first axiom,

∠c=∠a

(from 1 and 2)

But either ∠a is a part of ∠b or ∠b is part of ∠a ( Euclid’s fifth axiom)

∴∠a≠ ∠b

So Plafair’s Axiom is true

Explanation:

True

It is true that intersecting lines cannot be parallel to each other, so they cannot be parallel to the same line.

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 straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. Since the above statement is equivalent to Euclid's fifth postulate, it is true.

Explanation:

True

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 straight lines, if produced indefinitely, meet on that side on which the sum of angles is taken together less than two right angles”.

There were several attempts to prove Euclid’s fifth postulate using the other postulates and axioms which led to the discovery of various other new geometries, which are quite different from Euclidean geometry, that is known as non-Euclidean geometry.

Explanation:

Ram and Ravi both have the same weight.

Now, we have to compare their new weights,i.e 2kgs

Let's assume the weight of Ram be x kg

So, Ravi’s weight = x kg

When they both gain weight by 2kg, 

Ram’s weight=(x+2kg)

Ravi’s weight=(x+2kg)

According to Euclid’s second axiom,” If equals are added to the equals, the wholes are equal”

Therefore, (x+2)=(x+2)

So, the weight of Ram is equal to the weight of Ravi.

Explanation:

 Here we have to use Euclid's second axiom, which states that, “If equals are added to the equals, the wholes are equal”.

Here the given equation is a -15=25

By adding 15 on both sides of the equation,

  a - 15+15=25+15

   a=25+15

Hence, a=40

So, the solution of the above equation is a=40.

Explanation:

Given, ∠1= ∠3, ∠ 2=∠4 and ∠ 3= ∠4

Now we have to write the relation between ∠1 and  ∠2, using Euclid’s axiom, which states that the things which are equal to the same thing are equal to one another.

∠1=∠3

∠3 =∠4

Now,∠1=∠3=∠4

Given,∠2=∠4

So,∠1=∠2=∠3=∠4

So, all the angles are equal to one another.

Explanation:

Here, AC=XD

Now C is the midpoint of AB

D is the midpoint of XY

Now we have to show that AB=XY by using Euclid’s axiom.

It states that “Things which are double of the same thing are equal to one another.

If C  is the midpoint of AB

Then AB=2AC

AC=AB/2

Since D is the midpoint of XY

XY=2XD

XD=XY/2

We know, AC=XD

So, AB/2=XY/2

Therefore, AB=XY.

Explanation:

Let's assume that each salesman makes a sale of Rs.x in the month of August.

And in the month of September, their sales doubled i.e 2x.

Euclid’s sixth axiom states that” things that are double the same thing are equal to one another.” Therefore, the sales of both the salesmen are equal in the month of September.

Explanation:

Euclid’s axiom states that “If equals are added to equals, the wholes are equal.

 x+y=10………(a)

And, x= z………(b)

So, if we use the above equation (a) and (b)

x+y=z+y…….( c) i.e(x=z)

From (a) and (c )

z+y=10

So it is proved that, z+y=10.

Explanation:

Euclid’s axiom states that “The whole is greater than the part”.

From the above figure,

AB+BC+CD=AD

That means AD is a part of AH.

i,e, AH>AD

So, the length of AH> the sum of the lengths of AB+BC+CD.

Explanation:

The above figure represents a triangle with ABC.

X and Y are the points that lie on the sides of AB and BC.

If, AB=BC……..(a)

And, BX=BY…….(b)

Now we have to show that AX=CY

From the above figure,

AB=AX+BX

So, AB- BX=AX……..(c )

Likewise, BC=BY+CY

BC-BY=CY…….(d)

If we use Euclid’s axiom which states that” If equals be subtracted from equals, the remains are equal”.

By using (a) and (b) in (c ) and (d),

AB-BX=BC-BY

So, AX=CY.

Explanation:

In the above triangle ABC, the X, and Y points lie on the sides, AB and BC.

If AC=BC………..(a)

Also, AX=CY……(b)

Now we have to show that AC=BC

Since the X is the midpoint of AC,

AC=2AX=2CX……..(c )

Since  Y is the midpoint of BC

BC=2BY=2CY………….(d)

By using Euclid’s axiom, “Things which are double of the same thing are equal to one another”,

If we use (b) and (c ),

2AX=2CY……….(e)

By using (e) in ( c) and (d),

AC=2AX=2CY

BC=2CY=2AX

So, AC=BC.

Explanation:

In the above figure, the triangle ABC, the X and Y lie on the sides AB and BC.

Now, BX= AB/2……..(a)

          BY=BC/2……..(b)

 And also, AB=BC…..(c )

Now we have to show that, BX=BY

From(a), AB=2BX, this implies, X is the midpoint of AB

From(b), BC=2BY, this implies,  Y is the midpoint of BC

According to Euclid’s seventh axiom, “the things which are double  of the same things are equal to one another”

By using this axiom, 

From(c ), 2BX=2BY

Therefore , BX=BY.

Explanation:

The above figure is a quadrilateral ABCD.

Given,∠1=∠2……….(a)

Also,∠2=∠3………..(b)

Now we have to prove that ∠1=∠3

From (a) and (b), 

∠1=∠2=∠3

Euclid’s axiom states that “ The things which are equal to the same thing are equal to one another”. So, ∠1=∠3

Explanation:

The above figure is a quadrilateral ABCD

If, ∠1=v3………(a)

And, ∠2=∠4……(b)

Now we have to prove that ∠A=∠C

By adding (a) and (b),∠1+∠2=∠3+∠4

Now let's use Euclid’s axiom,” If equals are added to equals, the wholes are equal”.

So from the above figure,∠1+∠2=vA

∠3+∠4=∠C

Hence ,∠A=∠ C.

Explanation:

In the above figure, there are two triangles ABC and BDC with a common base i.e BC

Now ∠ABC=∠ACB…….(a)

And, ∠3=∠4……(b)

Now we have to prove that ∠1=∠2

In the above figure, ∠ABC =∠1+∠4

                                ∠1=vABC-∠4…………(c )

                                ∠ACB=∠3+∠2

                                 ∠2=∠ACB-∠3………(d)

Euclid’s axiom states that” If equals be subtracted from equals, the remains are equal”

So from(c ) and (d)

∠ABC-∠4=∠ACB-∠3

So, ∠1=∠2.

Explanation:

AC=DC……(a)

And, CB=CE……(b) 

By using Euclid’s axiom,” If equals are added to equals, then the wholes are also equal”,

Now let's add (a) and (b)

We get, AC+CB=DC+CE

Therefore, AB=DE

Explanation:

It is given that OX=PX

Now OX= ½ XY and PX=1/2XZ

Therefore, 1/2XY=1/2XZ

According to Euclid’s seventh axiom” things which are halves of the same thing are equal to one another”  therefore,XY=XZ

According to Euclid’s sixth axiom, “things which are double of the same thing are equal to one another”

Explanation:

Here, M is the midpoint of AB

AM=BM=1 /2 AB…..(a)

N is the midpoint of BC

BN=NC= 1/2BC……(b)

AB=BC

According to Euclid’s axiom” If two things are equal then their halves are also equal”

So, 1/2AB =1/2BC

AM=NC from(a) and (b)

Explanation:

Polygon: It is defined as a two-dimensional and also simple closed figure that is made up of three or more line segments.

Line segment: It's part of a line that has two endpoints. The length of the line segments is always fixed and also this is w=equal to the distance between two endpoints.

Right angle: Angle whose measure is ninety degrees.

Here line and point are undefined.

Angle: A figure formed by two rays with a common initial point.

Ray: It's part of a line that has one endpoint.

According to Euclid’s fourth postulate, “ All the right angles are equal to one another”

So in a square, all the angles are right angles, so all angles are equal

Three-line segments are equal to the fourth-line segment

By Euclid’s first axiom, Things which are equal to the same thing are equal to one another”.

Therefore, all four sides of a square are equal.

Explanation:

The undefined terms are line and point. Euclid’s axiom states that” Things that are equal to the same thing are equal to one another”.

In this, the angles that are present in an equilateral triangle are sixty degrees, and the two line segments are also equal to the third one.

The defined terms are:

Equilateral triangle: It's a triangle with all three sides of equal length and all the angles are equal.

Polygon; It's a plane-closed geometric figure which has a minimum of three straight sides and angles.

Line segments: It's part of a line that consists of two endpoints.

Angle: It's a shape formed by two rays that meet at a common endpoint called an angle.

Acute angles: The angles whose measures are less than ninety degrees are known as acute angles.

So, by using Euclid’s axiom, all three sides and all the angles of an equilateral triangle are equal.

Explanation:

Two equivalent versions of Euclid’s fifth postulate are:

1. For every line l and for every point P not lying on l, there exists another unique line m that passes through P and is also parallel to l.

2. The two distinct intersecting lines cannot be parallel to the same line.

   Hence, the given statement is not an equivalent version of Euclid’s fifth postulate.

Explanation:

This type of axiom is not consistent.

1. When two parallel lines that are intersected by a transversal, then the corresponding angles will become equal. So here, ∠1=∠5,∠2=v6,v3=v7 and v4=∠8

2. In a parallel line, the interior angles will always be equal.

Explanation:

 The above-given system of axioms is not consistent, that is because, if a ray stands on the line and the sum of the two adjacent angles is formed, is equal to 180 degrees. Hence for two lines that intersect each other, the vertically opposite angles become equal.

Explanation:

According to Euclid’s axiom” Things which are equal to the same thing are equal to one another.” So, the given axiom is consistent.

According to Euclid’s axiom “If equals are added to equals, the wholes are equal” Therefore the given axiom is also consistent.

“Things which are double of the same thing are equal to one another” This is also Euclid’s axiom, hence the given axiom is consistent.