Introduction To Euclids Geometry

Reason: The three steps from solids to point are solids-surfaces-lines-points.

Reason: A solid shape has shape, size, position and can be moved from one place to another. So, solid has three dimensions, e.g. Cuboid.

Reason: Boundaries of a solid are called surfaces. A surface(plane) has only length and breadth. So, it has two boundaries.

Reason: A point is that which has no part i.e., no length, no breadth and no height. So, it has no dimension.

Reason: Euclid divided his famous treatise The Elements’ into 13 chapters.

Reason: The statements that can be proved are called proportions or theorems. Euclid deduced 465 proportions in a logical chain using his axioms, postulates, definitions and theorems.

Reason: The boundaries of surface are curves.

Reason: The boundaries of surfaces are curves.

Reason: In Indus valley civilization, the bricks used for construction work were having dimensions in the ratio Length: breadth: thickness = 4:2:1

Reason: A pyramid is solid figure, the base of which is a triangle or square or some other polygon.

Reason: The side faces of a pyramid are always triangles.

Reason: The Euclid’s axiom that illustrate the given statement is second axiom, according to which, if equals are added to equals, the wholes are equals.

Reason: In ancient India, squares and circular altars were used for household rituals.

Reason: The Syriyantra (in the Atharvaveda) consists of nine interwoven isosceles triangles.

Reason: Greek’s emphasized on deductive reasoning.

Reason: In ancient India altars whose shapes were combinations of rectangles, Triangles and trapezium were used for public worship.

Reason: Euclid belongs to the country Greece.

Reason: Thales belong to the country.

Reason: Pythagoras was a student of Thales.

Reason: The statements that were proved are called propositions or theorem’s.

Reason: Euclid stated that all right angles are equal to each other in the form of a postulate.

Reason: ‘Line as parallel, if they do no intersect’ is the definition of parallel lines.

False

Reason: Because Euclidean geometry is valid only for the figures in the plane but on the curved surfaces it fails.

False

Reason: Because the boundaries of the solids are surfaces.

False

Reason: Because the edges of surfaces are lines.

True

Reason: Since, it is one of the Euclid’s axiom.

True

Reason: Since, it is one of the Euclid’s axiom

False

Reason: Because the statements that are proved are called theorems.

True

Reason: Since, it is an equivalent version of Euclid’s fifth postulate and it is known as Playfair’s axiom.

True

Reason: Since, it is an equivalent version of Euclid’s fifth postulate.

True

Reason: All attempts to prove the fifth postulate as a theorem led to a great achievement in the creation of several other geometries. These geometries are quite different from Euclidean geometry and called non-Euclidean geometry.

Let the equal sale of two salesmen in August be x.

In September, each salesman doubles his sales of August.

Thus, sales of first salesmen is 2x and sales of second salesman is 2x.

According to Euclid’s axioms, things which are double of the same things are equal to one another.

So, in September their sales are again equal.

Note: Thinking process be like, apply the Euclid’s axiom, if equals are added to equals, the wholes are equal, to show the given result.

We have, x+y=10 …(i)

and x=z …(ii)

According to the Euclid’s axiom, if equals are added to equals, the wholes are equal.

So, from Eqs. (i) and (ii)

x+y=z+y ….(iii)

From Eqs. (i) and (iii)

Z+Y=10

From the given figure, we have

AB+BC+CD =AD [AB, BC and CD are the parts of AD]

Here, AD is also the part of AH.

According to Euclid’s axiom, the whole is greater than the part i.e., AH > AD

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

We, have AB =AC …(i)

and BX=BY …(ii)

According to Euclid’s axiom, if equals are subtracted from equasl, the remainders are equal.

So, on subtracting Eq.(ii) from (i) ,

We get, AB-BX = BC-BY

⇒ AX = CY [from figure]

Given, X is the mid-point of AC

AX = CX=

⇒ 2AX =2CX = AC …(i)

Y is the mid-point of BC.

BY = CY =

⇒ 2BY = 2CY= BC …(ii)

Also, given AX=CY …(iii)

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

From Eq. (iii),

2AX = 2CY

⇒ AC=BC [from Eqs. (i) and (ii)]

*Note: Thinking process should be, things which are double of the same things are equal to one another.

Given, BX = and BY =

⇒ 2BX = AB ….(i)

⇒ 2BY = BC ….(ii)

and AB = BC …(iii)

On putting the values from Eqs. (i) and (ii) in eq. (iii)

We get,

2BX = 2BY

According to Euclid’s axiom, things which are double of same things are equal to one another.

BX =BY

Given, ∠1 = ∠2 …(i)

and ∠2 = ∠3 …(ii)

According to Euclid’s axiom, things which are equal to the same things are equal to one another.

From Eqs. (i) and (ii),

∠1 = ∠3

Given: ∠1= ∠3 …(i)

and ∠2 = ∠4 …(ii)

According to Euclid’s axiom, if equals are added to equals, then wholes are equal.

On adding Eq. (i) and (ii)

∠1 + ∠2 = ∠3 + ∠4

⇒ ∠A = ∠C

Given: ∠ ABC = ∠ ACB …(i)

and ∠4 = ∠3 …(ii)

According to Euclid’s axiom, if equals are subtracted from equals, then remainders are also equal.

On subtracting Eq. (ii) from (i) ,

We get,

∠ ABC - ∠4 = ∠ ACB - ∠3

⇒ ∠1 = ∠2

Now, in ABDC, ∠1 = ∠2

⇒ DC =BD [sides opposite to equal angles are equal]

Given: AC =DC …(i)

and CB = CE …(ii)

According to Euclid’s axiom, if equals are added to equals, then wholes are also equal.

So, on adding Eq.(i) and (ii),

We get,

AC + CB = DC + CE

⇒ AB = DE

Given: OX =

⇒ 2OX = XY …(i)

PX =

⇒ 2PX = XZ

and OX =PX …(iii)

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

On multiplying Eq. (iii) by 2,

We get,

2 OX = 2PX

∴ XY = YZ [from Eq. (i) and (ii)]

i. Given, AB = BC …(i)

M is the mid-point of AB

∴ AM = MB = …(ii)

and N is the mid-point of BC.

∴ BN = NC = …(iii)

According to Euclid’s axiom, things which are halves of the same things are equal to one another.

From Eq. (i), AB =BC

On multiplying both sides by ,

We get,

⇒ AM = NC [using Eq. (ii) and (iii)]

ii. Given, BM =BN …(i)

M is the mid-point of AB.

∴ AM = BM = AB

⇒ 2AM = 2BM = AB …(ii)

and N is the mid-point of BC.

∴ BN = NC = BC

⇒ 2BN = 2NC = BC …. (iii)

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

On multiplying both sides of Eq.(i) by 2,

We get, 2BM = 2BN

⇒ AB = BC [using Eq. (i) and (ii)]

The terms need to be defined are.

i: Polygon: A closed figure bounded by three or more-line segments.

ii: Line segment: part of line with two end points.

iii: Line: undefined term

iv: Point: undefined term

v: Angle: in a figure is formed by two rays with one common initial point.

vi: Acute angle: is an angle whose measure is between 0° to 90°.

Here undefined terms are line and point.

All the angles of equilateral triangle are 60° each (given)

Two-line segments are equal to third one (given)

Therefore, all three sides of an equilateral triangle are equal.

According to Euclid’s axiom, things which are equal to the same thing are equal to one another.

Two equivalent version of Euclid’s fifth postulate are:

• For every line L and for every point P not lying on the L, there exists a unique line M passing through P and parallel to L.

• Two distinct intersecting lines cannot be parallel to the same line.

From above two segments it is clear that given statement is not an equivalent version to the Euclid’s fifth postulate.

A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axioms.

We know that, if a transversal intersects two parallel line, then each pair of corresponding angles are equal, which is a theorem. So, statement (i) is false and not an axiom.

Also, we know that, if a transversal intersects two parallel line, then each pair of alternate interior angles are equal. It is also a theorem. So, statement(ii) is true and an axiom.

Thus, in given statements, first is false and second is an axiom.

Hence, given of axioms is not consistent.

We know that, if two lines intersect each other, then the vertically opposite angles are equal. It is a theorem. So, given statement I is false and not an axiom.

Also, 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. So, given statement II is true and an axiom.

Thus, in given statement, first is false and second is an axiom. Hence, given system of axioms is not consistent.

Some of the Euclid’s axioms are:

i: Things which are equal to same thing are equal to one another.

ii: If equals are added to equals, the wholes are equal.

iii: Things which are double of the same things are equal to one another.

Thus, given three axioms are Euclid’s axioms. So, here we cannot deduce any statement from these axioms which contains any axiom. So, given system of axioms is inconsistent.