Introduction To Euclid's Geometry

A theorem is a statement that has been proven true while an axiom is a statement that is accepted as true.

Axiom is a statement that is assumed to be true without proof.

Theorem is a statement that has been proven through testing or calculation.

(i) In line segment two points are connected with a straight line.

A line segment has two end points with a definite length.

(ii) A ray is a part of line with one end point and infinitely extends in one direction. It can show by drawing an arrow at one end of the ray.

(iii) Parallel lines are lines that never cross one another. Parallel lines do not intersect to each other.

(iv) A straight line extends from a point indefinitely in one direction only. It is the set of all points on a line on a given side of a given point of the line.

Notation:

is the half-line.

(v) When three or more points lie on a straight line called collinear points. If the points lie on the same line then the points are called collinear points.

A, B and C the collinear points.

(vi) A plane is a flat surface with no thickness. A plane is two dimensional.

(i) A, B, C, D, E, F

A point has location and it has no size.

(ii) , , ,

A line segment has two end points with a definite length.

(iii) , , ,

A ray is a part of line with one end point and infinitely extends in one direction.

(iv) , , ,

A line has no beginning point or end point.

(v) M, E, G, B

If three or more points lie on a straight line called collinear points.

(i) {, , R},{,,P}

Lines EF and GH cross to each other at point R, therefore EF and GH are intersecting lines at point R.

Similarly lines AB and CD cross to each other at point P, therefore EF and GH are intersecting lines at point P.

(ii) ,,,R

If the set of lines or curves intersect at the same point called concurrent lines. In this figure lines AB, EF and GH intersect at the point R.

(iii) ,,

A ray is a part of line with one end point and infinitely extends in one direction.

(iv) ,

A line segment has two end points with a definite length.

(v) { AB, EF, GH}, { CD, GH}, { AB, CD}

When two or more lines meet at a point, these are called intersecting lines. Intersecting lines share exactly one point.

(vi) ,,

If the set of lines or curves intersect at the same point, it is called concurrent lines.

(i) Infinite number of lines can be drawn to pass through a given point.

(ii) Only one line can be drawn to pass through two given points.

(iii) Two lines intersect in at most one point.

(iv) , ,

A line segment has two end points with a definite length.

If three or more points lie on a straight line, it is called collinear points.

(i) False

Because a line segment has two end points with a definite length.

(ii) False

Because a ray has one end point and infinitely extends in one direction.

(iii) False

Because a line has no beginning point or end point therefore a line has not definite length.

(iv) True

Because a line has no beginning point or end point therefore line is the same as line .

(v) False

Because a ray has one end point and infinitely extends in one direction therefore ray is not same as the ray .

In this ray has the end point A.

In this ray has the end point B.

(vi) True

Because a line has no beginning point or end point.

(vii) True

Because the set of lines intersect at the same point called concurrent lines.

Lines l, m, and n are concurrent at the common point P.

(viii) True

Two lines in a plane that do not intersect or touch each other at any point, they are called parallel lines.

(ix) True

Because parallel lines never intersect or cross one another.

(x) False

Because half-line is the set of all points on a line on a given side of a given point of the line while ray is a part of line with one end point and infinitely extends in one direction.

(xi) False

Because two lines intersect at only one point.

Lines l and n intersect at only one point P.

(xii) False

Two lines are said to be parallel when

(a) they never meet

(b) they are coplanar

In ancient India, the shapes of altars used for household rituals were Squares and circles.

The geometry of the Vedic period originated with the construction of altars (or vedis) and fireplaces for performing Vedic rites. Square and circular altars were used for household rituals, while altars, whose shapes were combinations of rectangles, triangles and trapeziums, were required for public worship.

The number of interwoven isosceles triangles in Sriyantra is 9.

The Sri Yantra is a form of mystical diagram, known as a yantra. It consists of nine interlocking triangles that radiate out from the central point.

Thales belongs to the country Greece. He was a Greek philosopher, mathematician and astronomer. He was one of the seven sages of Greece.

Euclid belongs to the country Greece

Euclid was born in 300 BCE, Alexandria, Egypt. He was the most prominent mathematician of Greece and was best known for his treatise on geometry.

Pythagoras was a student of Thales

Pythagoras was born in about 570 BC on the Greek island of Samos. His father was a merchant. Pythagoras was taught mathematics by Thales, who brought mathematics to the Greeks from Ancient Egypt.

In Indus Valley Civilization (about 300 BC) the bricks used for construction work were having dimensions in the ratio 4:2:1

In Indus Valley Civilization, the bricks used for construction work were having dimensions in the ratio length : breadth : thickness = 4 : 2 : 1.

A theorem needs a proof.

A theorem is a mathematical statement proved by different steps of mathematical reasoning.

Axioms are assumed to be universal truths in all branches of mathematics.

Axioms are the derived and accepted true statements.

Lines are parallel if they do not intersect’ is stated in the form of a definition.

Definition is a formal statement of the meaning of a word or a set of words.

Euclid stated that ‘all right angles are equal to each other’, in the form of an axiom.

A right angle means 90°. Thus, All right angles are equal to 90°. Therefore, all right angles ae equal is an axiom. This is because it is derived from a true statement.

Greeks emphasized ondeductive reasoning.

The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning.

A solid has 3 dimensions.

A solid figure is three dimensional because it has length, width and height.

A surface has 2 dimensions.

A surface is two dimensional because it has length and width only.

A point has 0 dimension.

A point has no dimensions, only position

Boundaries of solids are surfaces.

The solids are three dimensional but their surfaces are two dimensional.

Boundaries of surfaces are curves.

Boundaries of surfaces are curves because surfaces are two dimensional figures and their boundaries are one-dimensional i.e. curves.

The side faces of a pyramid are triangles

A pyramid is a figure with triangular surfaces which converge to one single point.

The base of a pyramid is any polygon.

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.

The number of planes passing through three non-collinear points is 1.

If the points are collinear then an infinite number of planes can be made to pass through them. If three distinct points are non-collinear then exactly one plane passes through them.

Euclid divided his book ‘Elements’ into 13 chapters.

The book name is: “The Elements”

It is divided into thirteen chapters. The chapters are as follows: -

Books VII-IX -- Theory of Numbers

Book X -- Incommensurables

Book XI-XIII -- Solid Geometry

The floor and the ceiling of a room are parallel planes.

The floor and the ceiling of the room are parallel planes.

If two circles are equal, then their radii are equal.

A circle is formed by taking a radius. If the radius of two circles are same, means the two circles are equal.

Ray AB =ray BA.

Because ray is a part of a line that has one endpoint and extends in one direction without ending.

A point C is called the midpoint of a line segment AB, if C is an interior point of AB such that AC=CB

A point C is said to lie between the points A and B if points A, C and B are collinear.

Let us consider, a line segment AB.
Assume that it has two midpoints say C and D

Recall that the midpoint of a line segment divides it into two equal parts
That is AC = BC and AD = DB
Since C is midpoint of AB, we have A, C and B are collinear
∴ AC + BC = AB → (1)
Similarly, we get AD + DB = AB → (2)
From (1) and (2), we get
AC + BC = AD + DB
2 AC = 2AD
∴ AC = AD
This is a contradiction unless C and D coincide.
Therefore our assumption that a line segment AB has two midpoints is incorrect.
Thus every line segment has one and only one midpoint.

Let us consider, a line segment AB.
A and B are two different points from which infinite number of lines can be drawn.

A sentence that can be judged to be true or false is called a statement.

3+7 = 9 is false statement.

A line goes without end in both directions but a ray has one endpoint and goes without end in one direction.

The line AB is same as the line BA. But the ray AB is different than the ray BA.

A rectilinear figure is a figure all of whose edges meet at right angles.

So, Assertion false but the reason is true.

(A)-(q), (B)-(r), (C)-(p), (D)-(s)

(A)-(q): A line segment has definite length.

Its length can be measured. A line segment AB has two end points A and B. It starts from point A and ends at point B. One and only one line-segment can be between two given points A and B.

(B) – (r): A ray has the end point “B”

A ray is a line with one end. It starts at a given point and goes off in a certain direction forever. Here, the end point is B and it extends infinitely in the direction A.

(C)-(p) Through a point, infinitely many lines can be drawn.

In 1 dimension, only one line can be drawn passing through a single point. However, in 2 or more dimension, uncountably many lines can be draw which pass through one point.

(D)-(s) Only one lines can be drawn to pass through two given points.

Through two given points only one unique line can be drawn.

(A) Concurrent lines passthrough a given point.

If three or more lines pass through the same point then they are called concurrent lines and the common point is called the point of concurrency or concurrent point.

(B) Two distinct lines in a plane cannot have more than one point in common.

Let us suppose that the two lines intersect at 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.

(C) Two distinct points in a plane determine a unique line.

For any two distinct points in space there is a unique line that passes through both of them.

(D) A line segment has two end points

A line segment has definite length. Its length can be measured. A line segment AB has two end points A and B. It starts from point A and ends at point B. One and only one line-segment can be between two given points A and B.

According to question, C lies between points A and B and AC = BC

Adding AC both side we get,

AC + AC = BC + AC

According to definition of Euclid, if equals are added to equals, whole will equal.

Here, (BC + AC) will coincides with AB.

2AC = AB

So, AC = AB

Let us consider, a line segment AB.
Assume that it has two midpoints say C and D

Midpoint of a line segment divides it into two equal parts
So, AC = BC and AD = DB
Since, C is midpoint of AB, we have A, C and B are collinear
Thus, AC + BC = AB ……………… (i)

Similarly, we get AD + DB = AB ……………(ii)

From eq(i) and (ii), we get
AC + BC = AD + DB
2 AC = 2AD
AC = AD
This is a contradiction unless C and D coincide.
Therefore our assumption that a line segment AB has two midpoints is incorrect.
Thus every line segment has one and only one midpoint.

From the above figure we get that,

AC = AB + BC

BD = BC + CD

And it is given is that AC = BD

So, AB + BC = BC + CD ………….(i)

According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.

Subtracting BC from both side in eq(i), we get

AB + BC − BC = BC + CD − BC

AB = CD

Assume L will not intersect N.

Then, L ||N and it is given that M || N.

According to our assumption, L || M which is contradictory to given statement that L intersects M.

So, our assumption is wrong.

Thus, L will intersect N also.

Let x be the angle.

According to question, x is 20° more than its complement.

So, another angle is x - 20°

As we know that, sum of compliment is 90°

So,

x + x - 20° = 90°

2x = 110°

x =55°

Let x be the angle.

According to question, x is 20° less than its supplement.

So, another angle is x + 20°

As we know that, sum of compliment is 180°

So,

x + x + 20° = 180°

2x = 160°

x =80°

Let x be the angle.

According to question,

(90 - x)° is the complementary angle.

(180 - x)° is the supplementary angle.

According to question,

5 X (90 - x) =2 X (180 -x) -12

450 – 5x = 360 – 2x – 12

3x = 450 – 348

3x = 102

x = 34°

A theorem needs a proof.

If three distinct points are non-collinear then exactly one plane passes through them

(A) and (B)

(A) Infinite lines can be drawn through a given point.

(B) Only a single line can be drawn through two given points.

Three, AB, BC, AC

If A, B and C are three collinear points. Then there will be only 3 line segments.

AB, BC and AC.

From the above figure we get that,

AC = AB + BC

BD = BC + CD

And it is given is that AC = BD

So, AB + BC = BC + CD ………….(i)

According to Euclid’s axiom,when equals are subtracted from equals, the remainders are also equal.

Subtracting BC from both side in eq(i), we get

AB + BC − BC = BC + CD − BC

AB = CD

Let us consider, a line segment AB.
Assume that it has two midpoints say C and D

Midpoint of a line segment divides it into two equal parts
So, AC = BC and AD = DB
Since, C is midpoint of AB, we have A, C and B are collinear
Thus, AC + BC = AB ……………… (i)

Similarly, we get AD + DB = AB ……………(ii)

From eq (i) and (ii), we get
AC + BC = AD + DB
2 AC = 2AD
AC = AD
This is a contradiction unless C and D coincide.
Therefore our assumption that a line segment AB has two midpoints is incorrect.
Thus every line segment has one and only one midpoint.

(A) Parallel Lines: Two lines in a plane that do not intersect or touch each other at any point are said to be parallel lines.

(B) Intersecting Lines: Two lines in a plane that intersect or cross each other at any point are said to be intersecting lines.

(C) Concurrent Lines: If three or more lines intersect each other at a single point then they are said to be concurrent.

L || M M|| L.

Now, M||L and L||N M||N.

Assume M will intersect N.

It is given that L ||M and L ||N

According to our assumption, M will intersect N which is contradictory to parallel axiom.

So, our assumption is wrong.

Thus, M|| N

If three or more lines pass through the same point then they are called concurrent lines and the common point is called the point of concurrency or concurrent point.

Ray is a part of a line that has one endpoint.

Infinite number of lines can be passed through a single point.

(A)

(A) Lines will be AB, PQ and RS.

(B) CEFG

A rectilinear figure is a figure all of whose edges meet at right angles.

(C) Concurrent Points: A, E, F, B

If a set of lines pass through the same point then they are called concurrent lines and the common point is called the point of concurrency or concurrent point.

(C)

A point C is called the midpoint of a line segment AB, if C is an interior point of AB such that AC=CB

(C)

From the above figure we get that,

AE = AD + DE

CB = CD + DB

And it is given is that AE = CB

So,

AD + DE = CD + DB

AD + CD= CD + DB………….(i) [DE = CD as given]

According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.

Subtracting CD from both side in eq (i), we get

AD + CD − CD = CD + DB − CD

AD = DB

So, D is the mid-point of line segment AB.

(A) one

If all the 4 points are collinear then exactly one line can be drawn from them

(B)

6 lines can be drawn when no three of the four lines are collinear.

Suppose lines “l” and “m” intersect at two points P and Q. Then, line P must contain both the points P and Q.

Also, line m must contain both the points P and Q.

But only one line can pass through two different points.

Thus, the assumption we started with that two lines can pass through two distinct point is wrong.

(B)

It is given in the question as hint that: (A) and (C) are false sentences and (D) is a true sentence.

Then, “Kunal is a tall boy” is not a statement because this sentence is true for someone and it is false for other one. So, it is not a statement.

The basic facts which are taken for granted without proof are called axioms.

Some Euclid's axioms are:

1. The things which are equal to the same thing are equal to one another.

2. If equals be added to the equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equals.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

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

7. Things which are halves of the same thing are equal to one another

(A)-(q)

An infinite number of lines can be drawn to pass through a given point.

(B)-(p)

Only one line can be drawn to pass through two given point.

(C)-(s)

A line is a straight set of points that extend in opposite directions without ending.

(D)-(r)

Line segment has two end points A and B

A rectilinear figure is a figure all of whose edges meet at right angles.

So, Assertion& Reason both are true.

According to Euclid’s Fourth Postulate Assertion is right but reason given is not linked to right angle.