Introduction to Euclid's Geometry, Class 9 Mathematics Detailed Chapter Notes PDF Download

INTRODUCTION

The word 'geometry' comes from the Greek words 'geo', meaning 'earth' and 'metrein', meaning 'to measure'. Thus, the word 'Geometry' means 'earth measurement'. Geometry appears to have originated from the need for measuring land. This branch of mathematics was studied in Egypt, Babylonia, China, India, Greece etc. The people of these civilizations faced several problems which led to the development of geometry.

ORIGIN OF GEOMETRY

(1) In Egypt : Ancient Egyptians were the first people to study geometry. For example, when ever river Nile overflowed, it wiped out the boundries between the adjoining fields of different land owners. After such flooding, these boundaries had to be redrawn. For this purpose, Egyptian developed a number of geometrical techniques and rules for calculating areas of plane figures such as triangle, rectangle etc. The knowledge of geometry was also used by them for consulting volumes of granaries and for contructing canals and pyramids.

(2) In Indian Sub continent : In Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc shows that the Indus valley civilization (about 3000 B.C.) made extensive use of geometry. The cities were highly developed and very well planned.

For example, the roads were parallel to each other and there was an underground drainage system. Their huses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and pratical arithmetic.

(3) Unsystematic development : Above examples shows that geometry was being developed and applied every where in the word. But this was happening in an unsystematic manner.

(4) Systematic development : A knowledge of geometry passed from the Egyptians to the Greeks and many

Greek mathematicians worked on geometry.

Thales (640 BC – 546 BC) : A Greek mathematicians Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts) by its diameter. One of 'Thales' most famous pupil was 'pythagorus'. Pythagorus (572 BC) : Pythagorus and his group discovered many geometric properties and develped the theory of geometry to a great extent. This process continue till 300 B.C.

Euclid (325 BC – 265 BC) : A Greek Mathematicians Euclid collected and developed all the known facts about geometry in his famous treaties called 'Elements'. He divided elements into thirteen chapters called Book. This was the first systematic study of the subject geometry, hence ideas influenced by the Euclid's approach is generally referred as Euclid's Geometry. The Greek mathematicians of Euclid's time expressed some basic terms in geometry such as point, line, plane, solid, etc. according to what they observed in the world around them. From their observations a solid is an object in space which has three dimensions called length, breadth and thickness.
It has shape, size, position and place. It can be moved from one place to another. The boundries of a solid are called surfaces. Surface has two dimension length and breadth. It has no thickness. The boundries of a surface are lines or curves. Lines and curves has no breadth and no thickness. The ends of a line or a curve are points. A point has no dimension. Euclid presented his work in the form of definitions., axioms, postulates and theorems. Some terms defined by Euclid and other mathematicians of that time may not be fully explained but still the observation were very strong and hence, from the basis for the further development of the subject.

EUCLID'S APPROACH TO GEOMETRY

Euclid's "Elements" do not admit of any defined terms. That is why his definitions are defective. He listed 23 definitions in Book 1 of "Element". A few on them are given below :

Euclid definition :

1. Point : A point is that which has no part.

2. Line : A line is breadthless length. The end of a line are points.

3. Surface : A surface is that which has length and breadth only.

4. Straight line : A straight line is a line which lies evenly with the point on itself.

5. Plane surface : A plane surface is a surface which lies evenly with the straight lines on itself.

6. Solid : A solid is that which has shape, size and position and can be moved from one place to another. The edges of a solid are surfaces.

(i) Undefined terms

The basic undefined terms in geometry are :

(a) Point (b) Line (c) Plane

(a) Point : A point has position only. It has no length, no width and no thickness.
(b) Line : A line has length but no width and no thickness.
(c) Plane : If any two points are taken anywhere on a surface and joined by a straight line, then if each and every point of this line lies in the surface, the surface is called a plane or a plane surface. Though Euclid defined a point, a line and a plane, the definitions are not accepted by today's mathematicians. They take terms as undefined terms. Because these terms can be represented intuitively or explain with the help of 'Physical Models'.
For example : The tip of a fine sharp pencil, or the tip of a needle represent a point. A thread hold tightly by two hands represent a line. The top of a table represent a plane etc.

(ii) Undefined properties (Axioms and Postulates)

Axioms : The assumptions, which are granted without proof and are used throught in mathematics which are obvious universal truths, and not specifically linked to geometry are termed as axioms. On the basis of above definitions and some more observations, Euclid stated some obvious universal truth as axioms and postulates. Euclid assumed these universal truths as such, which were not to be proved. In present time, the terms axioms and postulates canbe used interchangeably but Euclid made a fine distiction between the two terms.
For example : (i) a > b, b > c ⇒ a > c, (ii) Halves of equal are also equal, (iii) A line contains infinitely many points.
Some of the Euclid's Axioms are given below:

1. Things which are equal to the same thing are equal to one another.

For examples :-
(i) if x = y ans y = z, then x = z.
(ii) If area of a circle is equal to that of a square and the area of the square is equal to that of a rectangle,
then the area of the circle is equal to the area of the rectangle.

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

For example :-
If 5 = 5, then
5 + 2 = 5 + 2
⇒ 7 = 7

3. If equals are subtracted from equals, the remainder are equal.

For example :-
If 3 = 3, then
3 – 1 = 3 – 1
⇒ 2 = 2
Here magnitudes of same kinds can be compare and subtract. We cannot subtract a line from a triangle similarly
we cannot subtract kg from litres.

4. The things which coincide with one another are equal to one another.

For example :-

Here, two line segment AB and CD coincide with each other.
∴ AB = CD = 2 cm

Postulates : The assumptions, which are specifically linked to geometry and are obvious universal truths, are termed as postulates.

Euclid gave five postulates as stated below :

Postulate 1 : A straight line may be drawn from any one point to any other point.
Let A be a given point and B be some other point. If we draw several lines passing through the point A we see that only one of these lines passes through the point B also. Similarly, if we draw several lines passing through the point B we see that only one of these lines passes through the point A also. So, we can say a unique line passes through the points A and B. This result in the form of an axiom is as follows:

Axiom 5.1 : Given two distinct points, there is a unique line that passes through them.
Postulate 2 : A terminated line (i.e., a line segment) can be produced indefinitely on either side.
A terminated line is called line segment these days. The adjoining figure shows a terminated line (line segment) AB with two end points A and B.

According to the second postulate, a terminated line (line segment) can be produced to any length on both sides. Line segement AB when produced on both sides become line AB.

Postulate 3 : A circle can be drawn with any centre and any radius.
Postulate 4 : All right angles are equal to one another.
Postulate 5 : 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 two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

(a) The adjoining figure shows a line PQ falling on lines AB and CD such that the sum of the interior angles ∠1 and ∠2 is less than 180° or 2 right angle.Therefore, lines AB and CD if produced on left side will meet somewhere.

(b) The given figure shows a line MN falling on lines AB and CD such that the sum of the interior angles ∠3 and ∠4 is less than 180° or 2 right angles. Therefore, the Iines AB and CD if produced on right side will meet somewhere.

PROPOSITIONS OR THEOREMS :

Propositions or theorems are statements which are proved, using definition, axioms, previously proved statements and by deducitve reasoning.
For example :-

1. "The sum of all angles of a triangles is equal to 180°" is a theorem.
2. "The sum of all angles of a quadrilateral is 360°" is a theorem.

Euclid deduced 465 propositions (theorem) in a logical chain using his axioms, postulates, definitions and already proved theorems. When we proved a theorem, then it becomes a general statement for other theorem. A theorem and assumption (postulate) are both statement. The difference is that an assumption (postulate) is a accepted to be true without any proof while as theorem is accepted to be true only when it has been proved.

INCIDENCE AXIOMS ON LINES

Here, we shall assume some properties about lines and points without any proof but these properties are obvious universal truths. These properties are taken as axioms.

Axiom 1. A line contains infinitely many points.

Axiom 2. Through a given point, infinitely many lines can be drawn.

In figure, infinitely many lines pass through the point P.

Axiom 3. Given two distinct points, there exists one and only one line through them.

In figure, we observe that, out of all lines passing through the point P there is exactly one line 'ℓ' which also passes through P. Hence, we find exactly one line 'ℓ' which can be drawn through two point P and Q.

Axiom 4. (Play fair Axiom) : If P is a point outside a line ℓ, then one and only one line can be drawn through
P which is parallel to .

Clearly, through a point P outside the line ℓ, only a line m can be drawn through P which is parallel to ℓ.

EQUIVALENT VERSIONS OF EUCLID'S FIFTH POSTULATE

There are several equivalent versions of the fifth postulate of Euclid. One such version is stated as "Playfair's Axiom" which was given by Scotish mathematician John Play Fair in 1929 and was named as "Play Fair's Axiom".

Playfair's Axiom (Axiom for Parallel Lines)
For every line ℓ and for every point P not lying on ℓ, there exists a unique line m passing

through P and parallel to ℓ.
Another version of the above axiom is as stated below : Two disinct intersecting lines cannot be parallel to the same line. In figure, there are infinitely many straight line through P but there is exactly one line m which is parallel to ℓ. Thus, two intersecting lines cannot be parallel to the same line.

SOME TERMS RELATED TO GEOMETRY

(i) Line
(ii) Ray
(iii) Line segment
(iv) Collinear points
(v) Intersecting lines
(vi) Concurrent lines
(vii) Parallel lines.
(vii) Perpendicular lines
(ix) Radius

(i) Line : A line has length but no width or thickness.A line is unlimited in extent.

It extends in both the direction without end. The given figure shows a line AB. A

(ii) Ray : A straight line, generated by a point and moving in the same direction

is called a ray. The given figure shows a ray AB.

(iii) Line segment : It is the part of a line whose both the ends are fixed (terminated).

The given figure shows a line segment AB.

(iv) Collinear points : Three or more points lying on the same straight line are 

called collinear points. In the given figure, points A, B and C lie on the same straight line, so these points are collinear.

(v) Intersecting lines : If two lines have a common point, the lines are said to be

intersecting lines. In the given figure, line and m have common point O, m therefore these lines are intersecting lines.

(vi) Concurrent lines : Three or more lines in a plane are said to be concurrent o if all of them pass through the same point. In the given figure, four lines are passing through the same point O, therefore these lines are concurrent lines. The common point O is called the point
of concurrency.

(vii) Parallel lines : Two lines are said to be parallel to each other if they do not have common point. i.e. they do not intersect.

The given figure shows two parallel lines, AB and CD which will never intersect on producing both of these in any direction and upto any extent.

(viii) Perpendicular lines : Two lines which are at a right angle to each other are called perpendicular lines.

(ix) Radius : The length of the line-segment joining the centre of a circle to any point on its circumference is called its radius.

Ex. If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2
AB. Explain by drawing the figure.

Sol. According to the given statement, the figure will be as shown alongside in which the point C lies between two
points A and B such that AC = BC.
Clearly, AC + BC = AB

⇒ AC + AC = AB ⇒ [∴ AC = BC] 
⇒2AC = AB ⇒ AC = 1/2 AB

Theorem 1 : Two distinct lines cannot have more than one point in common.

Given : Two distinct line ℓ and m.
To prove : Lines ℓ and m have at most one point in common.
Proof : Two distinct lines ℓ and m intersect at a point P.
Let us suppose they will interect at another point, say Q (different from P). ℓ It means two lines ℓ and m passing through two distinct points P and Q. But it is contrary to the axiom 5.1 which states that "Given two distinct points, there exists one and only one line pass through them". So our supposition is wrong.
Hence, two distinct lines cannot have more than one point in common.

Theorem 2 : Two lines which are both parallel to the same line, are parallel to each other.
Given : Three lines ℓ, m, n in a plane such that m || ℓ and n || ℓ. m

To prove : m || n 
Proof : 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, there are two lines m and n both parallel to ℓ. This is a contradiction to the parallel axiom. So, our supposition is wrong. Hence m || n.

Theorem 3 : If , m, n are lines in the same plane such that ℓ intersects m and n|| m, then ℓ intersects n also.
Given : Three lines ℓ , m, n in the same plane such that ℓ intersect m and n|| m.
To prove: Lines ℓ and n are intersecting lines
Proof: Let ℓ and n be non intersecting lines. Then ℓ || n

But, n || m
and m are non intersecting lines.
This is a contradiction to the hypothesis that ℓ and m are intersecting lines.
So, our supposition is wrong. Hence, line ℓ intersects line n.

Ex. In figure, C is mid-point of the segment AB, P and Q are mid-point of the segment AC and BC respectively.
Prove that AP = BQ = 1/4 AB. A P C Q B

Sol. In figure, C is mid-point of the segment AB.

From (1) = 1/2 AB ......(1)

Similarly, we have 

Therfore, we hve

Ex. In figure, AD = BC, then prove that AC = BD.

Sol. In figure, we have

AD = BC AC + CD = BD + CD ... (1)

By the application of Euclid's axiom (2) when we add CD from both side of (1), the whole part sides of (1) are equal.
⇒ ∴ AC = BD.