Equivalent Versions of Euclids Fifth Postulate and Euclids Geometry, Class 9, Mathematics | Extra Documents & Tests for Class 9 PDF Download

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  and for every point P not lying on , there exists a unique line m passing

through P and parallel to .
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 . 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 = (1over2)
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 = (1over 2)AB

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

Given : Two distinct line and m.
To prove : Lines  and m have at most one point in common.
Proof : Two distinct lines and m intersect at a point P.
Let us suppose they will interect at another point, say Q (different from P).  It means two lines  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 , m, n in a plane such that m ||  and n || . m

To prove : m || n 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 , there are two lines m and n both parallel to . 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 intersects m and n || m, then  intersects n also.
Given : Three lines  , m, n in the same plane such that intersect m and n|| m.
To prove: Lines  and n are intersecting lines
Proof: Let  and n be non intersecting lines. Then || n

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

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

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.