chapter 1 ( basic concepts in geometry )

CHAPTER 1 : BASIC CONCEPTS IN GEOMETRY1.1 Points, Lines and Planes 1.2 Line Segment 1.3 Rays and Angles 1.4 Some Special Angles 1.5 Angles Made By A Transversal 1.6 Transversal Across Two Parallel Lines 1.7 Conditions For Parallelism

INTRODUCTIONFor hundreds of years man has been studying Geometry. Ancient civilizations like the Egyptians and the Babylonians discovered many properties through actual measurements. However it is the 3 Greek mathematician Euclid who is credited with giving a completely new outlook to the study of Geometry. He showed that all knowledge is not arrived at by physical measurements. If some basic facts that seem simple and obvious are accepted as true the remaining facts can be arrived at by logical reasoning. Those facts which, are to be simply accepted are called 'axioms' and those which can be questioned or proved are called 'theorems.'

1.1 Points, Lines & Planes The most fundamental geometric form is a point. It is represented

as a dot with a capital alphabet which is its name (Figure 1.1) A line is a set of points and it extends in opposite directions up to

infinity. It is represented by two points on the line and a double headed arrow or a single alphabet in the lower case (Figure 1.1)

A plane is a two dimensional (flat) surface that extends in all directions up to infinity.

Figure 1.1 shows points A, D & Q, line AB, line l and plane P.

Some axioms regarding points, lines and planes are given below.

1. An infinite number of lines can be drawn through any given point.

2. One and only one line can be drawn through two distinct points.

3. When two lines intersect they do so at only one point.

Collinear And Coplanar

Two or more points are said to be collinear if a single line contains all of them. Otherwise they are said to be non collinear. (Figure 1.2)

Figure 1.2 shows two lines l and m . Line l is such that it passes through A, B and C. Hence points A B and C are collinear. In the case of points P, Q and R there can be no single line containing all three of them hence they are called non-linear.

Axiom : A plane containing a line and a point outside it or by using the definition of a line, a plane can be said to contain three non-collinear points. Conversely, through any three non collinear points there can be one and only one plane (figure 1.3).

Axiom : If two lines intersect, exactly one plane passes through both of them (figure1.4).

Axiom : If two planes intersect their intersection is exactly one line (figure 1.5).

Axiom : If a line does not lie in a plane but intersects it, their intersection is a point (figure 1.7 ).

Point A is the intersection point of line l and plane P.

Example 1 Take any three non-collinear points A,B and C on a paper.

How many different lines can be drawn through different pairs of points ? Name the lines.

Solution : Three lines can drawn namely AB, BC & AC.

Example 2 Name lines parallel to line AB Are line AO and point R coplanar ? Why ? Are points A, S, B and R coplanar ? Why ? Name three planes passing through at A.

1.2 Line Segment A line segment is a part of a line. It has a fixed length

and consequently two end points. They are used to name the line segment (figure 1.8).

A line contains infinite segments and if two segments on a line have a common end point, they can be added

( figure 1.9).

Figure 1.8

Seg. PQ is a segment of line AB.Figure 1.9

Seg. PQ and Seg.QR are two segments on line l and they have a common end point Q. Therefore Seg.PQ + Seg.QR = Seg.PR.

Exercise: Are the following statements true or false ?

1. Any number of lines can pass through a single given point.

2. If two points lie in a plane the line joining them also lies in the same plane.

3. Any number of lines can pass through two given points.

4. Two lines can intersect in more than one point.

5. Two planes intersect to give two lines.

6. If two lines intersect only one plane contains both the lines.

7. A line segment has two end points and hence a fixed length.

8. The distance of the midpoint of a segment from one end may or may not be equal to its distance from the other.

a

81

61

a

28

9 10

Solve for a.

1.3 Rays and Angles A ray has one end point and extends in the other

direction up to infinity. It is represented by naming the end point and any other point on the ray with the symbol(figure 1.10 ).

J is the end point of a ray and K is a point on it. This ray is represented as .A ray can extend in any one direction only.

Angle

Two rays going in different directions, but having a common end point, form an angle. The common end point is called the vertex of the angle and the rays are called its sides or arms. An angle is represented by the symbol  and named, using either both the rays or just the vertex (figure 1.11).

Figure 1.11 represents Ð XYZ or Ð Y.

Interior and Exterior of an angle : 

The interior of  PQR is the shaded region in figure 1.12. S is a point in the interior of  Q because it lies on the R- side of ray PQ and the P - side of ray QR. The set of all such points is called the interior of  PQR.

In figure 1.13 the shaded region shows the exterior of   XYZ. The exterior of an angle is defined as the set of points in the plane of a given angle which are neither on the sides of the angle nor in its interior.

Measure of an angle :

Every angle has a measure. It is measured in degrees from 00to 1800 and is represented as m  . A line is also an angle because it satisfies the definition of having two rays going in different ( in this case opposite ) directions with a common end point ( figure 1.14 ).

Angle addition property : Two angles with a common side and a common vertex are adjacent if their interiors are disjoint . The measures of two adjacent angles can be added to find the measure of the resultant angle. This is called the angle addition property. With reference to figure 1.15 m   COB + m   DOC = m   DOB.

Angle Bisectors :  The ray which passes through the vertex on an

angle and divides the angle into two angles of equal measure is called the bisector of that angle. Of the two angles ( figure 1.16 )   AOB and   COB are equal in measure then  is called the bisector of   COA. Just as every line has only one midpoint every angle has only one bisector.

Types of angles :Depending on their measure, angles can be classified as acute angle, right angle or obtuse angle. Right angle : The right angle ( figure 1.17 ) has a

measure of 900. It is represented by the symbol

Types of angles :Depending on their measure, angles can be classified as acute angle, right angle or obtuse angle. Acute angle : Any angle whose measure is

between 0 and 900 is called an acute angle (figure 1.18).

0 < a < 90 \ a is an acute angle.

Types of angles :Depending on their measure, angles can be classified as acute angle, right angle or obtuse angle. Obtuse angle : An angle with a measure

between 900 and 1800 is called an obtuse angle (figure 1.19).

900 < b < 1800 \ b is an obtuse angle.

1.4 Some special angles Complementary angles : If the sum of two angles

equals 900 the two angles are called complementary. Complementary angles thus add up to a right angle.

Complementary angles are of two types. If they have one side in common they are called adjacent complementary angles ( figure 1.20 a ). If no side is common then they are called non-adjacent complementary angles ( figure 1.20 b ).

Figure 1.20a Figure 1.20b

Adjacent Angles – two angles with common side.

A

B D

C

Angle ABC, angle CBD, Angle ABD

Since Angle ABC and angle CBD with common side BC, so Angle ABC and angle CBD are adjacent angles.

A

B D

C

E

What angle is adjacent to angle ABC?

dc

ba

What angle is adjacent to angle a?

What angle is vertical to angle b?

Example Since the measures of complementary angles always sum

up to 900 if the measure of one angle is known that of the complement can be found easily. In figure 1.20 b   a and   b are complementary. Also it is known that m   a = 300.

m   a + m   b = 900

    300 + m   b = 900

              m   b = 900 - 300

              m   b = 600

Supplementary angles : If the measures of two angles sum up to 1800 they are called supplementary angles. Supplementary angles are of two types :

a) Non adjacent supplementary angles and b) Adjacent supplementary angles. Non adjacent supplementary angles are distinct and have

no arm in common (figure 1.21).

Supplementary angles : If the measures of two angles sum up to 1800 they are called supplementary angles. Supplementary angles are of two types :

Adjacent supplementary angles are called angles in a linear pair and have one arm in common ( figure 1.22 ).

Vertical angles : When two lines AB and CD intersect at O, four angles are formed with vertex O. Consider   AOC and   BOD. It is observed that and are opposite rays and so is and . In such a case   AOC and   BOD are called vertical angles ( figure 1.23 ). 

Theorem : Vertical angles are always equal in measure.

Proof : To prove m   AOC = m   BOD

m   AOC + m   COB = 1800 ( supplementary angles )

m   BOD + m   COB = 1800 ( supplementary angles )

i.e. m   AOC + m   COB = m   BOD + m   COB

or m   AOC = m   BOD.

Example 1 Measures of some angles are given below. Find the measures of

their supplements. a) 750   

ans. 1050  b) 1250

ans. 550     c) x0    

ans. (180 - x) 0

d) (180 - x) 0    

ans. x0

e) (90 + x) 0

ans. (90 - x) 0

Example 2 Measures of some angles are given below. Find the measures of

their complements. a) 350    

ans. 550

b) 450    

ans. 450

c) (90 - r) 0    

ANS. r0

d) x0

ANS. (90 - x) 0

Example 3 The measure of one angle is twice that of its

complement. Find its measure. Solution : 600

Example 4 The measure of an angle is four times the measure of its

supplementary angle. Find its measure. Solution : 1440

Angles made by a transversal Definition of a transversal : A line which intersects two or

more given coplanar lines in distinct points is called a transversal of the given lines. In figure 1.24 the line l is the transversal of lines a and b.

t

sr

Angles made by a transversal Interior and Exterior angles : Those angles which lie between

lines a and b are called interior angles, i.e.  3,  4,  5 and  6 .Exterior angles lie on opposite sides of lines a and b, i.e.  1, 2, 7 and 8 .

Angles made by a transversal Interior angles on opposite sides of the transversal are

called alternate interior angles and  4,  6 are alternate interior angles and so also   3, and  5 .

Angles made by a transversal Interior angles on same side of the transversal are called

consecutive interior angles.  4, and  5 are consecutive interior angles and so also   3 and 6 .

Angles made by a transversal Alternate exterior angles are on opposite sides of the

transversal and do not lie between lines a and b, i.e. 1 and  7and also   2 and  8 .

Angles made by a transversal Exterior angles on the same side of the transversal are

called consecutive exterior angles, i.e.  1 &  8 as also   2 and 

7 .

Corresponding angles : Angles that appear in the same relative position in each group are called corresponding angles, i.e.   1 and   5 are called corresponding angles. Similarly   2 &  6,   4 &   8 and   3 &   7 are pairs of corresponding angles.

8

7

64

3

5

2

1

m

n

l

Example 1

In figure 1.25 n is the transversal of lines l and m. Write down the pairs of :

a) corresponding angles ,

b) alternate interior angles, 

c) alternate exterior angles ,

d) consecutive interior angles & ,

e) consecutive exterior angles.

g

a b

d

c

e

f

102

Parallel lines are defined as those lines which are coplanar and do not intersect (figure 1.27). The figure merely show a part of the lines as the lines actually extend up to infinity. Hence although they do not intersect in the region that is observed it is possible that they will intersect when sufficiently extended.

1.6 Transversal across two parallel lines Corresponding Angles : If a transversal cuts two

parallel lines the corresponding angles are equal ( figure 1.26 )

l & m are two parallel lines cut by a transversal n to form angles 1 to 8.

Axiom : Corresponding angles are equal in measure if a transversal cuts parallel lines.

m   1 = m   5 m   2 = m   6 m   3 = m   7 m   4 = m   8 .

Alternate interior angles : If a transversal cuts two parallel lines the alternate interior angles are equal in measure. In figure 1.26 m   4 = m   6 and m   3 = m   5. This can be proved as follows :

m   3 = m   4 are supplementary and so also m Р6 = m   7 are supplementary. Since the sums of their measures are 1800 in both cases. m   3 + m   4 = m   6 + m   7. However m   3 = m   7 as they are corresponding angles

formed by a transversal across parallel lines. Therefore, m  3 + m   4 = m   6 + m   3 i.e. m   4 = m   6. Similarly it can be shown that m  3 = m   5 .