BASIC GEOMETRICAL CONCEPTS

                                         BASIC GEOMETRICAL CONCEPTS

                                                                                                                                                                                                                                                  

Point – A point is an exact location in space usually represented by a dot (.).

                A point has no dimensions (length & breadth).

•A

                                                                                                                                    Point A               

Line – A line is a straight path that extends without end in opposite directions.

                Or, A line is a straight path of points that extends endlessly in both the directions.

                A line has no end points. It does not have a definite length.

                A line is named by using any two points on a line.

                                                     •_A                                    •_B                 

                                                     _{AB                                                        BA}          

                                                         Line AB or line BA

                                Thus, the line above is named as    AB    (line AB) or    BA   (line BA).

                Sometimes, a line is represented by a small letters like etc.

                                In the figure given alongside  is line                                              

There are two types of lines –

                Straight line and Curved line

Straight lines are of three types –

                 
                                                Horizontal line                                                      

                                                                                                      Vertical line                                 Slanting line

                                                     A curved line is called a curve.                                          

Line segment – A line segment is a part of a line that extends from one end point to the other end point.

                Or, A part of a line is called a line segment. It has a fixed length that can be measured. The points at which the line segment begins and ends are called its end points. Every line segment has two end points.

                                                                •_{A                                      B}•                                                 

                                                                                                ^{AB  line segment AB}

Ray – A ray is a part of a line which has one end point and extends without end in one direction.

                Or, A ray is a straight path that has one end point and which extends endlessly in one direction.

•                                      •_{A                                                                                                                                                                                              AB     or ray AB}                                                                                                                                                   

Plane – A plane is a flat surface which extends infinitely in all directions. Two planes intersect in a line.

                Or, A plane is a flat surface that goes endlessly in all directions.

•C       A•                  •B

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               

                                                                                                                                  

Collinear – Three or more points which lie on the same line are called collinear.

                                                 •_A    •_B     •_C     •_D                 

Angle – Two rays with a common end point form an angle.

                Or, an angle is a figure formed by two rays with the same initial (end) point.

Ø  The common end point is called the vertex of the angle and the two rays the arms of the angle.

Ø  An angle is also thought of as a figure formed by rotating a single ray about its vertex.

             ^A_{•                                                                                                                                                                                                                                 O   or       AOB  or      BOA   or                                                                                                                                                                                              O}¹              _B•   

Interior and Exterior of an Angle –                                                                                                        

Interior – The inside of the angle, that is the region                                                         ^{K  •}                                                         

               Between the rays, is called the interior of the angle.                          •^X                            •^A                                                                                             

                                                                                                                                        ^D•

Exterior – The points of the plane that do not lie on the arms or        ^{L•       E•}          _Y•         ^•_M

              in the interior of the angle are in the exterior of the angle.                                                                      

                      The points x and y are in the exterior of  _ABC.

Ø  Whenever two rays have a common endpoint, we get an angle.

 

 

Kinds of angles –

                Angles are classified into various types according to their measure.

1.       zero angle – An angle whose measure is  is called zero angle.

2.       acute angle – An angle whose measure is less than  but greater than  is called an acute angle.

Here,

                                                                                                         ^P_•

                                                      

                                                                    

                                                                                                                                     _Q^{•                                 •}_R

                                                             

3.       Right angle – An angle whose measure is  is called a right angle.

             

4.       Obtuse angle – An angle whose measure is between  is called an obtuse angle.

5.       Straight angle – An angle whose measure is  is called straight angle.

6.       Reflexive angle – The angle whose measure is between  is called a reflexive angle.

7.       Complete angle – An angle whose measure is  is called a complete angle.

or, when the initial and terminal positions of an angle coincide, the measure of the angle is called a complete angle.

 A complete angle is made up of two straight angles.

      

Pairs of angles –

                (i)Complementry angle – Two angles are complementary if their sum is  

                (ii) Supplementry angle – Two angles are supplementary if their sum is .

              (iii)Congruent angle – Two angles are congruent if their measures are equal.

Adjacent angle – Two angles in the same plane that have a common vertex and a common side, but no common interior points are called adjacent angles.

Parallel line – Two lines in the same plane that never meet or intersect each other are called parallel lines.

Perpendicular lines –