STUDY NOTES FOR SSC CGL AND CPO 
      
      
                    
                    
                         www.mahendras.org           myshop.mahendras.org 
   Write us : content@mahendras.org   www.mahendraguru.com 
                                                    
                                                STUDY NOTES FOR SSC CGL AND CPO 
            
            
                    FUNDAMENTAL CONCEPTS OF                                                       In the figure above, the angle is represented as ∠AOB. OA and 
                                       GEOMETRY                                                   OB are the arms of ∠AOB. Point O is the vertex of ∠AOB. 
                Point:                                                                            The amount of turning from one arm (OA) to other (OB) is 
                It is an exact location. It is a fine dot which has neither length                called the measure of the  angle ( AOB).  
                nor  breadth  nor  thickness  but  has  position  i.e.,  it  has  no               Right angle:  
                magnitude.                                                                        An angle whose measure is 90  is called a right angle.     
             
                Line segment:  
                The straight path joining two points A and B is called a line 
                segment points and a definite length.   
                                                                 
                Ray:  
                A line segment which can be extended in only one direction is                                                                               
                called a ray.                                                                                                        
                                                                                                  Acute angle:  
                                                                                                  In angle whose measure is less than one right angle (i.e., less 
                Intersecting lines:                                                               than 90), is called an acute angle.    
                Two lines having a common point are called intersecting lines. 
                The common point is known as the point of intersection.  
                                                                                               
                                                                                                                                                               
                                                                                                  Obtuse angle:  
                Concurrent lines:                                                                 An angle whose measure is more than one right angle and less 
                If two or more lines intersect at the same point, then they are                   than two right angles (i.e., less than 180 and more than 90) is 
                known as concurrent lines.                                                        called an obtuse angle. 
                                                         
                Angles:  
                When two straight lines meet at a point they form an angle                                                                                   
                                                                                                                                     
                                                                                                  Reflex angle: An angle whose measure is more than 180 and 
                                                                                                  less than 360 is called a reflex angle.    
                                                                               
                                                              www.mahendras.org                                                        myshop.mahendras.org 
       Write us : content@mahendras.org                                                        www.mahendraguru.com 
                                                                                                                                  
                                          STUDY NOTES FOR SSC CGL AND CPO 
          
          
                                                                        
              Complementary angles: If the sum of the two angles is one                                                                     
              right angle (i.e., 90), they are called Complementary angles.          In the above figure, ∠1 and ∠3 and angles ∠2 and ∠4 are 
                                                                                     vertically opposite angles.     
              Therefore, the complement of an angle θ is equal to 90° − θ.           Note: Vertically opposite angles are always equal.  
                                                                                      Bisector of an angle: If a ray or a straight line passing through 
                                                                                     the vertex of that angle, divides the angle into two angles of 
                                                                                     equal measurement, then that line is known as the Bisector of 
                                                                                     that angle. 
                                                                                 
              Supplementary  angles:  Two  angles  are  said  to  be 
              supplementary, if the sum of their measures is 180.   Example:   
              Angles measuring 130 and 50 are supplementary angles. Two                                                               
              supplementary  angles  are  the  supplement  of  each  other.          A point on an angle is equidistant from both the arms.  
              Therefore, the supplement of an angle θ is equal to 180° − θ. 
                                                                                                                                      
              Vertically opposite angles: When two straight lines intersect          In the figure above, Q and R are the feet of perpendiculars 
               each other at a point, the pairs of opposite angles so formed         drawn from P to OB and OA. It follows that PQ = PR.  
                          are called vertically opposite angles                      Parallel lines: Two lines are parallel if they are coplanar and 
                                                                                     they do not intersect each other even if they are extended on 
                                                                                     either side.  
                                                                                     Transversal: A transversal is a line that intersects (or cuts) two 
                                                                                     or more coplanar lines at distinct points.  
                                                     www.mahendras.org                                              myshop.mahendras.org 
      Write us : content@mahendras.org                                            www.mahendraguru.com 
                                                                                                                 
                                              STUDY NOTES FOR SSC CGL AND CPO 
           
           
                                                                                              Answer: As 67° + 113° = 180°, lines P and S, R and S, and S 
                                                                                              and U are parallel. Therefore, lines P, R, S and U are parallel 
                                                                                              to each other. Similarly, lines Q and T are parallel to each other.  
                                                                                              Example- 
                                                                                              In the figure given below, PQ and RS are two parallel lines 
                                                                                              and AB is a transversal. AC and BC are angle bisectors of 
                                                                                              ∠BAQ and ∠ABS,  respectively.  If  ∠BAC  =  30°,  find 
                                                                                              ∠ABC and ∠ACB. 
                                                                     
               In the above figure, a transversal t is intersecting two parallel 
               lines, l and m, at A and B, respectively. 
               Angles formed by a transversal of two parallel lines:   
                                                                                                                                               
                                                                                              Answer: ∠BAQ + ∠ABS = 180° [Supplementary angles]   
                                                                                          
                                                                                              BAQ         ABS 180            00
                                                                                                  2         2     2 90 BACABC90
                                                                                               
                                                                                              Therefore, ∠ABC = 60° and ∠ACB = 90°. 
                                                                                              Example- 
               In the above figure, l and m are two parallel lines intersected            
               by a transversal PS. The following properties of the angles can                For what values of x in the figure given below are the lines 
               be observed:                                                                   P-A-Q and R–B-S parallel, given that AD and BD intersect 
               ∠3 = ∠5 and ∠4 = ∠6 [Alternate angles]                                         at D? 
               ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, ∠3 = ∠7 
               [Corresponding angles]  
               ∠4 + ∠5 = ∠3 + ∠6 = 180° [Supplementary angles]  
               In the figure given below, which of the lines are parallel to 
               each other? 
                                                                                                                                                 
                                                                                              Answer: We draw a line DE, parallel to RS, as shown in the 
                                                                                              figure below: 
                                                                           
                                                                                                                                                  
                                                           www.mahendras.org                                                     myshop.mahendras.org 
       Write us : content@mahendras.org                                                    www.mahendraguru.com