Mathematics Form 1: Pythagoras Theorem

LINES AND ANGLES


ANGLES
A) Identifying an Angle

angle is formed by two straight lines that meet at a point called the vertex.

For example : –

     
In the figure above,
(a) AOB is an angle.
(b) OA and OB are called the arms of the angle.
(c) O is the vertex, that is the point where the two arms meet.
Worked Example 1
Mark the angle in each case.
(a)                              (b)
                  
Solution
(a)                              (b)
                 

B) Naming an angle
An angle can be named by using one letter or three letters.
For example :-
        
     
Worked Example 2
        
      

C) Measuring Angles

1. Angles are measured in units called degrees ( 0 ).
2. To measure an angle, we can use an instrument called the protractor as shown below.
         
3. Note that if we read from left to right ( clockwise direction ), we use the inner scale.
4. To measure an angle less than 1800, <KLM, follow the steps below.
    Method 1 :
         
    Step 1
    Place the protractor that its centre is on the vertex L. Adjust the protractor until its base line corresponds with the arm LM.
        
 Step 2
    Read the value of <KLM using the inner scale. Therefore, <KLM = 300.

 Method 2 :
        
    Step 1
    Place the protractor so that its centre is on the vertex L. Adjust the protractor until its base line corresronds with the arm LK.
    
    Step 2
    Read the value of  <KLM using the outer scale.Therefore, .KLM = 300
5. To measure an angle which is more than 1800, follow the steps below :
    To measure <STU
    
    Step 1
    Produce the arm ST to V and measure <STV. <STV = 1800
    
 Step 2
    Place and adjust the protractor as shown to measure <VTU.
    
    Step 3
    <STU = <STV + <VTU
             =  1800 + 200
             = 2000
    
D) Drawing Angles Using a Protractor
1. We can also use a protractor to draw an angle.
2. To draw <RST =600, follow the steps below.
    Step 1
    Draw an arm ST with S as the vertex.
        
    Step 2
    Place the protractor so that its centre is on the vertex S and its base line is on ST.
             
    Step 3
    Find 600 at the inner scale and mark it with a point. Call this point R.
             
    Step 4
    Remove the protractor and draw a line to join R with S.

                 
Step 5
    Mark and label <RST as 600.
       
3. To draw <KLM = 2400 ( more than 1800 ), follow the steps below.
    Step 1
    Draw an arm KL with L as the vertex.
         
    Step 2
    Place the protractor so that its centre is on the vertex L and its base line is on KL. Mark the point M at 600 on the outer scale.

         
    Step 3
    Remove the protractor and join LM with a straight line.
         
    Step 4
    Label <KLM as 2400.
   
    
E) Identifying the Different Types of Angles
The table below shows the different types of angles.
Worked Example 3
Which of the following angles is acute, obtuse, reflex or right-angled?
(a) 1650    
(b) 900
(c) 2340
(d) 830

Solution
(a) 1650 is an obtuse angle.
(b) 900 is right angle.
(c) 2340 is a reflex angle.
(d) 830 is an acute angle.
G) Determining the Sum of Angles on a
 Straight Line 

1. Use a protractor to measure the angles on the straight line. 
Worked Example 4
Using a protractor, measure the angles on the straingh line KLM. Then, find the sum of the angles in each case.
(a)                                                                      (b)        
                                            
Solution
(a) x = 1200 , y = 600
     x + y = 1200 + 600
             = 1800
(b) p = 400 , q = 900 , r = 500
      p + q + r = 400 + 900 + 500   
                    = 180
2. In general, the sum of the angles on a straight line is 1800.
    For example :-
          
    AOB is a straight line.
    x + y + z = 1800
H) Determining the Sum of Angles in 
One Whole Turn

1. A protractor is used to measure the angles at a point.
Worked Example 5
Use a protractor to measure the angles in the figures. Then, find the sum of the angles in each case.
(a)                                (b)    
Solution
(a) x = 1100 , y = 2500
     x + y = 1100 + 2500
              = 3600
(b) p = 1300 , q = 600 , r = 700 , s = 1000
     p + q + r + s = 1300 + 600 + 700 + 1000
                         = 3600
2. In general, the sum of the angles that formed one whole turn is 3600.
    For example :-
    
    a + b + c + d + e = 360       
I) Calculating Angles involving One
 Whole Turn

Worked Example 6

Without measuring, calculate the angles marked.
(a)
     
(b)
    

Solution
PARALLEL LINES AND 
PERPENDICULAR LINES

A) Determining Parallel Lines
1. Parallel lines are lines that will not meet however far they are produced either way.
2. They are at the same distance apart from one other
    For example :-
    (a)
          
KL is parallel to RS or KL//RS
    (b)

        
     AB//CD
(c)
        
        EF//HG
        EH//FG
3. To determine whether two given lines are parallel or not, follow the steps below.
    Step 1
    Mark two points P and R on of two straight lines.The points should be as far apart as possible.
    
    Step 2
    Using a protractor ora set aquare draw the two perpendicular lines PM and RN as shown.
    
    Step 3
    Measure PM and RN. The given lines are parallel to each other if PM =RN.

B) Drawing Parallel Lines
There are three methods to draw parallel lines.
Method 1 : Using a ruler
   
(a)   

       
(b) 
       
Method 2 : Using a protractor
(a)
     

(b)
     
    Therefore, PM//RN
Method 3 : Using a set square 
(a) To draw a straight line through the point P and parallel to the straight line XY.
     
(b)
     
(c)
     
 (d)

      
C) Determining Perpendicular Lines

1. If two straight lines intersect at 90 , we say the two lines are perpendicular to each other.
   For example :-


 
3. We can use a protractor or a set square to determine whether two straight lines are perpendicular to each other or not.
    For example :-
    (a)
          
    (b)
            
        
D) Drawing Perpendicular Lines
1. To draw a line perpendicular ti PR from a point M on PR, follow the steps as shown below.
    Step 1
                       

 
  Step 2
   
              
    Join MN. The straight line MN will be perpendicular to PR at M.
2. To draw a line perpendicular to PR from a point M outside PR, follow the steps below.
   Step 1
                      
    Step 2
    

INTERSECTING LINES AND 
THEIR PROPERTIES
A) Identifying Intersecting Lines
We say the two straight lines intersect if they meet (or cut) at a point. This point is known as the point of intersection.
For example :-
   
B) Identifying Complementary Angle
 and Supplementary Angles
1. We know that when two lines are perpendicular, the angle formed by them is a right angle or 90 .
2. Two angles which add up to 90 are called complementary angles. Each is the complement of the other.
 For example :-
          
    
3. We know that the sum of the angles on a straight line is 180.
4. Two angles which add up to 180 are called supplementary angles. Each is the supplement of the other.
    For example :-

        
    
C) Determining Complementary and 
Supplementary Angles
Worked Example 7
Find the value of x in each of the following.
(a)                             (b) 
                
Solution

D) Identifying Adjacent Angles on a Straight Line

1. When two straight lines intersect, the sum of the adjacent angles on a straight line is 180 .
    For example :-
             
    The angles x and y which CE makes with the straight line ACB are called adjacect angles on a straight line.Therefore, x + y  = 180
2. When two adjacent angles together make up 180, they are called supplementary angles.

Worked Example 8
Identify the different pairs of adjacent angles in the following.
(a)                                      (b)
        
Solution
(a) To determine adjacent angles on a straight line, measure the angles marked. If the sum of the angles is 180 , then they are adjacent angles on a straight line.
     x = 60 , y = 120
     x + y = 60 + 120
     = 180
     Therefore, x and y are adjacent angles on the straight line DEF. 
(b) a = 110 , b = 50 , c = 130 , d = 70
     a + d = 110 + 70
             = 180
     LINES AND ANGLES
ANGLES

 A) Identifying an Angle

 angle is formed by two straight lines that meet at a point called the vertex.

 For example : –
     

 In the figure above,

 (a) AOB is an angle.

 (b) OA and OB are called the arms of the angle.

 (c) O is the vertex, that is the point where the two arms meet.

 Worked Example 1

 Mark the angle in each case.

 (a)                              (b)
                  

 Solution

 (a)                              (b)
                 

B) Naming an angle

 An angle can be named by using one letter or three letters.

 For example :-

         
     

 Worked Example 2
        
      
C) Measuring Angles

 1. Angles are measured in units called degrees ( 0 ).

 2. To measure an angle, we can use an instrument called the protractor as shown below.

          

 3. Note that if we read from left to right ( clockwise direction ), we use the inner scale.

 4. To measure an angle less than 1800, <KLM, follow the steps below.

     Method 1 :
         

     Step 1

     Place the protractor that its centre is on the vertex L. Adjust the protractor until its base line corresponds with the arm LM.

         

  Step 2

     Read the value of <KLM using the inner scale. Therefore, <KLM = 300.

 Method 2 :

          

     Step 1

     Place the protractor so that its centre is on the vertex L. Adjust the protractor until its base line corresronds with the arm LK.

     

     Step 2

     Read the value of  <KLM using the outer scale.Therefore, .KLM = 300

 5. To measure an angle which is more than 1800, follow the steps below :

     To measure <STU

     

     Step 1

     Produce the arm ST to V and measure <STV. <STV = 1800

     

  Step 2

     Place and adjust the protractor as shown to measure <VTU.

     

     Step 3

     <STU = <STV + <VTU

              =  1800 + 200

              = 2000

     

 D) Drawing Angles Using a Protractor

 1. We can also use a protractor to draw an angle.

 2. To draw <RST =600, follow the steps below.

     Step 1

     Draw an arm ST with S as the vertex.

         

     Step 2

     Place the protractor so that its centre is on the vertex S and its base line is on ST.

              

     Step 3

     Find 600 at the inner scale and mark it with a point. Call this point R.
             

     Step 4

     Remove the protractor and draw a line to join R with S.
                 
Step 5

     Mark and label <RST as 600.

        

 3. To draw <KLM = 2400 ( more than 1800 ), follow the steps below.

     Step 1

     Draw an arm KL with L as the vertex.

          

     Step 2

     Place the protractor so that its centre is on the vertex L and its base line is on KL. Mark the point M at 600 on the outer scale.

          

     Step 3

     Remove the protractor and join LM with a straight line.

          

     Step 4

     Label <KLM as 2400.
   
    

 E) Identifying the Different Types of Angles

 The table below shows the different types of angles.

Worked Example 3

 Which of the following angles is acute, obtuse, reflex or right-angled?

 (a) 1650    

 (b) 900

 (c) 2340

 (d) 830

 Solution

 (a) 1650 is an obtuse angle.

 (b) 900 is right angle.

 (c) 2340 is a reflex angle.

 (d) 830 is an acute angle.

 G) Determining the Sum of Angles on a
 Straight Line 

 1. Use a protractor to measure the angles on the straight line. 

 Worked Example 4

 Using a protractor, measure the angles on the straingh line KLM. Then, find the sum of the angles in each case.

 (a)                                                                      (b)        
                                            

 Solution

 (a) x = 1200 , y = 600

      x + y = 1200 + 600

              = 1800

 (b) p = 400 , q = 900 , r = 500

       p + q + r = 400 + 900 + 500   

                     = 180

 2. In general, the sum of the angles on a straight line is 1800.

     For example :-

           

     AOB is a straight line.

     x + y + z = 1800

 H) Determining the Sum of Angles in 
One Whole Turn

 1. A protractor is used to measure the angles at a point.

 Worked Example 5

 Use a protractor to measure the angles in the figures. Then, find the sum of the angles in each case.

 (a)                                (b)    

 Solution

 (a) x = 1100 , y = 2500

      x + y = 1100 + 2500

               = 3600

 (b) p = 1300 , q = 600 , r = 700 , s = 1000

      p + q + r + s = 1300 + 600 + 700 + 1000

                          = 3600

 2. In general, the sum of the angles that formed one whole turn is 3600.

     For example :-

     

     a + b + c + d + e = 360       

 I) Calculating Angles involving One
 Whole Turn

 Worked Example 6

 Without measuring, calculate the angles marked.

 (a)
     

 (b)
    

 Solution
PARALLEL LINES AND 
PERPENDICULAR LINES

 A) Determining Parallel Lines

 1. Parallel lines are lines that will not meet however far they are produced either way.

 2. They are at the same distance apart from one other

     For example :-

     (a)
          

 KL is parallel to RS or KL//RS

     (b)
        
     AB//CD

 (c)
        

         EF//HG

         EH//FG

 3. To determine whether two given lines are parallel or not, follow the steps below.

     Step 1

     Mark two points P and R on of two straight lines.The points should be as far apart as possible.

     

     Step 2

     Using a protractor ora set aquare draw the two perpendicular lines PM and RN as shown.

     

     Step 3

     Measure PM and RN. The given lines are parallel to each other if PM =RN.
B) Drawing Parallel Lines

 There are three methods to draw parallel lines.

 Method 1 : Using a ruler
   
(a)   
       

 (b) 

       

 Method 2 : Using a protractor

 (a)
     
(b)
     

     Therefore, PM//RN

 Method 3 : Using a set square 

 (a) To draw a straight line through the point P and parallel to the straight line XY.

      

 (b)
     

 (c)
     
 (d)
      

 C) Determining Perpendicular Lines

 1. If two straight lines intersect at 90 , we say the two lines are perpendicular to each other.

   For example :-





3. We can use a protractor or a set square to determine whether two straight lines are perpendicular to each other or not.

     For example :-

     (a)
          

     (b)

             

         

 D) Drawing Perpendicular Lines

 1. To draw a line perpendicular ti PR from a point M on PR, follow the steps as shown below.

     Step 1
                       

  Step 2
   
              

     Join MN. The straight line MN will be perpendicular to PR at M.

 2. To draw a line perpendicular to PR from a point M outside PR, follow the steps below.

    Step 1

                       

     Step 2
    
INTERSECTING LINES AND 
THEIR PROPERTIES

 A) Identifying Intersecting Lines

 We say the two straight lines intersect if they meet (or cut) at a point. This point is known as the point of intersection.

 For example :-
   

 B) Identifying Complementary Angle
 and Supplementary Angles

 1. We know that when two lines are perpendicular, the angle formed by them is a right angle or 90 .

 2. Two angles which add up to 90 are called complementary angles. Each is the complement of the other.

  For example :-

            

     

 3. We know that the sum of the angles on a straight line is 180.

 4. Two angles which add up to 180 are called supplementary angles. Each is the supplement of the other.

     For example :-

         

     
C) Determining Complementary and 
Supplementary Angles

 Worked Example 7

 Find the value of x in each of the following.

 (a)                             (b) 
                

 Solution

D) Identifying Adjacent Angles on a Straight Line

 1. When two straight lines intersect, the sum of the adjacent angles on a straight line is 180 .

     For example :-

              

     The angles x and y which CE makes with the straight line ACB are called adjacect angles on a straight line. Therefore, x + y  = 180

 2. When two adjacent angles together make up 180, they are called supplementary angles.

Worked Example 8

 Identify the different pairs of adjacent angles in the following.

 (a)                                      (b)
        
Solution

 (a) To determine adjacent angles on a straight line, measure the angles marked. If the sum of the angles is 180 , then they are adjacent angles on a straight line.

      x = 60 , y = 120

      x + y = 60 + 120

      = 180

      Therefore, x and y are adjacent angles on the straight line DEF. 

 (b) a = 110 , b = 50 , c = 130 , d = 70

      a + d = 110 + 70

              = 180

      Therefore, a and d are adjacent angles on the straight line PRT.

      b + c = 50 + 130

              = 180

 Therefore, b and c are adjacent angles on the straight line PRT.

 E) Identifying Vertically Opposite Angles

 When two straight lines intersect, either pair of opposite angles are called vertically opposite angles.

 For example :-

                 

 Intersection of the straight lines KL and RS. a and c are vertically opposite angles. b and d are vertically opposite angles.

 F) Determining the Size of  Vertically 
Opposite Angles

 If two straight lines intersect, the vertically opposite angles are equal.

 For example :-

               

 G) Finding the Values of Adjacent Angles
 on a Straight Line 

 Worked Example 9

           
KLM is a straight line . Find x.

 Solution

H) Problem Solving involving Angles
 formed by Intersecting Lines

 Worked Example 10
     
In the figure above, AB and CD are straight lines. Find the values of x and y.

 Solution
Therefore, a and d are adjacent angles on the straight line PRT.
     b + c = 50 + 130
             = 180
Therefore, b and c are adjacent angles on the straight line PRT.
E) Identifying Vertically Opposite Angles
When two straight lines intersect, either pair of opposite angles are called vertically opposite angles.

For example :-
                
Intersection of the straight lines KL and RS. a and c are vertically opposite angles. b and d are vertically opposite angles.
F) Determining the Size of  Vertically 
Opposite Angles

If two straight lines intersect, the vertically opposite angles are equal.
For example :-
              
G) Finding the Values of Adjacent Angles
 on a Straight Line 

Worked Example 9

          
KLM is a straight line . Find x.

Solution
H) Problem Solving involving Angles
 formed by Intersecting Lines

Worked Example 10
     
In the figure above, AB and CD are straight lines. Find the values of x and y.
Solution

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