MATHEMATICS - FORM 1

LINES AND ANGLES



ANGLES


A) Identifying an Angle

angle is formed by two straigth 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 instru-

    ment 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 protactor 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 wheter 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

    wheter 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 comple-

    mentary angles. Each is the complement of the

    other.

    For example :-
         

   

3. We know that the sum of the angles on a atraight line

    is 180.

4. Two angles which add up to 180 are called supplemen-

    tary 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  adjucent 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