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LINES AND ANGLE I
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.
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 1Place 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 corresponds with the arm LK.
Step 2 :Read the value of <KLM using the outer scale. Therefore,
KLM = 300
MEASURE AN ANGLE WHICH IS MORE THAN 1800
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
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
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 thefigures. 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)
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
EXAMPLE 1
KL is parallel to RS or KL//RS
AB//CD
EF//HG
EH//FG
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 or a set square 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.
C) DETERMINING PERPENDICULAR LINES
1. If two straight lines intersect at 90 , we say the two lines are perpendicular to each other.
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 :-
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 atraight 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
Find the value of x in each of the following.
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 .
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 anglesin the following.
(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.
WORKED EXAMPLE 9
KLM is a straight line . Find x.
WORKED EXAMPLE 10
In the figure above, AB and CD are straight lines. Find the values of x and y.
