Embedded Files
By studying this lesson you will be able to
identify the dynamic or static nature of an angle,
name angles,
measure and draw angles using the protractor, and
classify angles based on their magnitude.
9.1 Angles
You learnt in grade 6 that an angle is created when two straight line segments meet each other.
Below are a few types of angles we identified.
Do the following review exercise to recall the facts you have learnt about angles.
(1) Choose the figures that are angles and write down the corresponding letters.
(2) Identify the angles in the figure below and complete the table.
(3) Draw an angle of each type on a square ruled paper. Write the type of angle next to the corresponding figure.
Acute angle, Right angle, Obtuse angle, Straight angle, Reflex angle
9.2 The dynamic or static nature of an angle
Let us investigate more on angles.
If we observe our surroundings, we can identify many angles. A few examples are given below.
A common property of the above angles is that their magnitude does not change.
If the magnitude of an angle does not change, then it is static in nature.
So the angles in the above figures are static in nature.
Note that the magnitude of the angle between two spokes of a wheel does not change, even when the wheel is turning.
Let us now consider some situations that involve rotation.
In the examples given above, let us consider the arms of the angle involved.
We see that there is a rotation of both arms or of one of them. Therefore, the magnitude of the angle changes. This is the dynamic nature of such an angle.
Let us understand the dynamic nature of an angle further by doing the following activity.
You can see that the magnitude of the angle between the two parts of the ekel changes. That is, this angle is dynamic in nature.
When both parts of the ekel are rotated too the magnitude of the angle between the two parts changes.
A rotation which is in the same direction as the rotation of the arms of a clock is defined as a clockwise rotation. A rotation which is in the opposite direction to that of the rotation of the arms of a clock is defined as an anticlockwise rotation.
(1) (i) Write down 3 instances where you can observe angles which are dynamic in nature in your surrounding environment.
(ii) Write down 3 instances where you can observe angles which are static in nature in your surrounding environment.
(2) (i) Give an example of an angle which is static in nature where the positions of the arms of the angle are fixed.
(ii) Give an example of an angle which is static in nature where there is a change in the positions of the arms of the angle.
(iii) Give an example of an angle which is dynamic in nature where there is a change in the position of only one arm of the angle.
(iv) Give an example of an angle which is dynamic in nature where there is a change in the positions of both arms of the angle.
9.3 Naming Angles
Let us now consider how angles are named.
In Figure I, two angles have been created by the straight line segments AB and BC meeting.
The straight line segments AB and BC are defined as the “arms of the angle”. The point B where AB and BC meet is defined as the “vertex of the angle”.
The magnitude of the angle which is indicated in red is less than that of a straight angle; that is, less than the magnitude of two right angles.
The magnitude of the angle indicated in blue is greater than that of a straight angle.
The angle indicated in red is named as angle ABC and is written as " ABCor CBA
Here we write the letter which indicates the vertex in the middle and the other two letters beside it.
The angle indicated in blue is named as the reflex angle ABC and is written as reflex angle ABC or reflex angle CBA.
In some books angle ABC is written as .
(1) Write down the arms and the vertex of each of the angles given below.
(2) Copy each of the angles given below and name them using letters of the English alphabet.
(3) Draw and name an angle of your choice on a square ruled paper.
(4) Draw an obtuse angle with arms XY and YZ on a square ruled paper.
(5) Draw an angle and name it DEF. Name its arms and its vertex.
(6) Draw a reflex angle and name it.
(7) Draw a right angle on a square ruled paper and name it.
(8) Prabath has written the angle in the figure as XYZ.
Sumudu has written it as ZYX . Kasun says that both Prabath and Sumudu are correct. Do you agree with Kasun? Explain your answer.
9.4 Measuring angles
There are standard units and instruments to measure distance, mass, time and the volume of a liquid. You learnt about these in grade 6.
Now let us learn about a standard unit and an instrument used to measure angles.
The standard unit used to measure angles is degrees. One degree is written as 1o.
The angle that is formed when a straight line segment completes one full circle by rotating about a point is 360o.
The instrument used to measure angles is made of one half of a full circle. It is called a “protractor”. The figure of a protractor is shown below. It is numbered from 0o to 180o clockwise and anticlockwise. The line indicated by 0 - 0 is called the “base line”.
There are two scales indicated in the protractor. They are the inner scale and the outer scale.
The long line segments on the outer scale are marked as 0, 10, 20, ..., 180. The gap between every pair of long line segments is again divided into 10 similar parts using short line segments. As indicated in the figure, the magnitude of the angle between two long line segments is 10o.
Let us now see how we can use the protractor to measure the angle AOB in the figure'
Place the protractor on the figure such that the origin and the base line coincide with the vertex O and the arm OA respectively.
Then the arm OB coincides with the line indicated by 50o in the inner scale (Note that OA coincides with 0o on this scale). Therefore the magnitude of the angle AOB is 50o, and we write AOB= 50o.
By observing this figure, we see that an angle of 1o is a small angle which is difficult to draw.
(2)Draw each of the angles below on a square ruled paper. Measure and write the magnitude of each angle.
(3) Draw the following figures in your exercise book. Measure and write down the magnitude of each of the angles indicated by the English letters.
9.5 Drawing angles with given magnitude
Let us now draw angles when their magnitude is given.
Step 3 - Now find 35o in the outer scale. Place a dot mark on the paper at 350
Step 4 - Remove the protractor.
Name the dot marked in step 3 as R. Now draw a straight line from Q to R. Write the magnitude of the angle PQR as 350
As above, draw the following:
(6) Dasun says that the angle in Figure (II) is larger than the angle in Figure (I). Do you agree? Explain your answer.
9.6 Classification of angles
We learnt in grade 6 to classify angles using a right angle. Magnitude of a right angle is 90°. We can classify angles by comparing them with 90°.
Right Angles
Any angle of magnitude 90° is called a “right angle”. KLM is a right angle.
Acute Angles
Any angle of magnitude less than 90° is called an “acute angle”. PQR is an acute angle.
Obtuse Angles
Any angle of magnitude greater than 90° but less than 180° (that is an angle between 90°and 180° )is called an “obtuse angle”. ABC is an obtuse angle.
Straight Angles
Any angle of magnitude 180° is called a “straight angle”. XYZ is a straight angle.
Reflex Angles
Any angle of magnitude between 180°and 360° is called a “ reflex angle”. EFG is areflex angle.
9.7 Measuring and Drawing Reflex angles
The figure shows the reflex angle ABC.This angle cannot be measured directly using a protractor. So let us see how we can measure this reflex angle.
Method I :-
Let us use the ruler to extend AB and obtain the straight angle ABD.
That is, ABD = 180o.
(1) Copy the two groups (a) and (b) in your exercise book. Join each angle and its type with a straight line.
Group (a) (Magnitude of the angle) Group (b) (Type of angle)
18o Straight angle
135o Right angle
180o Acute angle
255o Obtuse angle
90o Reflex angle
(1) Copy the two groups (a) and (b) in your exercise book. Join each angle and its type with a straight line.
Group (a) (Magnitude of the angle) Group (b) (Type of angle)
18o Straight angle
135o Right angle
180o Acute angle
255o Obtuse angle
90o Reflex angle
(3) Choose and write down the most appropriate magnitude for each of the angles below, from the values given in brackets.
(4) Draw the following reflex angles using the protractor.
(1) (a) Simplify the following.
(i) 15 + 13 + 12 (ii) 18 - 12 + 6 (iii) 9 + 6 - 8
(iv) 8 × 7 - 12 (v) 7 × 3 + 5 (vi) 24 - 18 ÷ 3
(vii) 15 + 18 ÷ 3 (viii) 16 + 5 × 3 (ix) 15 - 9 ÷ 3
(b) Hasintha says “when we simplify 91 - 35 ÷ 7, we get 8 as the answer”.
Explain why Hasintha’s answer is incorrect.
(2) (i) What is a bilaterally symmetric plane figure?
(ii) Write the number of axes of symmetry in each of the symmetric figures given below.
(iii) Draw the following symmetric figures in your square ruled exercise book.
Draw their axes of symmetry and name them.
(a) A rectilinear plane figure with only one axis of symmetry
(b) A rectilinear plane figure with only two axes of symmetry
(c) A rectilinear plane figure with more than two axes of symmetry
(iv) If the plane figure is cut along the dotted line, then it will be divided into two parts which coincide with each other. Is the figure bilaterally symmetric about this line? Explain your answer giving reasons.
(v) Copy each of the following figures onto a square ruled paper and complete each figure such that the two dotted lines become axes of symmetry of the completed figure.
(3) (i) Set A is given below by listing its elements.
A = {2, 3, 5, 7}
Write A using a common property of its elements.
(ii) Re-write P = {factors of 12} by listing its elements.
(iii) Let A = {multiples of 3 that lie between 8 and 20}
(a) Write A by listing its elements.
(b) Represent A in a Venn diagram.
(iv) Write the set represented by the Venn diagram,
(a) using a common property of its elements,
(b) by listing its elements
(4) (i) Write the factors of 44.
(ii) Write the prime factors of 44.
(iii) Write 56 as a product of its prime factors.
(iv) Find the highest common factor of 18, 30, 42.
(v) Find the least common multiple of 18, 30, 42.
(5) (i) What is the digital root of 522?
(ii) Using the digital root, explain why 522 is divisible by 3.
(iii) Using the digital root explain why 522 is divisible by 9.
(iv) How do we find out without dividing a number whether it is divisible by 4 or not?
(v) 4 3 2 1 are four numbers written on four cards. How many numbers which are divisible by 4 can be made using all these cards? Write down all such numbers.
(vi) If the number 53 which has 3 digits is divisible by 9, then what is the digit in the units place?
(vii) If the number 53 which has 3 digits is divisible by 6, then what is the digit in the units place?
(6) (a) (i) Find the value of 62'
(ii) Write all the factors of the number corresponding to the value found in (i).
(iii) There are only two prime factors among the factors written in (ii). Write down three more numbers where each of them has only two prime factors.
(iv) Write each of the three numbers as a power of a prime number.
(b) (i) Expand a2 b3 .
(ii) Evaluate x3 y2 when x = 5 and y = 4
(7) Write whether the following statements are true or false.
(i) Any multiple of 2 has only one prime factor.
(ii) Any number which can be written as a power of 2, has 2 as its only prime factor.
(iii) Any multiple of 3 has only one prime factor.
(iv) Any number which can be written as a power of 3, has only one prime factor.
(v) Any number which can be written as a power of 5, has 5 as its only prime factor.
(vi) The highest common factor of any two positive integers is less than or equal to their least common multiple.
(vii) The highest common factor of any two distinct prime numbers is 1.
(viii) The highest common factor of 12 and 13 is 1.
(8) (i) Explain, giving reasons whether AD 1892 is a leap year or not.
(ii) Explain, giving reasons whether AD 2100 is a leap year or not.
(iii) To which decade does the year AD 2100 belong to?
(9) (a) Add the following.
(i) years months days (ii) years months days
(b) Subtract the following.
(i) years months days (ii) years months days
(10) The fifth birthday of a child fell on 2002 - 08- 26. His mass was 20 kg and 700 go on that day.
(i) When was his birthday?
(ii) On his eighth birthday his mass was 30kg and 600g. What is the increase in his mass during the three years?
(iii) What was his age on 2012 - 03 - 25?
(iv) On 2012 - 03 - 25, the mass of the child was 12kg and 800g more than his mass on his fifth birthday. Find the mass of the child on 2012 - 03 - 25.
(11) (a) Using the number line, determine each of the following sums.
(i) (-6) + (-4) (ii) (-5) + (+5) (iii) (+8) + (-9)
(b) Simplify
(12) (a) Complete the table given below by considering person whose journey starts at A and ends at F.
(b) Measure the magnitude of each angle given below using a protractor and write it down.
(c) Draw the following angles using the protractor and the ruler.
(i) ABC = 65o (ii) PQR = 130o (iii) MNR = 145o
(13) (i) Two parallel lines are shown below. How far apart are they?
(ii) (a) Draw a straight line segment and name it XY.
(b) Mark a point A which is a distance of 4.8 cm from XY.
(c) Draw a straight line segment through the point A parallel to XY.
(iii) (a) Draw the parallelogram ABCD.
Draw parallel lines to diagonal AC throught B and D.
(14) (i) Nimal’s birthday is 2002 -11 -25. Find Nimal’s age in years, months and days on 2016 - 08 - 20.
(ii) Write the time that has elapsed between 12:35 of 2015 - 01 - 01 and 19:20 of 2015 - 02 - 05 in days, hours and minutes.
Page updated
Report abuse