Question 16
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
If a figure looks the same after turning by a small angle, it will also look the same after turning by multiples of that angle.
Step 1 — Smallest angle of symmetry
The problem tells us the smallest angle of symmetry is 60°. This means the figure looks exactly the same. It looks the same after we turn it by 60°.
Step 2 — Finding other angles
If the figure looks the same after a 60° turn, it will also look the same if we turn it again by 60°. This means it will look the same after a total turn of . Let us calculate this total turn.
So, 120° is another angle of symmetry. We can keep turning it by 60° each time. Each new turn will make it look the same. Let us find the next angle.
This means 180° is also an angle of symmetry. Let us find the next angle.
So, 240° is another angle. Let us find the next angle.
This means 300° is also an angle of symmetry. Let us find the last angle for a full circle.
A 360° turn means the figure is back to its starting position. Every figure looks the same after a 360° turn. So, 360° is always an angle of symmetry.
Answer
The other angles of symmetry are 120°, 180°, 240°, 300°, and 360°.
More questions in FIO
Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
For each of the following figures, identify the line(s) of symmetry if it exists.
Punching Game
The fold is a line of symmetry. Punch holes at different locations of a folded square sheet of paper using a punching machine and create different symmetric patterns.
In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole. How was the paper folded?
Given the line(s) of symmetry, find the other hole(s):
Here are some questions on paper cutting. Consider a vertical fold. We represent it this way:
Vertical Fold
Similarly, a horizontal fold is represented as follows:
Horizontal Fold
After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.
Trace each figure and draw the lines of symmetry, if any:
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Find the angles of symmetry for the given figures about the point marked •.
(a) (b) (c)
Which of the following figures have more than one angle of symmetry?
Give the order of rotational symmetry for each figure:
Colour the sectors of the circle below so that the figure has:
(i) 3 angles of symmetry
(ii) 4 angles of symmetry
(iii) What are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
Draw, wherever possible, a rough sketch of:
a. A triangle with at least two lines of symmetry and at least two angles of symmetry.
b. A triangle with only one line of symmetry but not having rotational symmetry.
c. A quadrilateral with rotational symmetry but no reflection symmetry.
d. A quadrilateral with reflection symmetry but not having rotational symmetry.
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
Can we have a figure with rotational symmetry whose smallest angle of symmetry is:
a. 45°?
b. 17°?