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?

Question

Homi · Accepted Answer

All angles of symmetry are multiples of the smallest angle of symmetry.

Step 1 — Understanding angles of symmetry

An angle of symmetry means a figure looks the same. It looks the same after turning it by that angle. We turn it around its center point.

Step 2 — Finding the smallest angle

Let us call the smallest angle of symmetry $x$ . All other angles of symmetry are multiples of $x$ . So, $x$ , $2x$ , $3x$ , and so on, are all angles of symmetry.

Step 3 — Using the given information

We know that 60° is an angle of symmetry. So, 60° must be a multiple of $x$ . Let us write this as: $60 = n \times x$ Here, $n$ is a whole number.

Step 4 — Counting the smaller angles

The angles of symmetry smaller than 60° are: $x$ $2x$ ... $(n-1)x$ The problem says there are two such angles. This means $(n-1)$ must be equal to 2.

Step 5 — Calculating the value of n

We found that: $n - 1 = 2$ To find $n$ , we add 1 to both sides. $n = 2 + 1$ $n = 3$ So, 60° is the third multiple of $x$ .

Step 6 — Calculating the smallest angle

We know that $60 = n \times x$ . We found that $n = 3$ . So, we can write: $60 = 3 \times x$ To find $x$ , we divide 60 by 3. $x = \frac{60}{3}$ $x = 20$ The smallest angle of symmetry is 20°.

$\boxed{\text{Smallest angle of symmetry} = 20^{\circ}}$

Answer

(i) Smallest angle of symmetry = 20°.

Question 17

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