In a figure, is an angle of symmetry. The figure has two angles of symmetry less than . What is its smallest angle of symmetry?

Question

Homi · Accepted Answer

Understand the Question

An angle of rotational symmetry is an angle through which a figure is rotated about its center to look identical to its original position.
All angles of rotational symmetry for a regular or symmetric figure are positive integer multiples of its smallest angle of rotational symmetry, $x$ .
If there are $2$ angles of symmetry less than $60^\circ$ , then $60^\circ$ must be the $3^{\text{rd}}$ multiple of $x$ .

Step 1 · Find the Multiple Position of $60^\circ$

Let the smallest angle of symmetry be $x$ .

All angles of symmetry are multiples of $x$ : $x, 2x, 3x, \dots, nx$

Given that $60^\circ$ is an angle of symmetry: $60^\circ = n \times x$

The angles of symmetry smaller than $60^\circ$ are $x, 2x, \dots, (n - 1)x$ .

Since there are $2$ such angles:

\begin{aligned} n - 1 &= 2 \\[0.6em] n &= 2 + 1 \\[0.6em] &= 3 \end{aligned}

Thus, $60^\circ$ is the $3^{\text{rd}}$ multiple of $x$ .

Step 2 · Calculate the Smallest Angle of Symmetry

Substitute $n = 3$ into the equation $60^\circ = n \times x$ :

$60^\circ = 3 \times x$

\begin{aligned} x &= \dfrac{60^\circ}{3} \\[0.6em] &= 20^\circ \end{aligned}

Answer

$20^\circ$

Common Mistakes

Dividing by $2$ instead of $3$ : Assuming that having "two angles less than $60^\circ$ " means dividing $60^\circ$ by $2$ ( $30^\circ$ ). The angles less than $60^\circ$ are the $1^{\text{st}}$ and $2^{\text{nd}}$ multiples, making $60^\circ$ the $3^{\text{rd}}$ multiple.
Non-Multiple Assumption: Forgetting that all angles of rotational symmetry of a figure must be uniform integer multiples of the smallest angle ( $x, 2x, 3x, \dots$ ).

Question 17