Question

the decagon has a -fold reflectional symmetry. the order of rotational symmetry for the decagon is. the smallest angle of rotational symmetry for the decagon is degrees.

Explanation:

Step1: Reflectional symmetry

A decagon has \(10\) sides. The number of lines of reflectional symmetry of a regular polygon is equal to the number of its sides. So, a decagon has \(10 -\)fold reflectional symmetry.

Step2: Order of rotational symmetry

The order of rotational symmetry of a regular polygon is equal to the number of its sides. Since a decagon has \(n = 10\) sides, the order of rotational symmetry is \(10\).

Step3: Smallest angle of rotational symmetry

The formula for the smallest angle of rotational symmetry \(\theta\) of a regular polygon is \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For \(n = 10\), \(\theta=\frac{360^{\circ}}{10}=36^{\circ}\)

Answer:

The decagon has a \(10 -\)fold reflectional symmetry. The order of rotational symmetry for the decagon is \(10\). The smallest angle of rotational symmetry for the decagon is \(36\) degrees.