Question

name ____ period ____ unit 1 lesson 16 additional practice problem set additional practice problems

1. for each figure, identify any angles of rotation that create symmetry.
2. a hexagon has rotational symmetry that can take any of its vertices to any of its other vertices. select all conclusions that we can reach from this.

a. all rotations take one half of the hexagon to the other half of the hexagon.
b. all angles of the hexagon have the same measure.
c. all sides of the hexagon have the same length.

1. select all the angles of rotation that produce symmetry for this flower.

a. 60 degrees
b. 72 degrees
c. 144 degrees
d. 180 degrees
e. 216 degrees
f. 288 degrees

Explanation:

Step1: Analyze rotational symmetry of hexagon

A regular hexagon has 6 - fold rotational symmetry. The angles of rotation that map a regular hexagon onto itself are multiples of $\frac{360^{\circ}}{6}=60^{\circ}$. Rotations do not necessarily take one - half of the hexagon to the other half. A regular hexagon has congruent angles and congruent sides, but this is not related to the property of rotational symmetry taking vertices to vertices. So, for a hexagon, all sides have the same length and all angles have the same measure, which are properties of a regular hexagon, not directly related to the rotational - symmetry vertex - to - vertex property.

Step2: Analyze angles of rotation for flower

The flower has 6 "petals", so the angles of rotation that produce symmetry are multiples of $\frac{360^{\circ}}{6}=60^{\circ}$.
$60^{\circ}\times1 = 60^{\circ}$, $60^{\circ}\times3=180^{\circ}$, $60^{\circ}\times 2 = 120^{\circ}$, $60^{\circ}\times4 = 240^{\circ}$, $60^{\circ}\times5=300^{\circ}$, $60^{\circ}\times6 = 360^{\circ}$.

Answer:

1. B. All angles of the hexagon have the same measure, C. All sides of the hexagon have the same length
2. A. 60 degrees, D. 180 degrees