date
period
6. find the angle(s) of rotation that will carry the 12 - sided polygon below onto itself.
image of a 12 - gon
7. what are the angles of rotation for a 20 - gon? how many lines of symmetry (lines of reflection) will it have?
8. what are the angles of rotation for a 15 - gon? how many line of symmetry (lines of reflection) will it have?
9. a regular polygon has rotational symmetry for an angle of 18°. how many sides does it have? explain.

Question

Accepted Answer

### Problem 6
# Explanation:
## Step1: Recall rotational symmetry formula
For a regular $ n $-sided polygon, the smallest angle of rotation that maps it onto itself is $ \frac{360^\circ}{n} $, and all multiples of this angle (up to $ 360^\circ $) are also angles of rotation.
## Step2: Apply formula for 12 - sided polygon
Here, $ n = 12 $. So the smallest angle is $ \frac{360^\circ}{12}=30^\circ $. The angles of rotation are $ 30^\circ k $, where $ k = 1,2,\cdots,11 $ (since $ 30^\circ\times12 = 360^\circ $, which is a full rotation and maps the polygon to itself trivially, but we usually consider non - trivial rotations first and include the full rotation conceptually).
# Answer:
The angles of rotation are $ 30^\circ,60^\circ,90^\circ,120^\circ,150^\circ,180^\circ,210^\circ,240^\circ,270^\circ,300^\circ,330^\circ $ (all multiples of $ 30^\circ $ from $ 30^\circ $ to $ 330^\circ $)

### Problem 7
# Explanation:
## Step1: Find angles of rotation for 20 - gon
For a regular $ n $-sided polygon, the smallest angle of rotation is $ \frac{360^\circ}{n} $. For $ n = 20 $, the smallest angle is $ \frac{360^\circ}{20}=18^\circ $. The angles of rotation are $ 18^\circ k $, where $ k = 1,2,\cdots,19 $.
## Step2: Find number of lines of symmetry
A regular $ n $-sided polygon has $ n $ lines of symmetry. For $ n = 20 $, the number of lines of symmetry is 20.
# Answer:
Angles of rotation: $ 18^\circ,36^\circ,\cdots,342^\circ $ (all multiples of $ 18^\circ $ from $ 18^\circ $ to $ 342^\circ $); Number of lines of symmetry: 20

### Problem 8
# Explanation:
## Step1: Find angles of rotation for 15 - gon
Using the formula for the smallest angle of rotation of a regular $ n $-sided polygon $ \theta=\frac{360^\circ}{n} $. For $ n = 15 $, $ \theta=\frac{360^\circ}{15} = 24^\circ $. The angles of rotation are $ 24^\circ k $, where $ k = 1,2,\cdots,14 $.
## Step2: Find number of lines of symmetry
A regular $ n $-sided polygon has $ n $ lines of symmetry. For $ n = 15 $, the number of lines of symmetry is 15.
# Answer:
Angles of rotation: $ 24^\circ,48^\circ,\cdots,336^\circ $ (all multiples of $ 24^\circ $ from $ 24^\circ $ to $ 336^\circ $); Number of lines of symmetry: 15

### Problem 9
# Explanation:
## Step1: Recall the formula for rotational symmetry angle
For a regular polygon with $ n $ sides, the smallest angle of rotational symmetry $ \theta=\frac{360^\circ}{n} $.
## Step2: Solve for $ n $
We know that $ \theta = 18^\circ $. From $ \theta=\frac{360^\circ}{n} $, we can re - arrange the formula to solve for $ n $: $ n=\frac{360^\circ}{\theta} $. Substituting $ \theta = 18^\circ $, we get $ n=\frac{360^\circ}{18^\circ}=20 $.
# Answer:
The polygon has 20 sides. Because for a regular polygon, the measure of the smallest angle of rotational symmetry is given by $ \frac{360^\circ}{n} $, and solving $ \frac{360^\circ}{n}=18^\circ $ gives $ n = 20 $.

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QUESTION IMAGE

Question

Explanation:

Problem 6

Step1: Recall rotational symmetry formula

Step2: Apply formula for 12 - sided polygon

Step1: Find angles of rotation for 20 - gon

Step2: Find number of lines of symmetry

Step1: Find angles of rotation for 15 - gon

Step2: Find number of lines of symmetry

Answer:

Problem 7