Symmetry in Polygons

Show your work for each problem.

1.
Which shape has exactly $2$ lines of symmetry and rotational symmetry of order $2$ ?
1. a)Kite
2. b)Regular hexagon
3. c)Rectangle
4. d)Square
Answer: Rectangle
A rectangle has $2$ lines of symmetry (through the midpoints of opposite sides) and order $2$ . A square has $4$ and $4$ , a kite $1$ and $1$ , a regular hexagon $6$ and $6$ .
2.
Which shape has point symmetry?
1. a)Equilateral triangle
2. b)Isosceles trapezoid
3. c)Kite
4. d)Parallelogram
Answer: Parallelogram
A parallelogram fits onto itself after a half turn about the point where its diagonals cross: order $2$ , so it has point symmetry. The equilateral triangle has order $3$ (odd), the kite and the isosceles trapezoid have order $1$ .
3.
Where do the lines of symmetry of a regular pentagon go?
1. a)A regular pentagon has no lines of symmetry
2. b)Each through the midpoints of two opposite sides
3. c)Each through two opposite vertices
4. d)Each through a vertex and the midpoint of the opposite side
Answer: Each through a vertex and the midpoint of the opposite side
A regular pentagon has $5$ lines of symmetry. Because $5$ is odd, each one goes through a vertex and the midpoint of the opposite side. There is no pair of opposite vertices and no pair of opposite sides.
4.
Which shape has rotational symmetry but no line of symmetry?
1. a)Kite
2. b)Rectangle
3. c)Parallelogram (not a rectangle or rhombus)
4. d)Isosceles triangle
Answer: Parallelogram (not a rectangle or rhombus)
A parallelogram (not a rectangle or rhombus) has order $2$ but $0$ lines of symmetry. The kite and the isosceles triangle have $1$ line and order $1$ , the rectangle has $2$ lines and order $2$ .
5.
Which statement is always true?
1. a)Every polygon with point symmetry has at least one line of symmetry
2. b)A shape with rotational symmetry of order $3$ has point symmetry
3. c)A regular polygon with $n$ sides has rotational symmetry of order $n$
4. d)A kite has rotational symmetry of order $2$
Answer: A regular polygon with $n$ sides has rotational symmetry of order $n$
A regular polygon with $n$ sides fits onto itself after every turn of $\frac{360°}{n}$ , so its order is $n$ . The others are false: a parallelogram has point symmetry but no line; order $3$ is odd, so a half turn does not fit; a kite has order $1$ .
6.
How many lines of symmetry does a regular hexagon have?
Answer: 6
A regular hexagon has $6$ lines of symmetry: $3$ through pairs of opposite vertices and $3$ through the midpoints of pairs of opposite sides.
7.
What is the smallest angle of rotation of an equilateral triangle, in degrees?
Answer: 120
An equilateral triangle has order $3$ , so the smallest angle is $\frac{360°}{3} = 120°$ .
8.
What is the order of rotational symmetry of a rhombus that is not a square?
Answer: 2
After $90°$ the long diagonal would lie on the short one, so it does not fit. After $180°$ it fits, and after $360°$ again: order $2$ .
9.
A regular polygon fits onto itself after a turn of $30°$ , and no smaller turn works. How many lines of symmetry does it have?
Answer: 12
The order is $\frac{360°}{30°} = 12$ , so it is a regular dodecagon with $12$ sides and $12$ lines of symmetry.
10.
A regular decagon has $10$ lines of symmetry. How many of them pass through two opposite vertices?
Answer: 5
$10$ vertices form $\frac{10}{2} = 5$ pairs of opposite vertices, so $5$ lines go through two vertices. The other $5$ go through the midpoints of opposite sides.
11.
Describe the symmetry of an isosceles triangle that is not equilateral.
Answer: 1
1. How many lines of symmetry does it have? 1
2. What is its order of rotational symmetry? 1
12.
Find the symmetry of a regular pentagon.
Answer: 72°
1. How many lines of symmetry? 5
2. What is the order of rotational symmetry? 5
3. What is the smallest angle of rotation in degrees? 72
13.
A rectangle measures $8$ cm by $5$ cm. Find its symmetry.
Answer: 180°
1. How many lines of symmetry? 2
2. What is the order of rotational symmetry? 2
3. What is the smallest angle of rotation in degrees? 180
14.
Describe the lines of symmetry and the rotational symmetry of a regular dodecagon ( $12$ sides).
Answer: 30°
1. How many lines go through two opposite vertices? 6
2. How many lines go through the midpoints of two opposite sides? 6
3. How many lines of symmetry in total? 12
4. What is the smallest angle of rotation in degrees? 30
15.
Check the symmetry of a kite (that is not a rhombus).
Answer: 0
1. How many lines of symmetry does the kite have? 1
2. What is its order of rotational symmetry? 1
3. Does it have point symmetry? Type $1$ for yes, $0$ for no. 0
16.
A regular hexagon is turned by $240°$ about its center. How many of its smallest rotation steps is that?
Answer: 4
1. What is the order of rotational symmetry of a regular hexagon? 6
2. What is the smallest angle of rotation in degrees? 60
3. How many smallest steps make $240°$ ? 4
17.
Each interior angle of a regular polygon measures $140°$ . Find its number of lines of symmetry and its smallest angle of rotation.
Answer: 40°
1. What is each exterior angle in degrees? 40
2. How many sides does the polygon have? 9
3. How many lines of symmetry does it have? 9
4. What is the smallest angle of rotation in degrees? 40
18.
All lines of symmetry of a regular pentagon pass through its center. What is the angle between two neighboring lines?
Answer: 36°
1. How many lines of symmetry does a regular pentagon have? 5
2. Each line makes two half-lines from the center. How many half-lines are there? 10
3. What is the angle between neighboring half-lines in degrees? 36