for questions 19-22, use the figure below.
19. what is the name of the shape on the left?
20. how many lines of symmetry are there on the shape to the left?
21. list all of the degrees of rotational symmetry for the shape to the left.
22. how many diagonals does the regular polygon have?
for questions 23-26 draw the following
23. draw all lines of symmetry on the square. (square with vertices labeled a, b, c, d)
24. draw all the diagonals on the square. (square with vertices labeled a, b, c, d)
25. draw all lines of symmetry on the parallelogram. (parallelogram with vertices labeled a, b, c, d)
26. draw all diagonals on the parallelogram. (parallelogram with vertices labeled a, b, c, d)

Question

Accepted Answer

### Question 19
# Brief Explanations:
The shape on the left is a regular nonagon (9 - sided polygon) as it has 9 equal - length sides and 9 equal - angle vertices.
# Answer:
Regular nonagon

### Question 20
# Explanation:
## Step1: Recall the formula for lines of symmetry of a regular polygon
For a regular $n$ - sided polygon, the number of lines of symmetry is equal to the number of sides $n$.
## Step2: Identify the number of sides of the polygon
The polygon (regular nonagon) has $n = 9$ sides. So the number of lines of symmetry is 9.
# Answer:
9

### Question 21
# Explanation:
## Step1: Recall the formula for rotational symmetry of a regular polygon
The formula for the angle of rotational symmetry of a regular $n$ - sided polygon is $\theta=\frac{360^{\circ}}{n}$, and the rotational symmetries are the multiples of this angle.
## Step2: Calculate the basic rotational angle
For a regular nonagon, $n = 9$, so the basic rotational angle $\theta=\frac{360^{\circ}}{9} = 40^{\circ}$.
## Step3: Find all multiples of the basic angle
The multiples of $40^{\circ}$ within $360^{\circ}$ (excluding $360^{\circ}$ as it is the same as $0^{\circ}$ rotation) are: $40^{\circ},80^{\circ},120^{\circ},160^{\circ},200^{\circ},240^{\circ},280^{\circ},320^{\circ}$. And including $360^{\circ}$ (which is equivalent to the identity rotation), but usually we list the non - trivial ones and sometimes $360^{\circ}$ is considered as well. But the general way is to list the angles $\theta_k=k\times\frac{360^{\circ}}{n}$ for $k = 1,2,\cdots,n - 1$ and $k=n$ (which is $360^{\circ}$). So for $n = 9$, the rotational symmetries are $40^{\circ},80^{\circ},120^{\circ},160^{\circ},200^{\circ},240^{\circ},280^{\circ},320^{\circ},360^{\circ}$ (or we can say the non - zero rotational symmetries are the multiples of $40^{\circ}$ up to $360^{\circ}$).
# Answer:
$40^{\circ},80^{\circ},120^{\circ},160^{\circ},200^{\circ},240^{\circ},280^{\circ},320^{\circ},360^{\circ}$ (or multiples of $40^{\circ}$ from $40^{\circ}$ to $360^{\circ}$)

### Question 22
# Explanation:
## Step1: Recall the formula for the number of diagonals of a polygon
The formula for the number of diagonals $d$ of a polygon with $n$ sides is $d=\frac{n(n - 3)}{2}$.
## Step2: Substitute $n = 9$ into the formula
For a nonagon, $n=9$, so $d=\frac{9\times(9 - 3)}{2}=\frac{9\times6}{2}=27$.
# Answer:
27

### Question 23
# Brief Explanations:
A square has 4 lines of symmetry: 2 lines that pass through the mid - points of opposite sides (horizontal and vertical) and 2 lines that pass through opposite vertices (the two diagonals). To draw them, draw a horizontal line through the mid - point of the top and bottom sides, a vertical line through the mid - point of the left and right sides, and the two diagonal lines connecting $A$ to $C$ and $B$ to $D$.
# Answer:
(Description of drawing: Draw 2 lines through mid - points of opposite sides (horizontal and vertical) and 2 lines through opposite vertices (diagonals) of the square $ABCD$)

### Question 24
# Brief Explanations:
A square has 2 diagonals. These are the lines connecting vertex $A$ to vertex $C$ and vertex $B$ to vertex $D$.
# Answer:
(Description of drawing: Draw the two diagonals $AC$ and $BD$ of the square $ABCD$)

### Question 25
# Brief Explanations:
A non - rectangular parallelogram (the general parallelogram) has 0 lines of symmetry. If it were a rectangle (a special case of parallelogram), it would have 2 lines of symmetry, but for a general parallelogram (with non - right angles), there are no lines that divide it into two congruent mirror - image halves.
# Answer:
(Description of drawing: A general parallelogram has 0 lines of symmetry, so no lines to draw. If it were a rectangle (a type of parallelogram), draw 2 lines through mid - points of opposite sides)

### Question 26
# Brief Explanations:
A parallelogram has 2 diagonals. These are the lines connecting vertex $A$ to vertex $C$ and vertex $B$ to vertex $D$.
# Answer:
(Description of drawing: Draw the two diagonals $AC$ and $BD$ of the parallelogram $ABCD$)

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QUESTION IMAGE

Question

Explanation:

Question 19

Step1: Recall the formula for lines of symmetry of a regular polygon

Step2: Identify the number of sides of the polygon

Step1: Recall the formula for rotational symmetry of a regular polygon

Step2: Calculate the basic rotational angle

Step3: Find all multiples of the basic angle

Answer:

Question 20