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Transformations can change the size, position, or orientation of a figure. They can also map one figure onto another. But, can a transformation map a figure onto the same figure? Throughout the lesson, the answer to this question will be developed.

### Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

## Rotational Symmetry in Rectangles

Consider a rectangle Can the rectangle be rotated such that the image exactly matches the preimage? Not so fast, there is one catch; can it be done where the angle of rotation is greater than and less than Try it out by rotating around the movable point If accomplished, where was point located? What was the angle of rotation?

## Line Symmetry in Squares

In the applet, square can be reflected along line Is it possible for the image of under a reflection to match the preimage exactly? Try it out! If it was possible, along which points of did the line pass? Could there be more than one line that works?

## Defining Symmetry

For some geometric figures, it is possible to find a transformation that maps the figures onto themselves. In such cases, the transformation is called a symmetry of the figure.

## Symmetry

A symmetry is a rigid motion that maps a figure onto itself. It is a non-trivial congruence of a figure with itself, that is, the sides and angles do not correspond with themselves. When a figure has a symmetry it is said to be a symmetric figure.

## Defining Rotational Symmetry

When a figure is rotated less than the final image can look the same as the initial one — as if the rotation did nothing to the preimage. In such a case, the figure is said to have rotational symmetry.

## Rotational Symmetry

A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation between and about the center of the figure. This point is called the center of symmetry. When a figure has rotational symmetry, it is said to be rotationally symmetric. As shown, a square, an equilateral triangle, and the digit all have rotational symmetry, each with a particular angle measure. The order of symmetry of a figure is the number of times it maps onto itself while rotating from to

## Identifying Rotational Symmetry

While walking downtown, Heichi and Paulina saw a store with the following logo. They began to discuss whether the logo has rotational symmetry. To figure it out, they went into the store and took a business card each. At their respective homes, and using patty paper, each performed rotations on the logo. The same night, Heichi called Paulina to tell her that the logo is rotationally symmetric, but Paulina disagreed. Who is correct?
In the event that the figure has rotational symmetry, what is the order of symmetry?

### Hint

Rotate the logo about its center. If possible, verify where along the way the rotation matches the original logo.

### Solution

To determine whether the logo has rotational symmetry, check for a rotation of less than about the center that maps the logo onto itself. By the use of patty paper or the following applet, rotational symmetry can be checked. As shown, a rotation of maps the logo onto itself. Therefore, the logo is rotationally symmetric, which means Heichi was right. Additionally, two other rotations of and also map the logo onto itself. This means that the order of symmetry is

## Identifying Rotational Symmetry in Parallelograms

In the first exploratory applet, it was studied whether a rectangle has rotational symmetry. Now, a parallelogram will be studied. Consider a parallelogram Does it have rotational symmetry?
If the answer is affirmative, consider the following questions.
a What is the center of rotation?
b What is the order of symmetry?
c What conclusions can be drawn about the sides and angles of the parallelogram?

Yes, the parallelogram has rotational symmetry.

a The point of intersection of the diagonals.
b
c The opposite sides of a parallelogram are congruent as well as the opposite angles.

### Hint

a How can the center of the parallelogram be found?
b Rotate the parallelogram around its center and count the number of times it matches the original one.
c When the parallelogram is rotated opposite vertices and sides are swapped. What does this mean in terms of the angle measures and side lengths?

### Solution

To determine whether the parallelogram has rotational symmetry, it needs to be checked if a rotation of less than about the center maps the parallelogram onto itself. In this case, the center of the parallelogram is the point of intersection of the diagonals. As shown, a rotation of about the intersection point of diagonals maps the parallelogram onto itself. Consequently, it is rotationally symmetric.
a The center of rotation is the point of intersection of the diagonals.
b Because rotations of and map the parallelogram onto itself, the order of symmetry is
c When is rotated about its center, the following events occur.
• The opposite vertices are swapped. That is, and are swapped as well as and
• The opposite sides are swapped. That is, and are swapped as well as and
The first point implies that the measures of and are equal and the measures of and as well. Therefore, in a parallelogram, the opposite angles are congruent angles.
Similarly, the second point implies that the lengths of and are equal and the lengths of and as well. Therefore, in a parallelogram, the opposite sides are congruent segments.
On the parallelogram, these congruences can be represented as follows. ## Properties of Parallelograms with Rotational Symmetry

The symmetries of a figure help determine the properties of that figure. For instance, since a parallelogram has rotational symmetry, its opposite sides and angles will match when rotated which allows for the establishment of the following property.

The opposite sides and angles of a parallelogram are congruent.

Symmetries can be used to characterize classes of figures. For example, all parallelograms have rotational symmetry. The same is true for rectangles, rhombi, and squares. In consequence, it can be said that rectangles, rhombi, and squares are special kinds of parallelograms. Upon taking a closer glance, notice that all squares have rotational symmetry, which implies squares belong to a particular class of parallelograms. Even more, this rotational symmetry implies that all squares have congruent sides and angles.

## Defining Line Symmetry

Is there another type of symmetry apart from the rotational symmetry? The answer is yes. Some figures can be folded along a certain line in such a way that all the sides and angles will lay on top of each other. In this case, it is said that the figure has line symmetry.

## Line Symmetry

A figure in the plane has line symmetry if the figure can be mapped onto itself by a reflection in a line. This line of reflection is called the line of symmetry. When a figure has line symmetry, it is said to be reflectionally symmetric or line symmetric. Some figures can have more than one line of symmetry.
Because of the symmetry definition, it seems logical to think that figures with congruent parts are symmetric. However, each figure needs to be studied separately before drawing conclusions.

## Line Symmetry in Parallelograms

After learning about line symmetry, Paulina and Heichi decided to study whether some quadrilaterals are line symmetric. To start, they picked a parallelogram. Without thinking too much, Paulina said it has no line symmetry. However, Heichi said that the parallelogram is symmetric along the line connecting the midpoints of and Who is correct?

### Hint

Try to find a line along which the parallelogram can be bent so that all the sides and angles are on top of each other. The lines containing the diagonals or the lines connecting the midpoints of opposite sides are always good options to start.

### Solution

To determine whether the parallelogram is line symmetric, it needs to be checked if there is a line such that when is reflected on it, the image lies on top of the preimage. Before start testing lines, mark the midpoints of each side. In the event that is line symmetric, the lines connecting the midpoints of opposite sides are good candidates of being lines of symmetry. Also, the lines containing the diagonals could work. All possible lines can be tried in the following applet. As can be seen, no reflection worked! In fact, there is no line along which the parallelogram can be reflected onto itself. This means that is not line symmetric. In conclusion, Paulina is right.

## Line Symmetry in Rectangles and Rhombi

Despite the previous example showing a parallelogram with no line symmetry, other types of parallelograms should be studied first before making a general conclusion. Consider a rectangle and a rhombus. Study whether or not they are line symmetric. If both polygons are line symmetric, compare their lines of symmetry.

Both rectangles and rhombi have line symmetry. Rectangles have lines of symmetry connecting midpoints of opposite sides, while rhombi have lines of symmetry along its diagonals.

### Hint

For each polygon, consider the lines along the diagonals and the lines connecting midpoints of opposite sides.

### Solution

When studying a polygon's line symmetries, it is always good to consider the lines along the diagonals and the lines connecting midpoints of opposite sides. For simplicity, consider one polygon at a time, starting with the rectangle. From the diagram, it can be seen that the rectangle is line symmetric. It has two lines of symmetries, each connecting the midpoints of opposite sides. Next, repeat the same steps with the rhombus. As can be seen, the rhombus is line symmetric and also has two lines of symmetries. In contrast to the rectangle, however, the lines of symmetries are the lines containing its diagonals.

## Line Symmetry in Rectangles and Rhombi

Before, it was said that rectangles, rhombi, and squares are special kinds of parallelograms because they all have rotational symmetry. In addition, rectangles and rhombi have line symmetry.

Polygon Line Symmetry
Rectangles Along the lines connecting midpoints of opposite sides
Rhombi Along the lines containing the diagonals
Consequently, based on the line symmetries, it can be concluded that rectangles and rhombi are not directly related. Therefore, they belong to different parallelogram classes. The next natural question is, what about the squares?

## Line Symmetry in Squares

Does a square have line symmetry? If so, how many different lines of symmetry does it have?

Yes, squares are line symmetric. They have four lines of symmetries — two of them connect the midpoints of opposite sides and the other two contain its diagonals.

### Hint

The lines containing the diagonals or the lines connecting the midpoints of opposite sides are always good options.

### Solution

To determine whether the square is line symmetric, it is always good to consider the lines along the diagonals and the lines connecting midpoints of opposite sides. All possible lines can be tried in the following applet. Give it a go! As might be verified, the square is indeed line symmetric. It has four lines of symmetry — two connect the midpoints of opposite sides and two others contain its diagonals.

## Summarizing Line Symmetry in Parallelograms

The line symmetries of a square confirm the claim made previously that squares belong to a particular class of parallelograms.

Polygon Number of Line Symmetries Line Symmetry
Rectangles Along the lines connecting midpoints of opposite sides
Rhombi Along the lines containing the diagonals
Squares Two along the lines connecting midpoints of opposite sides and two along the lines containing the diagonals

Notice that two symmetries of the square correspond to the rectangle's symmetries and the other two correspond to the rhombus symmetries. This suggests that squares are a particular case of rectangles and rhombi. ## Line Symmetry in a Trapezoid

Paulina and Heichi have the task of investigating whether trapezoids have some symmetry. The teacher gave them the diagram below, but she told them to try any trapezoid they could imagine. What conclusion should Paulina and Heichi reach?

Trapezoids have no rotational symmetry and have line symmetry only when the trapezoid is isosceles. In this case, the line of symmetry is the line passing through the midpoints of the bases.

### Hint

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Then, parallelograms are not trapezoids. Do not forget to try with isosceles trapezoids.

### Solution

Consider any trapezoid where and are parallel. First, start investigating whether has rotational symmetry. To figure it out, set different dimensions for the trapezoid using the following applet. But remember, parallelograms are not trapezoids. As investigated, regardless of the dimensions, no trapezoid has rotational symmetry. Next, repeat the same investigation but now checking for line symmetry. From this last investigation, it may be seen that some trapezoids have line symmetry. Going a bit deeper, it can be concluded that these trapezoids are isosceles trapezoids.

A trapezoid has line symmetry only when it is isosceles trapezoid.

In this case, the line of symmetry is the line passing through the midpoints of each base.

## Symmetry in Three-Dimensional Figures

Symmetries are not defined only for two-dimensional figures. The definition can also be extended to three-dimensional figures. In the real world, there are plenty of three-dimensional figures that have some symmetry. For example, sunflowers are rotationally symmetric while butterflies are line symmetric. External credits: Alejandro Santillana
A three-dimensional figure is said to have axis symmetry if it can be rotated about a line onto itself by a rotation between and This is the equivalent of rotational symmetry. A three-dimensional figure has plane symmetry if there is a plane that divides the figure into two halves and each half is a reflection of the other across the plane. This is the equivalent of line symmetry. {{ subexercise.title }}