The hierarchy of quadrilaterals
In plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides"). Simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral add up to 360 degrees. We will elaborate more on simple quadrilaterals. Any quadrilateral will tile the plane by repeated rotation of 180 ° around the midpoints of its edges. The quadrilaterals that we will focus on are squares, rectangles, rhombi, isosceles trapezoids (trapezium), parallelograms and kites. These are all standard conventional terms that will be related to symmetry. In our discussion below, we assume that the reader knows what all these types of quadrilaterals are, and we will emphasize the symmetry aspects of the various types.
A square has all the characteristics of the rhombus and of the rectangle. It has four lines of symmetry, which results in eight types of symmetry. That is, four reflective symmetries and a 4-fold rotational symmetry.
Where:
R0 = Rotation at 0° which means do nothing (also known as Identity),
R1 = Rotation at 90°, R2 = Rotation at 180°, R3 = Rotation at 270°
M1 =Reflection along the given line
M2 = Reflection along the given line
D1 = Reflection along the given diagonal
D2 = Reflection along the given diagonal
A rectangle is a special type of parallelogram, whose angles are all right angles. It has two lines of reflection symmetry and rotational symmetry through 180°, as well as the identity, for a total of four symmetries.
A rhombus is another special type of parallelogram. By definition, a quadrilateral is a rhombus if and only if all sides are congruent.
Thus a rhombus is an equilateral quadrilateral. The diagonals of a rhombus are perpendicular bisectors of each other. A rhombus has 2 lines of symmetry (namely, the two diagonals), which also results in four symmetries. That is, two reflective and two rotational symmetries.
In geometry, a trapezoid (trapezium) is a four-sided figure with one pair of parallel sides. We will take the point of view that a trapezoid is not a parallelogram because only one pair of sides is parallel, figure 1.
In an isosceles trapezoid (trapezium), the base angles have the same measure, and the other pair of opposite sides also has the same length. Hence you get a figure that looks like figure 2. It is called an isosceles trapezoid (trapezium) if the sides that are not parallel are equal in length. Then both angles coming from a parallel side are equal also, as shown. An alternative definition is: "a quadrilateral with an axis of symmetry bisecting one pair of opposite sides". This means that an isosceles trapezoid (trapezium) has only two symmetries, the identity and a reflection in the perpendicular bisector of the two parallel sides.
A Kite has two pairs of adjacent sides of equal length. This implies that one diagonal divides the kite into congruent triangles, by the SSS congruence condition. We call this diagonal the bisecting diagonal.
It has two pairs of sides, one pair on either side of the bisecting diagonal. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, the bisecting diagonal bisects (cuts equally in half) the other. In fact, the bisecting diagonal is a line of symmetry of the kite. With one line of symmetry, it means a kite also has two symmetries, reflection across the bisecting diagonal, and the identity.
Lastly, we look at the parallelogram. The typical parallelogram has no reflection symmetries. However, if you rotate it 180 o around the midpoint of one of the diagonals, the two vertices that are end points of the diagonal will be interchanged. Since any line is sent to a parallel line by rotation of 180 o, the two sides intersecting at one vertex will be taken to the opposite parallel sides intersecting at the other vertex. Therefore, the whole parallelogram is preserved, with pairs of opposite sides (and therefore, both pairs of opposite vertices) being interchanged. (This also shows that the center of rotation is also the midpoint of the other diagonal, and therefore, the diagonals of the parallelogram bisect each other in the center of rotation of the parallelogram.) Thus a (typical) parallelogram has two symmetries, rotation by 180 o, and the identity. Running this argument in reverse shows that a quadrilateral that is invariant by a rotation of 180 o must be a parallelogram.
Generally, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, a square can be rotated about 4 times and it will still look the same in all four rotations and hence it is said that a square has rotational symmetry of order 4. The parallelogram has a rotational symmetry of order 2. Also, if a shape only matches itself once as you go around (i.e. it matches itself after one full rotation) there is really no symmetry at all. This is also what is known as "do nothing" (identity).
With all these a really great hierarchy of quadrilaterals can be developed, based on their symmetry properties. The chart below shows the various quadrilaterals grouped according to the number of symmetries they have. All the arrows point to the square because the square has the most symmetries (eight in all). This is followed by the rhombus and the rectangle with four symmetries. The kite, parallelogram and isosceles trapezoid (trapezium) follow with 2 symmetries. (The kite really should be on the same line with the other two.) The arrows are seen originating from the quadrilateral because all the shapes are examples of quadrilaterals.
This will be a great tool in helping your students understand the quadrilaterals a lot better. Some of my struggling students are very happy when I am able to summarize my lessons. They say it helps them "tie it all together". To make sure that, summarizing indeed really works, I quizzed them on four lessons, one summarized and two not summarized. I realized that they did extremely well on the summarized lesson and not so well on the lessons that were not summarized.
Symmetry Hierarchy of quadrilaterals
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