Lesson 5 Symmetries of Quadrilaterals Develop Understanding
Jump Start
Which One Doesn’t Belong?
A square
A rhombus
A rectangle
An equilateral triangle
Learning Focus
Identify transformations that carry an image onto itself.
What does it mean to say that a figure is symmetrical?
How is symmetry related to rigid transformations?
Open Up the Math: Launch, Explore, Discuss
A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation has rotational symmetry.
Every four-sided polygon is a quadrilateral. Some quadrilaterals have additional properties and are given special names like squares, parallelograms, and rhombuses. A diagonal of a quadrilateral is formed when opposite vertices are connected by a line segment. Some quadrilaterals are symmetric about their diagonals. Some are symmetric about other lines. In this task you will use rigid-motion transformations to explore line symmetry and rotational symmetry in various types of quadrilaterals.
For each of the following quadrilaterals you are going to try to answer the question, “Is it possible to reflect or rotate this quadrilateral onto itself?” As you experiment with each quadrilateral, record your findings in the following chart. Be as specific as possible with your descriptions.
1.
Defining features of the quadrilateral | Lines of symmetry that reflect the quadrilateral onto itself | Center and angles of rotation that carry the quadrilateral onto itself |
|---|---|---|
A rectangle is a quadrilateral that contains four right angles. | ||
A parallelogram is a quadrilateral in which opposite sides are parallel. | ||
A rhombus is a quadrilateral in which all sides are congruent. | ||
A square is both a rectangle and a rhombus. |
Pause and Reflect
Consider the set of quadrilaterals that are trapezoids, with exactly one pair of opposite sides parallel. Is it possible to reflect or rotate such a trapezoid onto itself?
Draw a trapezoid with exactly one pair of parallel sides. Then see if you can find:
any lines of symmetry, or
any centers of rotational symmetry,
that will carry the trapezoid you drew onto itself.
If you were unable to find a line of symmetry or a center of rotational symmetry for your trapezoid, see if you can sketch a different trapezoid with exactly one pair of parallel sides that might possess some type of symmetry.
Ready for More?
Can you find other polygons that have rotational or line symmetry? For example:
1.
Can you draw a quadrilateral that has only one line of symmetry?
2.
Can you draw a polygon that has three lines of symmetry?
3.
Can you draw a polygon that has rotational symmetry of
Takeaways
We have found that many different quadrilaterals possess lines of symmetry and/or rotational symmetry. In the following chart, write the names of the quadrilaterals that are being described in terms of their symmetries.
What do you notice about the relationships between quadrilaterals based on their symmetries and highlighted in the structure of the above chart?
Adding Notation, Vocabulary, and Conventions
We use the notation to represent the line segment with endpoints at point
If two line segments, and , are the same length, they can be moved using translations, reflections, and rotations until their endpoints match. We say that the two line segments are congruent, and we write this symbolically as . However, we use the equal sign to show that the line segments are the same length, , since lengths are numbers rather than geometric objects.
Angles are represented by , using a single letter, the vertex of the angle; or by , using three letters, including the vertex, if there might be more than one angle with the same vertex. The measure of the angle is referred to by the symbol .
Vocabulary
- congruent (CPCTC)
- diagonal
- equilateral, equilateral triangle
- hexagon
- line of symmetry
- midpoint
- octagon
- opposite angles, opposite vertices
- opposite sides (in a parallelogram or an even-sided polygon)
- parallelogram
- pentagon
- polygon
- quadrilaterals: types
- rectangle
- regular polygon
- rhombus
- rotational symmetry
- square
- symmetry
- trapezoid
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we explored line and rotational symmetry in different types of quadrilaterals. A figure is symmetric if a figure can be reflected across a line or rotated about a point onto itself. We found that diagonals and lines connecting the midpoints of opposite sides of a quadrilateral might be lines of symmetry, depending on the quadrilateral, and the point of intersection of the diagonals is the center of rotation for parallelograms, rectangles, rhombuses, and squares. The possible angles of rotation vary depending on the quadrilateral, but are always multiples of
Give the name of a geometric figure that fits the following characteristics:
1.
A polygon with five sides
2.
A polygon with six sides
3.
A polygon with eight congruent sides
Given the following table, write equations of the lines described.
4.
The line that contains all of the points in the table
5.
The line that is parallel to the line given by the table, but has a
6.
The line that is perpendicular to the line given by the table, that has a