Unit 1 Rigid Transformations and Congruence (Family Materials)

Section A Rigid Transformations

This week your student will learn to describe the movement of two-dimensional shapes with precision. Here are examples of a few of the types of movements they will investigate. In each image, Shape A is the original and Shapes B, C, and D show three different types of movement:

Two figures, one labelled A, and its translation, labelled B.

Two figures, one labelled A, and its rotation, labelled C.

Two figures, one labelled A, and its reflection, labelled D

Students will also experiment with shapes and drawings to build their intuition by:

cutting shapes out
tracing shapes on tracing paper to compare with other shapes
drawing shapes on grid paper
measuring lengths and angles
folding paper

Here is a task to try with your student:

1. Describe how the shape changes from one panel to the next.

2. Draw a fourth panel that shows what the image would look like if the shape in the third panel is rotated 180 degrees counterclockwise around the middle of the panel.

Solution:

Turn it 90 degrees clockwise then move the shape to the right side.

Section B Properties of Rigid Transformations

This week your student will investigate rigid transformations, which is the name for moves (and sequences of moves) that preserve length and angle measures like translations, rotations, and reflections. For example, in this image the triangle $A B C$ was reflected across the line $A C$ and then translated to the right and up slightly.

Triangle A, B, C and its image after reflection and translation.

When we construct figures using rigid transformations, we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles.

Here is a task to try with your student:

Reflect triangle $A B C$ across side $A C$ to form a new triangle $A B^{'} C$ .
What is the measure of angle $B^{'} A C$ ?
Name two side lengths that have the same measure.

A triangle A, B, C where the interior angle at A has measure 36 degrees.

Solution:

Triangle A, B, C, with angle with measure 36 degrees at A. It has been reflected in the side A, B.

2. 36 degrees. Angle $B^{'} A C$ corresponds to angle $B A C$ .

3. Sides $A B^{'}$ and $A B$ have the same length as do sides $B^{'} C$ and $B C$ .

Section C Congruence

This week your student will learn what it means for two figures to be congruent. Let’s define congruence by first looking at two figures that are not congruent, like the two shown here. What do these figures have in common? What is different about them?

Two polygons that look similar but have different side lengths.

If two figures are congruent, that means there is a sequence of rigid transformations we could describe that would make one of the figures look like the other. Here, that isn’t possible. While each has 6 sides and 6 vertices and we can make a list of corresponding angles at the vertices, these figures are not considered congruent because their side lengths do not correspond. The figure on the left has side lengths 3, 2, 1, 1, 2, 1. The figure on the right has side lengths 3, 3, 1, 2, 2, 1.

For the last part of this unit, students will use the congruence to investigate some interesting facts about parallel lines and about the angles in a triangle.

Here is a task to try with your student:

1. Explain why these two ovals are not congruent. Each grid square is 1 unit along a side.

2. Draw two new ovals congruent to the ones in the image.

Solution:

While each oval has a horizontal measurement of 5 units and a vertical measurement of 4 units, the oval on the left’s “tallest” measurement is halfway between the left and right sides while the oval on the right’s “tallest” measurement is closer to the right side than the left side.
There are many possible ways to draw new ovals congruent to the original two. If a tracing of the original oval lines up exactly when placed on top of the new image (possibly after some rotation or flipping of the paper the tracing is on), then the two figures are congruent.

Section E Drawing Polygons with Given Conditions

This week your student will be drawing shapes based on a description. What options do we have if we need to draw a triangle, but we only know some of its side lengths and angle measures?

Sometimes we can draw more than one kind of triangle with the given information. For example, “sides measuring 5 units and 6 units, and an angle measuring $32 °$ ” could describe two triangles that are not identical copies of each other.

Two triangles each with two given side lengths labeled 5 and 6, and an angle labeled 32 degrees. For the triangle on the left, the angle labeled 32 degrees is between the adjacent side lengths 5 and 6. The triangle on the right has the angle labeled 32 degrees between the side length labeled 5 and the third side of the triangle that is not labeled.

Sometimes there is only one unique triangle based on the description. For example, here are two identical copies of a triangle with two sides of length 3 units and an angle measuring $60 °$ . There is no way to draw a different triangle (a triangle that is not an identical copy) with this description.

Sometimes it is not possible to draw a triangle with the given information. For example, there is no triangle with sides measuring 4 inches, 5 inches, and 12 inches. (Try to draw it and see for yourself!)

Here is a task to try with your student:

Using each set of conditions, can you draw a triangle that is not an identical copy of the one shown?

1. A triangle with sides that measure 4, 6, and 9 units.

2. A triangle with a side that measures 6 units and angles that measure $45 °$ and $90 °$ .

Solution:

1. There is no way to draw a different triangle with these side lengths. Every possibility is an identical copy of the given triangle. (You could cut out one of the triangles and match it up exactly to the other.) Here are some examples:

2. You can draw a different triangle by putting the side that is 6 opposite from the $90 °$ angle instead of next to it. This is not an identical copy of the given triangle, because it is smaller.