The purpose of this task is to use rigid transformations to explore the properties of a parallelogram. In an earlier course, students composed and rearranged two copies of a triangle to form a parallelogram. In this activity, students use the more precise language describing 180-degree rotations to describe this figure. 

Students must also use an important property of 180-degree rotations, namely that the image of a line after a 180-degree rotation is parallel to that line. This is what allows them to conclude that the shape they have built is a parallelogram (MP7).

Students may struggle to see the 180-degree rotation using center . This may be because they do not understand that is the center of rotation or because they struggle with visualizing a 180-degree rotation. Offer these students patty paper, a transparency, or the rotation overlay from earlier in this unit to help them see the rotated triangle.

The purpose of this discussion is for students to connect properties of 180-degree rotations with features of a parallelogram. Ask students, “What happens to points and under the rotation?” (They end up at and , respectively.) This type of rotation and analysis will happen several times in upcoming lessons.

Next ask, “How do you know that the lines containing opposite sides of are parallel?” (They are taken to one another by a 180-degree rotation.) As seen previously, the image of a 180-degree rotation of a line is parallel to . Students also saw that when 180-degree rotations were applied to a pair of parallel lines it resulted in a (sometimes) new pair of parallel lines which were also parallel to the original lines. The logic here is the same, except that only one line is being rotated rather than a pair of lines. This does not need to be mentioned unless it is brought up by students.

Finally, ask students “How is the area of parallelogram  related to the area of triangle ?” (The area of the parallelogram is twice the area of triangle because it is made up of and , which has the same area as .) 

The purpose of this activity is for students to identify rigid transformations in a figure composed of multiple congruent triangles, then use properties of rigid transformations to identify corresponding line segments and angles. This particular figure will be important later in this unit when students show that the sum of the three angles in a triangle is , so it is particularly important that students recognize corresponding angles with the same measure.

Students draw conclusions about the angle measures and segment lengths by applying the structure of rigid transformations to the composed figure, particularly the property that rigid transformations preserve angles and side lengths (MP7).

Ask students to list as many different pairs of matching line segments as they can find. Then, do the same for angles. Record these for all to see. Consider displaying the image from the student task and highlighting or coloring corresponding sides and angle measures. Students may wonder why there are fewer pairs of line segments: this is because of shared sides and . If they don’t ask, there’s no reason to bring it up.

Display this statement for all to see: “Under any rigid transformation, lengths and angle measures are preserved.” Invite students to restate this statement using their own words and share aloud with the whole class or with a partner.

This activity is optional because it provides additional practice for performing rigid transformations on a triangle to compose a new figure. Students may notice different properties of the composed figure, including that two of the sides of the triangles (one side of the original and one side of the 6th) lie on the same line. Students may also notice the right angle made by 3 triangles and reason that they can complete a circle with 4 right angles, or they may notice that the 6 triangle pattern can be reflected over this line to make it “complete” with 12 copies of the original triangle.

If students are stuck with the first reflection, suggest that they use tracing paper. If needed, show them the first reflected triangle, then have them continue to answer the problems and do the next reflection on their own.

Some students may have difficulty with the unlabeled length of , since it uses the initial information that triangle is isosceles. Ask students what other information is given and if they can use it to figure out the missing length.

The purpose of this discussion is to apply and reinforce students’ belief that rigid transformations preserve distances and angle measures.

Invite students to share how they know the angle measures and side lengths of each part of their completed figure. Highlight student responses that use reasoning, such as the following:

• Angle measures are the same after applying a rigid transformation.
• The image of a segment after a reflection is the same length as the original.
• Since each triangle is made using a sequence of rigid transformations from triangle , the central angle of each triangle in the figure must be the same, which is 30 degrees.

Select a few students, each with a different response, to share their description of the pattern they saw in the last question. The way in which each student visualizes and explains this shape may give insight into the different strategies used to create the final pattern.

Note: It is not important nor required that students know or understand how to find the base angle measures of the isosceles triangles, or even that the base angles have the same measure. Later in this unit, students will show that the sum of the angle measures in a triangle is 180 degrees.