Lesson 9

Isosceles triangles are triangles with (at least) one pair of congruent sides. Isosceles triangles also have (at least) one pair of congruent angles. In this warm-up, students show why this is the case using rigid motions by exploiting the fact that rigid motions of the plane do not change angle measures (MP7).

Give students 3 minutes of quiet work time followed by whole-class discussion.

Select students to share their images and conclusions about the measures of angles B and C.

Time permitting, mention that it is also true that when a triangle has two angles with the same measure then the sides opposite those angles have the same length (i.e., the triangle is isosceles). This can also be shown with rigid transformations. Reflect the triangle first and then line up the sides containing the pairs of angles with the same measure.

The purpose of this task is to use rigid transformations to describe an important picture that students have seen in grade 6 when they developed the formula for the area of a triangle. They first found the area of a parallelogram to be \(\text{base} \boldcdot \text{height}\) and then, to find \(\frac{1}{2}\text{base} \boldcdot \text{height}\) for the area of a triangle, they “composed” two copies of a triangle to make a parallelogram. The language “compose” is a grade 6 appropriate way of talking about a \(180^\circ\) rotation. The focus of this activity is on developing this precise language to describe a familiar geometric situation.

Students need to remember and 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).

Arrange students in groups of 2. Provide access to geometry toolkits. A few minutes of quiet work time, followed by sharing with a partner and a whole-class discussion.

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

Begin the discussion by asking, "What happens to points \(A\) and \(C\) under the rotation?" (They end up at \(C\) and \(A\), 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 \(ABCD\) are parallel?" (They are taken to one another by a 180 degree rotation.) As seen in the previous lesson, the image of a \(180^\circ\) rotation of a line \(\ell\) is parallel to \(\ell\). Students also saw that when \(180^\circ\) rotations were applied to a pair of parallel lines it resulting in a (sometimes) new pair of parallel lines which are also parallel to the original lines. The logic here is the same, except that only one line is being rotated \(180^\circ\) 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 \(ABCD\) is related to the area of triangle \(ABC\)?" (The area of the parallelogram \(ABCD\) is twice the area of triangle \(ABC\) because it is made up of \(ABC\) and \(CDA\) which has the same area as \(ABC\).) Later in this unit, area of shapes and their images under rigid transformations will be studied further.

This activity continues the previous one, building a more complex shape this time by adding an additional copy of the original triangle. The three triangle picture in the task statement will be important later in this unit when students show that the sum of the three angles in a triangle is \(180^\circ\). To this end, encourage students to notice that the points \(E\), \(A\), and \(D\) all lie on a line.

As with many of the lessons applying transformations to build shapes, students are constantly using their structural properties (MP7) to make conclusions about their shapes. Specifically, that rigid transformations preserve angles and side lengths.

Keep students in the same groups. Provide access to geometry toolkits. Allow for a few minutes of quiet work time, followed by sharing with a partner and a whole-class discussion.

Students may have trouble understanding which pairs of points correspond in the first two questions, particularly the fact that point \(A\) in one triangle may not correspond to point \(A\) in another. Use tracing paper to create a transparency of triangle \(ABC\), with its points labeled, and let students perform their rigid transformation. They should see \(A\), \(B\), and \(C\) on top of points in the new triangle.

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. Students may wonder why there are fewer pairs of line segments: this is because of shared sides \(AB\) and \(AC\). If they don’t ask, there’s no reason to bring it up.

If you create a visual display of these pairs, hang on to the information about angles that have the same measure. The same diagram appears later in this unit and is used for a proof about the sum of the angle measures in a triangle.

After this activity, ask students to summarize their understanding about lengths and angle measures under rigid transformations. If students don’t say it outright, you should: "Under any rigid transformation, lengths and angle measures are preserved."

This activity builds upon the ideas of the previous one. This time a pattern is built from a single triangle via reflections and the goal is to study the angles and side lengths in this pattern as it grows. If they build the pattern carefully, students may notice after putting together 6 triangles, like in the previous activity, that two of the sides of the triangles (one side of the original and one side of the 6th) lie on the same line. The 6 triangle pattern can be reflected over this line to make it “complete” with 12 copies of the original triangle. Alternatively, students may notice the right angle made by 3 triangles and reason that they can complete a circle with 4 right angles. Both of these arguments are good examples of MP8. Rather than repeating the reflecting procedure 12 times, it is possible to use the structure of what they have learned along the way to accurately predict how many copies make a circle.

Provide access to geometry toolkits. Allow for 8 minutes of work time, then a brief whole-class discussion.

If students are stuck with the first reflection, suggest that they use tracing paper. If you need to, 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 length of \(OT\), since it uses the initial information that triangle \(ONE\) is isosceles (otherwise, \(OT\) is unlabeled). Ask students what other information is given and if they can use it to figure out the missing length.

The main goal of this activity is to apply and reinforce students’ belief that rigid transformations preserve distances and angle measures. Watch and make sure students are doing well with this. If they are not, reinforce the concept. It is critical that this is understood by students before moving forward.

Identify 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 (though students do study this in the warm-up). Later in this unit, students will learn and prove that the sum of the angle measures in a triangle is 180 degrees.