In this lesson, students create composite shapes using translations, rotations, and reflections of polygons and continue to observe that the side lengths and angle measures do not change. They use this understanding to draw conclusions about the composite shapes. Later, they will use these skills to construct informal arguments, for example about the sum of the angles in a triangle.
When students rotate around a vertex or reflect across the side of a figure, it is easy to lose track of the center of rotation or line of reflection since they are already part of the figure. It can also be challenging to name corresponding points, segments, and angles when a figure and its transformation share a side. Students attend to these details carefully in this lesson (MP6).
Consider using the optional activity if you need to reinforce students’ belief that rigid transformations preserve distances and angle measures after main activities.
- Draw and label images of triangles under rigid transformations and then describe (orally and in writing) properties of the composite figure created by the images.
- Generalize that lengths and angle measures are preserved under any rigid transformation.
- Identify side lengths and angles that have equivalent measurements in composite shapes and explain (orally and in writing) why they are equivalent.
Let’s use reasoning about rigid transformations to find measurements without measuring.
- I can find missing side lengths or angle measures using properties of rigid transformations.
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations, as is any sequence of these.
Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.
For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).
Angles \(AED\) and \(BEC\) are another pair of vertical angles.