One of the powerful things about the definition of similarity in terms of transformations is that we can talk about whether two figures are similar even when they are not composed of straight lines. For example, we can show that all circles are similar, because we can translate one so they have the same center and then dilate one until it matches the other.
In the case of polygons, we can understand similarity by examining side lengths and angle measures. Since the transformations we have studied (translations, rotations, reflections, dilations) do not change angle measures, similar polygons have congruent corresponding angles. Only dilations change side lengths and they change them all by the same scale factor. This means that similar polygons have proportional corresponding side lengths. In general, both side lengths and angle measures are important to determine whether or not two polygons are similar. The next lesson will examine the special case of triangles where it turns out that congruent corresponding angles is all that is needed to conclude that two triangles are similar.
The focus in this lesson is on quadrilaterals, and students determine efficient ways to decide whether certain types of quadrilaterals are similar:
- Two rectangles are similar if the side lengths are proportional.
- Two rhombuses are similar if the angles are congruent.
- All squares are similar.
- Comprehend the phrase “similar polygons” (in written and spoken language) to mean the polygons have congruent corresponding angles and proportional side lengths.
- Critique (orally) arguments that claim two polygons are similar.
- Justify (orally) the similarity of two polygons given their angle measures and side lengths.
Let’s look at sides and angles of similar polygons.
Make 1 copy of the blackline master for every 10 students, and cut them up ahead of time.
- I can use angle measures and side lengths to conclude that two polygons are not similar.
- I know the relationship between angle measures and side lengths in similar polygons.
Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.
In this figure, triangle \(ABC\) is similar to triangle \(DEF\).
If \(ABC\) is rotated around point \(B\) and then dilated with center point \(O\), then it will fit exactly over \(DEF\). This means that they are similar.
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