In the previous lesson, students found that, in order to check if two quadrilaterals are similar, it is important, in general, to check that corresponding angles are congruent and that corresponding side lengths are proportional. This lesson focuses on triangles. Triangles are special since it is possible to determine whether or not they are similar by looking only at the angles. If two triangles share three corresponding angle measurements, then they are similar. In fact, since the sum of the angle measures in a triangle is 180 degrees, two angle measures determine the third. Hence for triangles, all that is needed to deduce similarity is having two corresponding angles with equal measure.
Students deduce the criteria for similarity in terms of angle measures by experimenting with triangles built out of pasta. As a result, they will need to make sense of measurements and account for possible inaccuracies.
Students will use the similarity criterion in future lessons to understand the concept of the slope of a line. Later on in high school, they will learn that three proportional sides (but not two) is also enough to deduce that two triangles are similar.
- Generalize a process for identifying similar triangles and justify (orally) that finding two pairs of congruent angles is sufficient to show similarity.
- Justify (orally) that two triangles are similar by finding a sequence of transformations that takes one triangle to the other or checking that two pairs of corresponding angles are congruent.
Let’s look at similar triangles.
Make 1 copy of the blackline master for every 4 students. Cut these into strips horizontally. Each student will receive one strip, which is a set of three angles labeled A, B, and C.
For the dried pasta that will be used to create the sides of the triangles, we recommend fettuccine or linguine so it doesn’t roll off the table and is easy to break as needed.
- I know how to decide if two triangles are similar just by looking at their angle measures.
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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