Common Core Standards: Math
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Students should already be familiar with transformations, at least to the extent that they know that it takes more than an, "Abracadabra!" to change or move a shape. It helps if they know what translation, reflection, rotation, and dilation are, though. Magic tricks are fun and all, but not exactly useful when determining similarity.
Students should already know that if they can perform any rigid transformation to carry one shape exactly onto another, the two shapes are congruent. If we throw dilation into the mix (whether it's contraction or expansion), we can safely say that the two shapes are similar.
While those basic rules apply to pretty much any two-dimensional shape you can imagine (although it might be a pain to verify that two dodecagons are similar), triangles get special treatment. In fact, triangles are like the royal family of geometry...only without the corgis.
Because they're so small and easy, we can use triangles to really get to the heart of what similarity is. Let's say we have two triangles, ΔABC and ΔPQR, that are congruent to each other.
Then, we can contract ΔABC so that it's half the size of ΔPQR. (That's a scale factor of 0.5. Hint, hint.) Dilation is still a similarity transformation, so even though they aren't congruent anymore, the two triangles are still similar.
Students should realize that even though the side lengths of ΔABC have changed, its angles have not. They might say something like, "Hey, those two triangles have similar-looking angle measures. They're just not the same size." If you hear something like that, it's a good sign.
This means two triangles are similar when all their corresponding angles are equal. In fact, students can take this to the next level. Whenever any two polygons of differing sizes have the same set of angle measures within them, the polygons are similar.
So that's angles. What about side lengths? Our scale factor tells us how we relate side lengths of similar triangles to each other. Ratios of corresponding side lengths should always give the same scale factor. If two shapes are similar, these ratios of corresponding side lengths will all equal the same number: the scale factor.
A good way to make sure students have cemented all this knowledge is to have the students measure the sides of similar shapes with a ruler and the angles with a protractor. Then, give another set of shapes that are almost similar. Students can then realize that the measures of the angles in the figure affect the scale ratio they calculate using the side lengths.