Lesson 12-5 of the U of Chicago text is about similar figures. There is not much for us to change about this lesson from last year, except for the definition of

*similar*itself. Recall the two definitions:

-- Two polygons are

*similar*if corresponding angles are congruent and sides are proportional.

-- Two figures are

*similar*if there exists a similarity transformation mapping one to the other.

The first definition is pre-Core, while the second is Common Core. The U of Chicago text, of course, uses the second definition. But that PARCC question I mentioned last week must be using the first definition, since it requires that we know what

*similar*means before we can define dilations and ultimately similarity transformations.

In the U of Chicago text, the Similar Figures Theorem is essentially the statement that the second definition implies the first definition. We would actually need to prove the converse -- that the first definition (at least for polygons) implies the second. But the proof isn't that much different -- suppose we have two figures

*F*and

*G*satisfying the first definition of similar -- that is, corresponding angles are congruent and sides are proportional, say with scale factor

*k*. Then use

*any*dilation with scale factor

*k*to map

*F*to its image

*F'*. Now

*F'*and

*G*have all corresponding parts congruent, so there must exist some isometry mapping

*F'*to

*G*. Therefore the composite of a dilation and an isometry -- that is, a similarity transformation -- maps

*F*to

*G*. QED

So I don't change the worksheet from last year.

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