Drag the triangles and move the sliders to explore how \triangle ABC and \triangle DEF change.

What do you notice about the corresponding parts of the triangles?

Is it possible to create triangles such that the ratios of their corresponding parts are not proportional?

In the previous lesson, we verified that similar figures will have corresponding sides that are proportional and corresponding angles that are congruent. We can build on this to formalize similarity theorems for triangles. Two triangles can be shown to be similar using several different theorems involving their angles and sides.

Explain why similarity theorems work using transformations.

a

Why can triangles be proven similar with only two corresponding congruent angles, and not all 3?

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b

Why can triangles be proven similar using only their side lengths?

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State whether Rectangle P\sim Rectangle Q. If so, describe the similarity transformation.

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Determine whether the given pairs of triangles are similar. If so, state the theorem which proves their similarity. If not, explain how you know.

a

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b

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c

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d

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Determine whether the pair of triangles are similar. If so, write a similarity statement and justify with a similarity theorem. If not, explain why not.

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Find x. Show your work and justify your steps.

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Idea summary

Use the theorems about the angles and side lengths of triangles to prove their similarity:

- Angle-angle similarity, or AA \sim: The two triangles have two pairs of congruent angles
- Side-side-side similarity, or SSS \sim: The two triangles have three pairs of sides whose lengths are in the same proportion
- Side-angle-side similarity, or SAS \sim: The two triangles have two pairs of sides whose lengths are in the same proportion, and the angles between these sides are also congruent