[G.SRT.3] Similarity, Right Triangles, and Trigonometry #3

Now, we know A' and C' will be the same angle as A and C because dilations don't change angles. Also, we know that A'C' is the same length as XZ because that's how we said we were going to dilate it.

Now if we compare A'B'C' and XYZ, we see they are congruent by SAS. Hmmmm. Well, for two shapes to be congruent, that means one can be moved onto the other by rigid motions

Soooo. We did a dilation, and then rigid motions will do the rest. But wait, what's it called when you do dilations and rigid motions to move shapes onto each other? That was our definition of similarity transformations! We've shown that if two triangles have two congruent angles, they can be moved onto each other through similarity transformations.

We're almost there. Now, how do similarity transformations change a shape? They keep the corresponding sides proportional and corresponding angles congruent. In other words, similarity transformations keep two shapes similar. And we've done it.

ΔABC ~ ΔXYZ

We've proven that two triangles are similar even if all they share is two congruent angles.

Let's do this in a 2-column proof

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In the figure above, ∠A ≅ ∠X and ∠C ≅ ∠Z. Prove that ΔABC ~ ΔXYZ

As you can see, all we know about these two triangles is that:

∠A ≅ ∠X
∠C ≅ ∠Z

From this, we need to prove they're similar. The trick is to dilate ΔABC so that side AC is the same length as side XZ. We chose this side because it is included between the congruent angles. You'll see why in a bit.

What would we use for our scale factor if we want to make A'C' the same length as XZ? it would be

k = XZ

because watch what happens when I dilate AC by this number

A'C' = k x AB = XZ x AB = XZ

Dilating, we get

Objective

Common Core Text:

- [G.SRT.3] Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar

Said Differently:

- Show why AA (Angle-Angle) guarantees similarity in triangles in terms of similarity transformations

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Explanation

When working with triangles, we found that we only needed 3 parts to determine if the triangles were congruent (ASA, SAS, or SSS)

For similarity, we only need 2 parts! Angle-Angle (AA) is enough to guarantee similarity. Let's prove it

Proof

Lets start with this picture

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