Similarity

As with the last section, there are just a few key facts to know. And, just like last section, we want you to know these facts absolutely cold, so we’re going to add them to your study deck immediately after you read about them.

If two figures are similar, then…

…their corresponding angles are congruent
…their corresponding lengths are proportional
…their corresponding areas (or surface areas) are proportional, per the square of their scale factor
…their corresponding volumes (if applicable) are proportional, per the cube of their scale factor

Note: most people know the first two of these. Far fewer know the last two. Therefore the AMC implicitly weights knowledge of the last two more heavily.

Also,

Alternate interior angles are congruent
If you get stuck while chasing angles, you should look for parallel lines cut by transversals

Examples

Example 1

I have two (similar) balloons. The first has six times as much air in it as the second. What’s the ratio of their radiuses?

(spoiler)

The correct answer is $36 : 1$ . (Did I trick you into saying $2 : 1$ though?)

Example 2

$△ A BC$ and $△ D EF$ are similar.

Their scale factor is $3 : 1$

$m ∠ A = \frac{π}{5}$

What is $m ∠ D$ ?

(spoiler)

$m ∠ D = \frac{π}{5}$ .

The two triangles have the same shape, and therefore their corresponding angles are the same.