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# Similar Triangles

## What are similar triangles?

Answer: Similar triangles have the same 'shape' but are just scaled differently. Similar triangles have congruent angles and proportional sides.

## What is true about the angles of similar triangles?

Answer: They are congruent. as the picture below demonstrates.

## What is true about the sides of similar triangles?

Answer: Corresponding sides of similar triangles are proportional. The example below shows two triangle's with their proportional sides ..

## What is the similarity ratio (aka scale factor)?

Answer: It's the ratio between corresponding sides. In the picture above, the larger triangle's sides are two times the smaller triangles sides so the scale factor is 2
 $$16 \cdot 2$$ = 32 $$22 \cdot 2$$ = 44 $$25 \cdot 2$$ = 50

Notation: $$\triangle ABC$$~$$\triangle XYZ$$ means that "$$\triangle ABC \text{ is similar to } \triangle XYZ$$"

## How do you find the similarity ratio?

Answer: Match up any pair of corresponding sides and set up a ratio. That's it!
Example
If $$\triangle ABC$$~$$\triangle WXY$$, then what is the similarity ratio?
 Step 1) Pick a pair of corresponding sides (follow the letters ) AB and WX are corresponding. Follow the letters: $$\triangle \color{red}{AB}C$$ ~ $$\triangle \color{red}{WX}Y$$ Step 2) Substitute side lengths into proportion $$\frac{AB}{WX} = \frac{7}{21}$$ Step 3) Simplify (if necessary) $$\frac{7}{21}=\frac{1}{3}$$

## Why is the following problem unsolvable?

If $$\triangle$$ JKL ~ $$\triangle$$ XYZ, LJ = 22 ,JK = 20 and YZ = 30, what is the similarity ratio?

Practice Problems

Problem 1 If $$\triangle$$ ABC ~ $$\triangle$$ADE , AB = 20 and AD = 30, what is the similarity ratio?
Problem 4) Below are two different versions of $$\triangle$$ HYZ and $$\triangle$$ HIJ . The only difference between the version is how long the sides are.