

Similar TrianglesAngles, Sides & similarity ratio
Notation: $$ \triangle ABC $$~$$\triangle XYZ $$ means that "$$ \triangle ABC \text{ is similar to } \triangle XYZ $$"
Example
If $$ \triangle ABC $$~$$\triangle WXY $$, then what is the similarity ratio?
If $$ \triangle $$ JKL ~ $$\triangle $$ XYZ, LJ = 22 ,JK = 20 and YZ = 30, what is the similarity ratio?
Answer: You are not given a single pair of corresponding sides so you cannot find the similarity ratio. Remember How to Find corresponding sides
Corresponding sides follow the same letter order as the triangle name so
YZ of $$ \triangle X\color{red}{YZ}$$ corresponds with side KL of$$\triangle J\color{red}{KL} $$ JK of $$ \triangle \color{red}{JK}L $$ corresponds with side XY of$$\triangle \color{red}{XY}Z $$ LJ of $$ \triangle \color{red}{J}K\color{red}{L} $$ corresponds with side ZX of$$\triangle \color{red}{X}Y \color{red}{Z}$$ Below is a picture of what these two triangles could look like Practice Problems
Problem 1
If $$ \triangle $$ ABC ~ $$\triangle $$ADE , AB = 20 and AD = 30, what is the similarity ratio?
Part B) If EA = 33, how long is CA?
EA and CA are corresponding sides ( $$ \triangle \color{red}{A}B\color{red}{C}$$ ~ $$\triangle \color{red}{A}D\color{red}{E}$$)
Since the sides of similar triangles are proportional, just set up a proportion involving these two sides and the similarity ratio and solve. $ \frac{EA}{CA} = \frac{3}{2} \\ \frac{33}{CA} = \frac{3}{2} \\ CA \cdot 3 = 2 \cdot 33 \\ CA \cdot 3 = 66 \\ CA = \frac{66}{3} = 22 $ DE = 27, how long is BC?
EA and AC are corresponding sides ( $$ \triangle \color{red}{ A}B\color{red}{C}$$ ~ $$\triangle \color{red}{A}D\color{red}{E}$$)
Since the sides of similar triangles are proportional, just set up a proportion involving these two sides and the similarity ratio and solve. $ \frac{DE}{BC} = \frac{3}{2} \\ \frac{27}{CA} = \frac{3}{2} \\ CA \cdot 3 = 2 \cdot 27 \\ CA \cdot 3 = 54 \\ CA = \frac{54}{3} = 18 $ Problem 3) Use your knowledge of similar triangles to find the side lengths below.
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. Only one of these two versions includes a pair of similar triangles.Can you identify which version represents similar triangles?
