### SSS Similarity Theorem

By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that all sides are proportional, that is enough information to know that the triangles are similar. This is called the **SSS Similarity Theorem**.

**SSS Similarity Theorem:** If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.

If \begin{align*}\frac{AB}{YZ} = \frac{BC}{ZX} = \frac{AC}{XY}\end{align*}, then \begin{align*}\triangle ABC \sim \triangle YZX\end{align*}.

What if you were given a pair of triangles and the side lengths for all three of their sides? How could you use this information to determine if the two triangles are similar?

### Examples

For Examples 1 and 2, use the following diagram:

#### Example 1

Is \begin{align*}\triangle DEF \sim \triangle GHI\end{align*}?

Is \begin{align*}\frac{15}{30} = \frac{16}{33} = \frac{18}{36}\end{align*}?

\begin{align*}\frac{15}{30} = \frac{1}{2}, \frac{16}{33} = \frac{16}{33}\end{align*}, and \begin{align*}\frac{18}{36} = \frac{1}{2}\end{align*}. \begin{align*}\frac{1}{2} \neq \frac{16}{33}, \triangle DEF\end{align*} is **not** similar to \begin{align*}\triangle GHI\end{align*}.

#### Example 2

Is \begin{align*}\triangle ABC \sim \triangle GHI\end{align*}?

Is \begin{align*}\frac{20}{30} = \frac{22}{33} = \frac{24}{36}\end{align*}?

\begin{align*}\frac{20}{30} = \frac{2}{3}, \frac{22}{33} = \frac{2}{3}\end{align*}, and \begin{align*}\frac{24}{36} = \frac{2}{3}\end{align*}. All three ratios reduce to \begin{align*}\frac{2}{3}\end{align*}, \begin{align*}\triangle ABC \sim \triangle GHI\end{align*}.

#### Example 3

Determine if the following triangles are similar. If so, explain why and write the similarity statement.

We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides.

\begin{align*}\frac{BC}{FD}=\frac{28}{20}=\frac{7}{5}\end{align*}

\begin{align*}\frac{BA}{FE}=\frac{21}{15}=\frac{7}{5}\end{align*}

\begin{align*}\frac{AC}{ED}=\frac{14}{10}=\frac{7}{5}\end{align*}

Since all the ratios are the same, \begin{align*}\triangle ABC \sim \triangle EFD\end{align*} by the SSS Similarity Theorem.

#### Example 4

Find \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, such that \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}.

According to the similarity statement, the corresponding sides are: \begin{align*}\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\end{align*}. Substituting in what we know, we have \begin{align*}\frac{9}{6} = \frac{4x-1}{10} = \frac{18}{y}\end{align*}.

\begin{align*}\frac{9}{6} &= \frac{4x-1}{10} && \quad \ \frac{9}{6} = \frac{18}{y}\\ 9(10) &= 6(4x-1) && \quad 9y =18(6)\\ 90 &= 24x-6 && \quad 9y = 108\\ 96 &= 24x && \quad \ y = 12\\ x &= 4\end{align*}

#### Example 5

Determine if the following triangles are similar. If so, explain why and write the similarity statement.

We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides.

\begin{align*}\frac{AC}{ED}=\frac{21}{35}=\frac{3}{5}\end{align*}

\begin{align*}\frac{BC}{FD}=\frac{15}{25}=\frac{3}{5}\end{align*}

\begin{align*}\frac{AB}{EF}=\frac{10}{20}=\frac{1}{2}\end{align*}

Since the ratios are not all the same, the triangles are not similar.

### Review

Fill in the blanks.

- If all three sides in one triangle are __________________ to the three sides in another, then the two triangles are similar.
- Two triangles are similar if the corresponding sides are _____________.

Use the following diagram for questions 3-5. *The diagram is to scale.*

- Are the two triangles similar? Explain your answer.
- Are the two triangles congruent? Explain your answer.
- What is the scale factor for the two triangles?

Fill in the blanks in the statements below. Use the diagram to the left.

- \begin{align*}\triangle ABC \sim \triangle\end{align*}_____
- \begin{align*}\frac{AB}{?} = \frac{BC}{?} = \frac{AC}{?}\end{align*}
- If \begin{align*}\triangle ABC\end{align*} had an altitude, \begin{align*}AG = 10\end{align*}, what would be the length of altitude \begin{align*}\overline{DH}\end{align*}?
- Find the perimeter of \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*}. Find the ratio of the perimeters.

Use the diagram to the right for questions 10-15.

- \begin{align*}\triangle ABC \sim \triangle\end{align*}_____
- Why are the two triangles similar?
- Find \begin{align*}ED\end{align*}.
- \begin{align*}\frac{BD}{?} = \frac{?}{BC} = \frac{DE}{?}\end{align*}
- Is \begin{align*}\frac{AD}{DB} = \frac{CE}{EB}\end{align*} true?
- Is \begin{align*}\frac{AD}{DB} = \frac{AC}{DE}\end{align*} true?

Find the value of the missing variable(s) that makes the two triangles similar.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.6.