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? After completing this Concept, you'll be able to use the SSS Similarity Theorem to decide if two triangles are similar.

### Watch This

CK-12 Foundation: Chapter7SSSSimilarityA

Watch this video beginning at the 2:09 mark.

James Sousa: Similar Triangles

Watch the first part of this video.

James Sousa: Similar Triangles Using SSS and SAS

### Guidance

If you do not know any angle measures, can you say two triangles are similar? Let’s investigate this to see.

##### Investigation: SSS Similarity

Tools Needed: ruler, compass, protractor, paper, pencil

- Construct a triangle with sides 6 cm, 8 cm, and 10 cm.
- Construct a second triangle with sides 9 cm, 12 cm, and 15 cm.
- Using your protractor, measure the angles in both triangles. What do you notice?
- Line up the corresponding sides. Write down the ratios of these sides. What happens?

To see an animated construction of this, click: http://www.mathsisfun.com/geometry/construct-ruler-compass-1.html

From #3, you should notice that the angles in the two triangles are equal. Second, when the corresponding sides are lined up, the sides are all in the same proportion, \begin{align*}\frac{6}{9}=\frac{8}{12}=\frac{10}{15}\end{align*}. If you were to repeat this activity, for a 3-4-5 or 12-16-20 triangle, you will notice that they are all similar. That is because, each of these triangles are multiples of 3-4-5. If we generalize what we found in this investigation, we have the SSS Similarity Theorem.

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

#### Example A

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 B

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 C

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.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7SSSSimilarityB

### Vocabulary

Two triangles are ** similar** if all their corresponding angles are

**(exactly the same) and their corresponding sides are**

*congruent***(in the same ratio).**

*proportional*### Guided Practice

Determine if any of the triangles below are similar. Compare two triangles at a time.

1. Is \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}?

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

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

**Answers:**

1. \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*}: Is \begin{align*}\frac{20}{15} = \frac{22}{16} = \frac{24}{18}\end{align*}?

Reduce each fraction to see if they are equal. \begin{align*}\frac{20}{15} = \frac{4}{3}, \frac{22}{16} = \frac{11}{8}\end{align*}, and \begin{align*}\frac{24}{18} = \frac{4}{3}\end{align*}.

\begin{align*}\frac{4}{3} \neq \frac{11}{8}, \triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*} are **not** similar.

2. \begin{align*}\triangle DEF\end{align*} and \begin{align*}\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*}.

3. \begin{align*}\triangle ABC\end{align*} and \begin{align*}\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 GIH\end{align*}.

### Practice

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.