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# SSS Similarity

## Triangles are similar if their corresponding sides are proportional.

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SSS Similarity

### SSSSimilarity

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

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

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, 69=812=1015\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.

Watch this video beginning at the 2:09 mark.

#### Determining if Two Triangles are Similar

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.

BCFD=2820=75\begin{align*}\frac{BC}{FD}=\frac{28}{20}=\frac{7}{5}\end{align*}

BAFE=2115=75\begin{align*}\frac{BA}{FE}=\frac{21}{15}=\frac{7}{5}\end{align*}

ACED=1410=75\begin{align*}\frac{AC}{ED}=\frac{14}{10}=\frac{7}{5}\end{align*}

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

Watch the first part of this video.

#### Solving for Unknown Values

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

According to the similarity statement, the corresponding sides are: ABDE=BCEF=ACDF\begin{align*}\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\end{align*}. Substituting in what we know, we have 96=4x110=18y\begin{align*}\frac{9}{6} = \frac{4x-1}{10} = \frac{18}{y}\end{align*}.

969(10)9096x=4x110=6(4x1)=24x6=24x=4 96=18y9y=18(6)9y=108 y=12\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*}

#### Explaining Similiarty and Writing a Similarity Statement

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.

ACED=2135=35\begin{align*}\frac{AC}{ED}=\frac{21}{35}=\frac{3}{5}\end{align*}

BCFD=1525=35\begin{align*}\frac{BC}{FD}=\frac{15}{25}=\frac{3}{5}\end{align*}

ABEF=1020=12\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.

### Examples

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

#### Example 1

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

ABC\begin{align*}\triangle ABC\end{align*} and DEF\begin{align*}\triangle DEF\end{align*}: Is 2015=2216=2418\begin{align*}\frac{20}{15} = \frac{22}{16} = \frac{24}{18}\end{align*}?

Reduce each fraction to see if they are equal. 2015=43,2216=118\begin{align*}\frac{20}{15} = \frac{4}{3}, \frac{22}{16} = \frac{11}{8}\end{align*}, and 2418=43\begin{align*}\frac{24}{18} = \frac{4}{3}\end{align*}.

43118,ABC\begin{align*}\frac{4}{3} \neq \frac{11}{8}, \triangle ABC\end{align*} and DEF\begin{align*}\triangle DEF\end{align*} are not similar.

#### Example 2

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

DEF\begin{align*}\triangle DEF\end{align*} and GHI\begin{align*}\triangle GHI\end{align*}: Is 1530=1633=1836\begin{align*}\frac{15}{30} = \frac{16}{33} = \frac{18}{36}\end{align*}?

1530=12,1633=1633\begin{align*}\frac{15}{30} = \frac{1}{2}, \frac{16}{33} = \frac{16}{33}\end{align*}, and 1836=12\begin{align*}\frac{18}{36} = \frac{1}{2}\end{align*}. 121633,DEF\begin{align*}\frac{1}{2} \neq \frac{16}{33}, \triangle DEF\end{align*} is not similar to GHI\begin{align*}\triangle GHI\end{align*}.

#### Example 3

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

ABC\begin{align*}\triangle ABC\end{align*} and GHI\begin{align*}\triangle GHI\end{align*}: Is 2030=2233=2436\begin{align*}\frac{20}{30} = \frac{22}{33} = \frac{24}{36}\end{align*}?

2030=23,2233=23\begin{align*}\frac{20}{30} = \frac{2}{3}, \frac{22}{33} = \frac{2}{3}\end{align*}, and 2436=23\begin{align*}\frac{24}{36} = \frac{2}{3}\end{align*}. All three ratios reduce to 23\begin{align*}\frac{2}{3}\end{align*}, ABCGIH\begin{align*}\triangle ABC \sim \triangle GIH\end{align*}.

### Review

Fill in the blanks.

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

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

3. What is the scale factor for the two triangles?

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

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

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

1. ABC\begin{align*}\triangle ABC \sim \triangle\end{align*}_____
2. Why are the two triangles similar?
3. Find ED\begin{align*}ED\end{align*}.
4. BD?=?BC=DE?\begin{align*}\frac{BD}{?} = \frac{?}{BC} = \frac{DE}{?}\end{align*}
5. Is ADDB=CEEB\begin{align*}\frac{AD}{DB} = \frac{CE}{EB}\end{align*} true?
6. Is ADDB=ACDE\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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

AAA Similarity Theorem

AAA Similarity Theorem states that if all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.

Congruent

Congruent figures are identical in size, shape and measure.

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.

SSS

SSS means side, side, side and refers to the fact that all three sides of a triangle are known in a problem.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.