How can you use the SSS similarity criterion to show that the triangles below are similar?

### SSS Triangle Similarity

If two triangles are **similar** it means that all corresponding angle pairs are congruent and all corresponding sides are proportional. However, in order to be sure that two triangles are similar, you do not necessarily need to have information about all sides and all angles.

**The SSS** **criterion for triangle similarity** **states that if** **three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.**

In the examples, you will use similarity transformations and criteria for triangle congruence to show why SSS is a criterion for triangle similarity.

Let's take a look at few problems regarding SSS triangle similarity.

1. Consider the triangles below with \begin{align*}\frac{AB}{FE}=\frac{CA}{DF}=\frac{CB}{DE}=k\end{align*}. Dilate \begin{align*}\Delta DEF\end{align*} with a scale factor of \begin{align*}k\end{align*} to create \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}. What do you know about the sides of \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}? How do they relate to the sides of \begin{align*}\Delta ABC\end{align*}?

Below, \begin{align*}\Delta DEF\end{align*} is dilated about point \begin{align*}P\end{align*} with a scale factor of \begin{align*}k\end{align*} to create \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}.

The scale factor is \begin{align*}k\end{align*}, which is equal to \begin{align*}\frac{AB}{FE}\end{align*} and \begin{align*}\frac{CA}{DF}\end{align*} and \begin{align*}\frac{CB}{DE}\end{align*}. This means:

- \begin{align*}D^\prime F^\prime=k\cdot DF=\frac{CA}{DF}\cdot DF=CA\end{align*}. Therefore, \begin{align*}\overline{D^\prime F^\prime} \cong \overline{CA}\end{align*}.
- \begin{align*}D^\prime E^\prime=k\cdot DE=\frac{CB}{DE}\cdot DE=CB\end{align*}. Therefore, \begin{align*}\overline{D^\prime E^\prime} \cong \overline{CB}\end{align*}.
- \begin{align*}F^\prime E^\prime=k\cdot FE=\frac{AB}{FE}\cdot FE=AB\end{align*}. Therefore, \begin{align*}\overline{F^\prime E^\prime} \cong \overline{AB}\end{align*}.

2. Use your work from the previous problem to prove that \begin{align*}\Delta ABC\thicksim\Delta FED\end{align*}.

You know that \begin{align*}\overline{D^\prime F^\prime}\cong \overline{CA}\end{align*}, \begin{align*}\overline{D^\prime E^\prime}\cong \overline{CB}\end{align*} and \begin{align*}\overline{F^\prime E^\prime}\cong \overline{AB}\end{align*}. This means \begin{align*}\Delta ABC \cong \Delta F^\prime E^\prime D^\prime\end{align*} by \begin{align*}SSS\cong\end{align*}.

Therefore, there must exist a sequence of rigid transformations that will carry \begin{align*}\Delta ABC\end{align*} to \begin{align*}\Delta F^\prime E^\prime D^\prime\end{align*}.

\begin{align*}\Delta ABC\thicksim\Delta FED\end{align*} because a series of rigid transformations will carry \begin{align*}\Delta ABC\end{align*} to \begin{align*}\Delta F^\prime E^\prime D^\prime\end{align*}, and then a dilation will carry \begin{align*}\Delta F^\prime E^\prime D^\prime\end{align*} to \begin{align*}\Delta ABC\end{align*}.

**All that was known about the original two triangles in previous problem was** **three pairs of proportional sides****.** **You** **have proved that** *SSS* is a criterion**for triangle similarity.**

Now, let's take a look at determining triangle similarity using SSS.

You want to show that the triangles below are similar by \begin{align*}SSS\thicksim\end{align*}. What additional information do you need?

You have three side lengths for \begin{align*}\Delta ABC\end{align*} with \begin{align*}\frac{AB}{FE}=\frac{AC}{FD}=3\end{align*}. Since \begin{align*}CB=6\end{align*}, you need to know that \begin{align*}DE=2\end{align*} (so that \begin{align*}\frac{CB}{DE}=3\end{align*}) in order to show that the triangles are similar by \begin{align*}SSS\thicksim\end{align*}.

**Examples**

**Example 1**

Earlier, you were asked how can you use the SSS similarity criterion to show that the triangles below are similar.

Only two pairs of sides and a pair of non-included angles are given. You can't say these triangles are similar by SSA because that is not a criterion for triangle similarity. However, because these are right triangles, you know that the third side of each triangle can be found with the Pythagorean Theorem.

- For the smaller triangle: \begin{align*}12^2+x^2=15^2\rightarrow x=9\end{align*}.
- For the larger triangle: \begin{align*}36^2+x^2=45^2\rightarrow x=27\end{align*}.

All three pairs of sides are proportional with a ratio of 3. Therefore, the triangles are similar due to \begin{align*}SSS\thicksim\end{align*}. See the practice exercises to generalize this proof.

#### Example 2

Are the triangles similar? Explain.

All three sides of each triangle are the same. The ratio between each pair of sides is \begin{align*}\frac{4x}{x}=4\end{align*}. Because three pairs of sides are proportional, the triangles are similar by \begin{align*}SSS\thicksim\end{align*}.

#### Example 3

Are the triangles similar? Explain.

Match the longest side with the longest side and the shortest side with the shortest side. Check all three ratios:

- \begin{align*}\frac{DF}{AC}=\frac{18}{6}=3\end{align*}
- \begin{align*}\frac{DE}{AB}=\frac{15}{5}=3\end{align*}
- \begin{align*}\frac{EF}{BC}=\frac{6}{4}=1.5\end{align*}

Because the three pairs of sides are not proportional, the triangles are not similar.

#### Example 4

You want to show that the triangles below are similar by \begin{align*}SSS\thicksim\end{align*}. What additional information do you need?

Look for two pairs of sides with lengths in the same ratio.

- \begin{align*}\frac{DE}{AB}=\frac{25}{15}=\frac{5}{3}\end{align*}
- \begin{align*}\frac{DF}{AC}=\frac{35}{21}=\frac{5}{3}\end{align*}

The common ratio is \begin{align*}\frac{5}{3}\end{align*}. This means, \begin{align*}\frac{EF}{BC}\end{align*} must equal \begin{align*}\frac{5}{3}\end{align*} as well.

\begin{align*}\frac{15}{BC}&=\frac{5}{3}\\ 45 &=5BC\\ BC &=9\end{align*}

For the triangles to be similar by \begin{align*}SSS\thicksim\end{align*}, you need to know that \begin{align*}BC=9\end{align*}.

### Review

1. What does SSS stand for? What does it have to do with similar triangles?

2. Draw an example of two triangles that must be similar due to SSS.

For each pair of triangles below state if they are similar, congruent, or if there is not enough information to determine whether or not they are congruent. If they are similar or congruent, write a similarity or congruence statement.

3.

4.

5.

6.

7.

8. What does the previous problem tell you about equilateral triangles?

9. How are the perimeters of two similar triangles related?

10. One triangle has side lengths of 6, 8, and 10. A similar triangle has a perimeter of 60. What are the lengths of the sides of the similar triangle?

11. One triangle has side lengths of 7, 8, and 14. A similar triangle has a perimeter of 87. What is the ratio between corresponding sides?

12. One triangle has side lengths of 2, 4, and 4. A similar triangle has a perimeter of 30. What are the lengths of the sides of the similar triangle?

13. Find the length of the unmarked side of each triangle in terms of \begin{align*}c\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}k\end{align*}.

14. Use your work from #13 to prove that the two triangles in #13 are similar. What does this tell you about one method for proving that right triangles are similar?

15. Show how the SSS criterion for triangle similarity works: use transformations to help explain why the triangles below are similar. *Hint: See Examples A and B for help.*

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.5.