### SAS Similarity

If we know that two sides are proportional AND the included angles are congruent, then are the two triangles are similar? Let's investigate.

#### Investigation: SAS Similarity

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

- Construct a triangle with sides 6 cm and 4 cm and the
*included*angle is \begin{align*}45^\circ\end{align*}. - Repeat Step 1 and construct another triangle with sides 12 cm and 8 cm and the included angle is \begin{align*}45^\circ\end{align*}.
- Measure the other two angles in both triangles. What do you notice?
- Measure the third side in each triangle. Make a ratio. Is this ratio the same as the ratios of the sides you were given?

**SAS Similarity Theorem:** If two sides in one triangle are proportional to two sides in another triangle and the included angle in the first triangle is congruent to the included angle in the second, then the two triangles are similar.

In other words, if \begin{align*}\frac{AB}{XY}=\frac{AC}{XZ}\end{align*} and \begin{align*}\angle A \cong \angle X\end{align*}, then \begin{align*}\triangle ABC \sim \triangle XYZ\end{align*}.

#### Determining Similarity

Are the two triangles similar? How do you know?

\begin{align*}\angle B \cong \angle Z\end{align*} because they are both right angles. Second, \begin{align*}\frac{10}{15}=\frac{24}{36}\end{align*} because they both reduce to \begin{align*}\frac{2}{3}\end{align*}. Therefore, \begin{align*}\frac{AB}{XZ}=\frac{BC}{ZY}\end{align*} and \begin{align*}\triangle ABC \sim \triangle XZY\end{align*}.

Notice with this example that we can find the third sides of each triangle using the Pythagorean Theorem. If we were to find the third sides, \begin{align*}AC = 39\end{align*} and \begin{align*}XY = 26\end{align*}. The ratio of these sides is \begin{align*}\frac{26}{39}=\frac{2}{3}\end{align*}.

#### Recognizing Similarity

Are there any similar triangles? How do you know?

\begin{align*}\angle A\end{align*} is shared by \begin{align*}\triangle EAB\end{align*} and \begin{align*}\triangle DAC\end{align*}, so it is congruent to itself. If \begin{align*}\frac{AE}{AD}=\frac{AB}{AC}\end{align*} then, by SAS Similarity, the two triangles would be similar.

\begin{align*}\frac{9}{9+3} &= \frac{12}{12+5}\\ \frac{9}{12} &= \frac{3}{4} \neq \frac{12}{17}\end{align*} Because the proportion is not equal, the two triangles are not similar.

#### Solving for Unknown Lengths

From the previous Example, what should \begin{align*}BC\end{align*} equal for \begin{align*}\triangle EAB \sim \triangle DAC\end{align*}?

The proportion we ended up with was \begin{align*}\frac{9}{12}=\frac{3}{4} \neq \frac{12}{17}\end{align*}. \begin{align*}AC\end{align*} needs to equal 16, so that \begin{align*}\frac{12}{16}=\frac{3}{4}\end{align*}. Therefore, \begin{align*}AC = AB + BC\end{align*} and \begin{align*}16 = 12 + BC\end{align*}. \begin{align*}BC\end{align*} should equal 4 in order for \begin{align*}\triangle EAB \sim \triangle DAC\end{align*}.

### Examples

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

#### Example 1

We can see that \begin{align*}\angle{B} \cong \angle{F}\end{align*} and these are both included angles. We just have to check that the sides around the angles are proportional.

\begin{align*}\frac{AB}{DF}=\frac{12}{8}=\frac{3}{2}\end{align*}

\begin{align*}\frac{BC}{FE}=\frac{24}{16}=\frac{3}{2}\end{align*}

Since the ratios are the same \begin{align*}\triangle ABC \sim \triangle DFE \end{align*} by the SAS Similarity Theorem.

#### Example 2

The triangles are not similar because the angle is not the included angle for both triangles.

#### Example 3

\begin{align*}\angle{A}\end{align*} is the included angle for both triangles, so we have a pair of congruent angles. Now we have to check that the sides around the angles are proportional.

\begin{align*}\frac{AE}{AD}=\frac{16}{16+4}=\frac{16}{20}=\frac{4}{5}\end{align*}

\begin{align*}\frac{AB}{AC}=\frac{24}{24+8}=\frac{24}{32}=\frac{3}{4}\end{align*}

The ratios are not the same so the triangles are not similar.

### Review

Fill in the blanks.

- If two sides in one triangle are _________________ to two sides in another and the ________________ angles are _________________, then the triangles are ______________.

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

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

Determine if the triangles are similar.

- \begin{align*}\Delta ABC\end{align*} is a right triangle with legs that measure 3 and 4. \begin{align*}\Delta DEF\end{align*} is a right triangle with legs that measure 6 and 8.
- \begin{align*}\Delta GHI\end{align*} is a right triangle with a leg that measures 12 and a hypotenuse that measures 13. \begin{align*}\Delta JKL\end{align*} is a right triangle with legs that measure 1 and 2.
- \begin{align*}\overline{AC} = 3\end{align*}

\begin{align*}\overline{DF} = 6\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 7.7.