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# SAS Similarity

## Triangles are similar if two pairs of sides are proportional and the included angles are congruent.

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SAS Similarity

What if you were given a pair of triangles, the lengths of two of their sides, and the measure of the angle between those two 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 SAS Similarity Theorem to decide if two triangles are similar.

### Watch This

Watch this video beginning at the 2:09 mark.

Watch the second part of this video.

### Guidance

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

1. Construct a triangle with sides 6 cm and 4 cm and the included angle is \begin{align*}45^\circ\end{align*}.
2. 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*}.
3. Measure the other two angles in both triangles. What do you notice?
4. 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*}.

#### Example A

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*}.

#### Example B

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.

#### Example C

From Example B, 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*}.

Watch this video for help with the Examples above.

### Vocabulary

Two triangles are similar if all their corresponding angles are congruent (exactly the same) and their corresponding sides are proportional (in the same ratio).

### Guided Practice

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

1.

2.

3.

Answers:

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.

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

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.

### Practice

Fill in the blanks.

1. 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.

1. \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.
2. \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.
3. \begin{align*}\overline{AC} = 3\end{align*}

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

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### Vocabulary Language: English

AA Similarity Postulate

If two angles in one triangle are congruent to two angles in another triangle, 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.

SAS

SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known.

SAS Similarity Theorem

The SAS Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

Similarity Transformation

A similarity transformation is one or more rigid transformations followed by a dilation.

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