7.7: 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
CK12 Foundation: Chapter7SASSimilarityA
Watch this video beginning at the 2:09 mark.
James Sousa: Similar Triangles
Watch the second part of this video.
James Sousa: Similar Triangles Using SSS and SAS
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
 Construct a triangle with sides 6 cm and 4 cm and the included angle is \begin{align*}45^\circ\end{align*}
45∘ .  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*}
45∘ .  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*}
Example A
Are the two triangles similar? How do you know?
\begin{align*}\angle B \cong \angle Z\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*}
Example B
Are there any similar triangles? How do you know?
\begin{align*}\angle A\end{align*}
\begin{align*}\frac{9}{9+3} &= \frac{12}{12+5}\\
\frac{9}{12} &= \frac{3}{4} \neq \frac{12}{17}\end{align*}
Example C
From Example B, what should \begin{align*}BC\end{align*}
The proportion we ended up with was \begin{align*}\frac{9}{12}=\frac{3}{4} \neq \frac{12}{17}\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Chapter7SASSimilarityB
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*}
\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*}
2. The triangles are not similar because the angle is not the included angle for both triangles.
3. \begin{align*}\angle{A}\end{align*}
\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.
 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*}
ΔABC is a right triangle with legs that measure 3 and 4. \begin{align*}\Delta DEF\end{align*}ΔDEF is a right triangle with legs that measure 6 and 8. 
\begin{align*}\Delta GHI\end{align*}
ΔGHI is a right triangle with a leg that measures 12 and a hypotenuse that measures 13. \begin{align*}\Delta JKL\end{align*}ΔJKL is a right triangle with legs that measure 1 and 2. 
\begin{align*}\overline{AC} = 3\end{align*}
AC¯¯¯¯¯¯¯¯=3
\begin{align*}\overline{DF} = 6\end{align*}
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Term  Definition 

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. 
Image Attributions
Here you'll learn how to decide whether or not two triangles are similar using SAS Similarity.