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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 Triangle Similarity

SAS is a criterion for both triangle similarity and triangle congruence. What's the difference between the two criteria?

### SAS 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 SAS criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

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

Let's take at a few example problems regarding SAS similarity.

1. Consider the triangles below with ACDE=BCFE=k\begin{align*}\frac{AC}{DE}=\frac{BC}{FE}=k\end{align*} and CE\begin{align*}\angle C \cong \angle E\end{align*}. Dilate ΔDEF\begin{align*}\Delta DEF\end{align*} with a scale factor of k\begin{align*}k\end{align*} to create ΔDEF\begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}. What do you know about the sides and angles of ΔDEF\begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}? How do they relate to the sides and angles of ΔABC\begin{align*}\Delta ABC\end{align*}?

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

Corresponding angles are congruent after a dilation is performed, so EE\begin{align*}\angle E\cong \angle E^\prime\end{align*}. Therefore, EC\begin{align*}\angle E^\prime \cong \angle C\end{align*} as well. The scale factor was k\begin{align*}k\end{align*}, which is equal to ACDE\begin{align*}\frac{AC}{DE}\end{align*} and BCFE\begin{align*}\frac{BC}{FE}\end{align*}. This means:

• DE=kDE=ACDEDE=AC\begin{align*}D^\prime E^\prime=k\cdot DE=\frac{AC}{DE}\cdot DE=AC\end{align*}. Therefore, DE¯¯¯¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯\begin{align*}\overline{D^\prime E^\prime}\cong \overline {AC}\end{align*}.
• FE=kFE=BCFEFE=BC\begin{align*}F^\prime E^\prime=k\cdot FE=\frac{BC}{FE}\cdot FE=BC\end{align*}. Therefore, FE¯¯¯¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{F^\prime E^\prime}\cong \overline {BC}\end{align*}.

2. Use your work from #1 to prove that ΔABCΔDFE\begin{align*}\Delta ABC\thicksim \Delta DFE\end{align*}.

From #1, you know that EC\begin{align*}\angle E^\prime \cong \angle C\end{align*}, DE¯¯¯¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯\begin{align*}\overline{D^\prime E^\prime}\cong \overline{AC}\end{align*} and FE¯¯¯¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯\begin{align*}\overline{F^\prime E^\prime} \cong \overline{BC}\end{align*}. This means ΔABCΔDFE\begin{align*}\Delta ABC \cong \Delta D^\prime F^\prime E^\prime\end{align*} by SAS\begin{align*}SAS\cong\end{align*}.

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

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

All that was known about the original two triangles in #1 was two pairs of proportional sides and included congruent angles. You have proved that SAS is a criterion for triangle similarity.

Now, let's take a look at determining if two triangles are similar.

Are the two triangles below similar? Explain.

First look at what is marked. CD\begin{align*}\angle C \cong \angle D\end{align*}. Also, ACDE=2\begin{align*}\frac{AC}{DE}=2\end{align*} and ABFE=2\begin{align*}\frac{AB}{FE}=2\end{align*}. Two sides are proportional but the congruent angle is not the included angle. This is SSA which is not a way to prove that triangles are similar (just like it is not a way to prove that triangles are congruent).

Look carefully at the two triangles. Notice that the longest side in ΔABC\begin{align*}\Delta ABC\end{align*} is BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*}, which is unmarked. The longest side in ΔEFD\begin{align*}\Delta EFD\end{align*} appears to be DE¯¯¯¯¯¯¯¯\begin{align*}\overline {DE}\end{align*}, which is marked. BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} and DF¯¯¯¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}will not be proportional to the other pairs of sides.

### Examples

#### Example 1

Earlier, you were asked what is the difference between triangle similarity and triangle congruence

SAS is a criterion for both triangle similarity and triangle congruence. What's the difference between the two criteria?

With SAS\begin{align*}SAS\cong\end{align*}, you must show that two pairs of sides are congruent and their included angles are congruent as well. With SAS\begin{align*}SAS\thicksim\end{align*}, you must show that two pairs of sides are proportional and their included angles are congruent. Two triangles that are similar by SAS\begin{align*}SAS\thicksim\end{align*} with a scale factor of 1 will be congruent.

#### Example 2

Are the triangles similar? Explain.

ACED=BCFD=2\begin{align*}\frac{AC}{ED}=\frac{BC}{FD}=2\end{align*}. CD\begin{align*}\angle C \cong \angle D\end{align*} so the included angles are congruent. Therefore, the triangles are similar by SAS\begin{align*}SAS \thicksim\end{align*}.

#### Example 3

Are the triangles similar? Explain.

While two pairs of sides are proportional and one pair of angles are congruent, the angles are not the included angles. This is SSA, which is not a similarity criterion. Therefore, you cannot say for sure that the triangles are similar.

#### Example 4

What additional information would you need to be able to say that ΔABCΔEBD\begin{align*}\Delta ABC\thicksim \Delta EBD\end{align*}?

Because BD¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯\begin{align*}\overline{BD}\cong \overline{DC}\end{align*}, \begin{align*}\frac{BC}{BD}=2\end{align*}. The two triangles share \begin{align*}\angle B\end{align*}, so that is a pair of congruent angles. To prove that the triangles are similar, you would need to know that \begin{align*}\frac{BA}{BE}=2\end{align*} as well. If you knew that \begin{align*}\overline{BE}\cong \overline{EA}\end{align*} or \begin{align*}E\end{align*} was the midpoint of \begin{align*}\overline{AB}\end{align*}, then you could say that the triangles are similar.

### Review

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

2. What does SSA stand for? What does it have to do with similar triangles?

3. Draw an example of two triangles that must be similar due to SAS.

4. Draw an example of two triangles that are not necessarily similar because all you know is SSA.

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.

5.

6.

7.

8.

9.

10.

11. What is the minimum additional information you would need in order to be able to state that the triangles below are similar by SAS?

12. What is the minimum additional information you would need in order to be able to state that the triangles below are similar by SAS?

13. What is the minimum additional information you would need in order to be able to state that the triangles below are similar by SAS?

14. AAS and ASA are both criteria for triangle congruence. Are they also criteria for triangle similarity? Explain.

15. Show how the SAS criterion for triangle similarity works: use transformations to help explain why the triangles below are similar given that \begin{align*}\frac{AC}{FD}=\frac{CB}{DE}\end{align*}. Hint: See Examples A and B for help.

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### Vocabulary Language: English

TermDefinition
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.