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

Below,

Corresponding angles are congruent after a dilation is performed, so

D′E′=k⋅DE=ACDE⋅DE=AC . Therefore,D′E′¯¯¯¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ .F′E′=k⋅FE=BCFE⋅FE=BC . Therefore,F′E′¯¯¯¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ .

2. Use your work from #1 to prove that

From #1, you know that

Therefore, there must exist a sequence of rigid transformations that will carry

**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. *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

**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 **congruent** and their included angles are congruent as well. With **proportional** and their included angles are congruent. Two triangles that are similar by

#### Example 2

Are the triangles similar? Explain.

#### 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

Because

### 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.

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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.*

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4.