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# SAS Triangle Congruence

## Two sets of corresponding sides and included angles prove congruent triangles.

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SAS Triangle Congruence

When two triangles have two pairs of sides and their included angles congruent, the triangles are congruent. What if the angles aren't included angles?

#### Guidance

If two triangles are congruent it means that all corresponding angle pairs and all corresponding sides are congruent. However, in order to be sure that two triangles are congruent, you do not necessarily need to know that all angle pairs and side pairs are congruent. Consider the triangles below.

In these triangles, you can see that GD\begin{align*}\angle G\cong \angle D\end{align*}, IG¯¯¯¯FD¯¯¯¯¯\begin{align*}\overline{IG}\cong\overline{FD}\end{align*}, and GH¯¯¯¯¯¯DE¯¯¯¯¯\begin{align*}\overline{GH}\cong\overline{DE}\end{align*}. The information you know about the congruent corresponding parts of these triangles is a side, an angle, and then another side. This is commonly referred to as “side-angle-side” or “SAS”.

The SAS criterion for triangle congruence states that if two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.

In the examples, you will use rigid transformations to show why the above SAS triangles must be congruent overall, even though you don't know the lengths of all the sides and the measures of all the angles.

Example A

Perform a rigid transformation to bring point G\begin{align*}G\end{align*} to point D\begin{align*}D\end{align*}.

Solution: Draw a vector from point G\begin{align*}G\end{align*} to point D\begin{align*}D\end{align*}. Translate GHI\begin{align*}\triangle GHI\end{align*} along the vector to create GHI\begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*}.

Example B

Rotate GHI\begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*} to map to GI¯¯¯¯¯¯¯\begin{align*}\overline{G^\prime I^\prime}\end{align*} to DF¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}.

Solution: Measure IDF\begin{align*}\angle I^\prime DF\end{align*}. In this case, mIDF=148\begin{align*}m\angle I^\prime DF=148^\circ\end{align*}.

Rotate GHI\begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*} counterclockwise that number of degrees about point G\begin{align*}G^\prime\end{align*} to create GHI\begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*}. Note that because GI¯¯¯¯DF¯¯¯¯¯\begin{align*}\overline{GI}\cong\overline{DF}\end{align*} and rigid transformations preserve distance, GI¯¯¯¯¯¯¯¯\begin{align*}\overline{G^{\prime \prime}I^{\prime \prime}}\end{align*} matches up perfectly with DF¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}.

Example C

Reflect GHI\begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*} to map it to DEF\begin{align*}\triangle DEF\end{align*}. Can you be confident that the triangles are congruent?

Solution: Reflect GHI\begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*} across GI¯¯¯¯¯¯¯¯\begin{align*}\overline {G^{\prime\prime}I^{\prime\prime}}\end{align*} (which is the same as DF¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}).

Because EDFHGI\begin{align*}\angle EDF\cong \angle{H^{\prime\prime} G^{\prime\prime} I^{\prime\prime}}\end{align*} and GH¯¯¯¯¯¯¯¯¯DE¯¯¯¯¯\begin{align*}\overline {G^{\prime\prime} H^{\prime\prime}}\cong \overline{DE}\end{align*}, the triangles must match up exactly (in particular, H\begin{align*}H^{\prime \prime \prime}\end{align*} must map to E\begin{align*}E\end{align*}), and the triangles are congruent.

This means that even though you didn't know all the side and angle measures, because you knew two pairs of sides and the included angles were congruent, the triangles had to be congruent overall. At this point you can use the SAS criterion for showing triangles are congruent without having to go through all of these transformations each time (but make sure you can explain why SAS works in terms of the rigid transformations!).

Concept Problem Revisited

Even though these triangles have two pairs of sides and one pair of angles that are congruent, the triangles are clearly not congruent. SSA is NOT a criterion for triangle congruence. In order to use two pairs of sides and one pair of angles to show that triangles are congruent, the pair of angles must be included between the pairs of congruent sides.

#### Vocabulary

SAS, or Side-Angle-Side, is a criterion for triangle congruence. The SAS criterion for triangle congruence states that if two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.

Rigid transformations are transformations that preserve distance and angles. The rigid transformations are reflections, rotations, and translations.

Two figures are congruent if a sequence of rigid transformations will carry one figure to the other. Congruent figures will always have corresponding angles and sides that are congruent as well.

#### Guided Practice

Are the following triangles congruent? Explain.

1.

2.

3. What additional information would you need in order to be able to state that the triangles below are congruent by SAS?

1. The triangles are congruent by SAS.

2. The triangles are not necessarily congruent. The given angle is not the included angle in both triangles.

3. You would need to know that AB¯¯¯¯¯AD¯¯¯¯¯¯\begin{align*}\overline{AB}\cong\overline{AD}\end{align*}.

#### Practice

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

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

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

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

For each pair of triangles below, state if they are congruent by SAS or if there is not enough information to determine whether or not they are congruent.

5.

6.

7.

8.

9.

10. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by SAS? Assume that points B\begin{align*}B\end{align*}, C\begin{align*}C\end{align*}, and E\begin{align*}E\end{align*} are collinear.

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

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

13. Do you think you always need at least three pairs of congruent sides/angles to show that two triangles are congruent? Explain.

14. If the two pairs of legs are congruent on two right triangles, are the triangles congruent? Explain. Draw a picture to support your reasoning.

15. Show how the SAS criterion for triangle congruence works: use rigid transformations to help explain why the triangles below are congruent.

### Vocabulary Language: English

Base Angles

Base Angles

The base angles of an isosceles triangle are the angles formed by the base and one leg of the triangle.
Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Equilateral Triangle

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length.
Included Angle

Included Angle

The included angle in a triangle is the angle between two known sides.
SAS

SAS

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

Side Angle Side Triangle

A side angle side triangle is a triangle where two of the sides and the angle between them are known quantities.
Triangle Congruence

Triangle Congruence

Triangle congruence occurs if 3 sides in one triangle are congruent to 3 sides in another triangle.
Rigid Transformation

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.