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# 4.11: Triangle Similarity using SAS

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Understand and apply the SAS Similarity Postulate.

## SAS for Similar Triangles

SAS (Side-Angle-Side) Similarity Postulate

If the lengths of two corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.

Two triangles are similar if two pairs of corresponding sides are __________________________ and the included angles are _____________________________.

Example 1

Cheryl made the diagram below to investigate similar triangles more.

She drew ΔABC\begin{align*}\Delta ABC\end{align*} first, with AB=40, AC=80\begin{align*}AB = 40, \ AC = 80\end{align*}, and mA=30\begin{align*}m \angle A = 30^\circ\end{align*}.

Then Cheryl did the following:

She drew MN¯¯¯¯¯¯¯¯¯¯\begin{align*}\overline{MN}\end{align*}, and made MN=60\begin{align*}MN = 60\end{align*}.

Then she carefully drew MP¯¯¯¯¯¯¯¯¯\begin{align*}\overline{MP}\end{align*}, making MP=120\begin{align*}MP = 120\end{align*} and mM=30\begin{align*}m \angle M = 30^\circ\end{align*}.

At this point, Cheryl had drawn two segments ( MN¯¯¯¯¯¯¯¯¯¯\begin{align*}\overline{MN}\end{align*} and MP¯¯¯¯¯¯¯¯¯\begin{align*}\overline{MP}\end{align*}) with lengths that are proportional to the lengths of the corresponding sides of ΔABC\begin{align*}\Delta ABC\end{align*}, and she had made the included angle, M\begin{align*}\angle M\end{align*}, congruent to the included angle (A)\begin{align*}(\angle A)\end{align*} in ΔABC\begin{align*}\Delta ABC\end{align*}.

Then Cheryl measured angles. She found that:

BN\begin{align*}\angle B \cong \angle N\end{align*} and CP\begin{align*}\angle C \cong \angle P\end{align*}

What could Cheryl conclude? Here again we have automatic results. The other angles are automatically congruent, and the triangles are similar by AA.

Cheryl’s work supports the SAS for Similar Triangles Postulate.

1. In the SAS for similar triangles postulate, which parts are congruent in the similar triangles?

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. In the SAS for similar triangles postulate, which parts are proportional in the similar triangles?

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

3. The two triangles below are similar because of the SAS for similar triangles postulate. Mark the SAS congruent parts with tic marks and/or arcs. Create numbers that are proportional for the similar parts.

## Similar Triangles Summary

We’ve explored similar triangles extensively in several lessons. Let’s summarize the conditions we’ve found that guarantee that two triangles are similar.

Two triangles are similar if and only if:

• the angles in the triangles are congruent.
• the lengths of corresponding sides in the polygons are proportional.

AA for Similar Triangles

It two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.

SSS for Similar Triangles

If the lengths of the sides of two triangles are proportional, then the triangles are similar.

SAS for Similar Triangles

If the lengths of two corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.

You can use the graphic organizer on the next page to keep all of this information in one place.

## Graphic Organizer for Lessons 9-10

Proving Similar Triangles
Type of Similarity What do the letters stand for? What does this mean? Draw a picture of two similar triangles and label parts Describe corresponding congruent parts Describe corresponding proportional parts
AA
SSS
SAS

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