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# 4.4: Triangle Congruence using SSS

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Understand and apply the SSS Congruence Postulate.

## Proving Triangles are Congruent

In the last section you learned that if two triangles are congruent, then the three pairs of corresponding sides are congruent and the three pairs of corresponding angles are also congruent.

In symbols, ΔCATΔDOG\begin{align*}\Delta CAT \cong \Delta DOG\end{align*} means:

• CD\begin{align*}\angle C \cong \angle D\end{align*}
• AO\begin{align*}\angle A \cong \angle O\end{align*}
• TG\begin{align*}\angle T \cong \angle G\end{align*}

and

• CA¯¯¯¯¯¯¯¯DO¯¯¯¯¯¯¯¯\begin{align*}\overline{CA} \cong \overline{DO}\end{align*}
• AT¯¯¯¯¯¯¯OG¯¯¯¯¯¯¯¯\begin{align*}\overline{AT} \cong \overline{OG}\end{align*}
• CT¯¯¯¯¯¯¯DG¯¯¯¯¯¯¯¯\begin{align*}\overline{CT} \cong \overline{DG}\end{align*}

Indeed, one triangle congruence statement contains six different congruence statements about sides and angles!

If two triangles are congruent, then they have three pairs of ______________________________ sides and three pairs of congruent _______________________.

In this section we show that proving two triangles are congruent does not necessarily require showing all six congruence statements are true. There are shortcuts for showing two triangles are congruent using certain combinations of three parts of two different triangles.

Side-Side-Side (SSS) Triangle Congruence Postulate

If three sides in one triangle are congruent to the three corresponding sides in another triangle, then the triangles are congruent to each other.

Do you remember the difference between a postulate and a theorem?

From Unit 1:

A postulate is a basic rule that we accept without proof.

A theorem is a statement that can be proven true using postulates, definitions, logic, and other theorems we’ve already proven to be true.

You have already learned many postulates and theorems; the ones in this lesson and the next few lessons are particular to congruent triangles.

Example 1

Write a triangle congruence statement based on the diagram below:

We can see from the tic marks that there are three pairs of corresponding congruent sides:

HA¯¯¯¯¯¯¯¯RS¯¯¯¯¯¯¯,AT¯¯¯¯¯¯¯SI¯¯¯¯¯¯,\begin{align*}\overline{HA} \cong \overline{RS}, \overline{AT} \cong \overline{SI},\end{align*} and TH¯¯¯¯¯¯¯¯IR¯¯¯¯¯¯\begin{align*}\overline{TH} \cong \overline{IR}\end{align*}

Matching up the corresponding sides, we can write the congruence statement:

ΔHATΔRSI\begin{align*}\Delta HAT \cong \Delta RSI\end{align*}

Don’t forget that ORDER MATTERS when writing triangle congruence statements. Here, we lined up the sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.

1. What do the letters SSS (in the SSS Congruence Postulate) stand for?

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

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

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

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

2. Use the congruence statement ΔABCΔFGH\begin{align*}\Delta ABC \cong \Delta FGH\end{align*} to name all congruent sides in the two triangles:

\begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

3. In your own words, describe the difference between a postulate and a theorem.

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

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

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

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

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