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# Congruence Statements

## Learn how to write congruence statements and use congruence statements to determine the corresponding parts of triangles.

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Congruence Statements

### Congruence Statements

When stating that two triangles are congruent, the corresponding parts must be written in the same order. For example, if we know that ABC\begin{align*}\triangle ABC\end{align*} and LMN\begin{align*}\triangle LMN\end{align*} are congruent then we know that:

Notice that the congruent sides also line up within the congruence statement.

AB¯¯¯¯¯¯¯¯LM¯¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯, AC¯¯¯¯¯¯¯¯LN¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{LM}, \ \overline{BC} \cong \overline{MN}, \ \overline{AC} \cong \overline{LN}\end{align*}

We can also write this congruence statement five other ways, as long as the congruent angles match up. For example, we can also write ABCLMN\begin{align*}\triangle ABC \cong \triangle LMN\end{align*} as:

ACBLNMCBANMLBCAMNLCABNLMBACMLN\begin{align*}&\triangle ACB \cong \triangle LNM && \triangle BCA \cong \triangle MNL && \triangle BAC \cong \triangle MLN\\ &\triangle CBA \cong \triangle NML && \triangle CAB \cong \triangle NLM && \end{align*}

What if you were told that FGHXYZ\begin{align*}\triangle FGH \cong \triangle XYZ\end{align*}? How could you determine which side in XYZ\begin{align*}\triangle XYZ\end{align*} is congruent to GH¯¯¯¯¯¯¯¯\begin{align*}\overline{GH}\end{align*} and which angle is congruent to F\begin{align*}\angle{F}\end{align*}?

### Examples

#### Example 1

If ABCDEF\begin{align*}\triangle ABC \cong \triangle DEF\end{align*}, what else do you know?

From this congruence statement, we know three pairs of angles and three pairs of sides are congruent. AD,BE,CF\begin{align*} \angle{A} \cong \angle{D}, \angle{B} \cong \angle{E}, \angle{C} \cong \angle{F}\end{align*}, AB¯¯¯¯¯¯¯¯DE¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯EF¯¯¯¯¯¯¯¯, AC¯¯¯¯¯¯¯¯DF¯¯¯¯¯¯¯¯\begin{align*}\overline{AB} \cong \overline{DE}, \ \overline{BC} \cong \overline{EF}, \ \overline{AC} \cong \overline{DF}\end{align*}.

#### Example 2

If KBPMRS\begin{align*}\triangle KBP \cong \triangle MRS\end{align*}, what else do you know?

From this congruence statement, we know three pairs of angles and three pairs of sides are congruent. KM,BR,PS\begin{align*} \angle{K} \cong \angle{M}, \angle{B} \cong \angle{R}, \angle{P} \cong \angle{S}\end{align*}, KB¯¯¯¯¯¯¯¯MR¯¯¯¯¯¯¯¯¯, BP¯¯¯¯¯¯¯¯RS¯¯¯¯¯¯¯, KP¯¯¯¯¯¯¯¯MS¯¯¯¯¯¯¯¯¯\begin{align*}\overline{KB} \cong \overline{MR}, \ \overline{BP} \cong \overline{RS}, \ \overline{KP} \cong \overline{MS}\end{align*}.

#### Example 3

Write a congruence statement for the two triangles below.

Line up the corresponding angles in the triangles:

RF, SE\begin{align*}\angle{R} \cong \angle{F}, \ \angle{S} \cong \angle{E}\end{align*}, and TD\begin{align*}\angle{T} \cong \angle{D}\end{align*}.

Therefore, one possible congruence statement is RSTFED\begin{align*}\triangle RST \cong \angle{FED}\end{align*}

#### Example 4

If CATDOG\begin{align*}\triangle CAT \cong \triangle DOG\end{align*}, what else do you know?

From this congruence statement, we know three pairs of angles and three pairs of sides are congruent.

#### Example 5

If BUGANT\begin{align*}\triangle BUG \cong \triangle ANT\end{align*}, what angle is congruent to N\begin{align*}\angle{N}\end{align*}?

Since the order of the letters in the congruence statement tells us which angles are congruent, NU\begin{align*}\angle{N} \cong \angle{U}\end{align*} because they are each the second of the three letters.

### Review

For questions 1-4, determine if the triangles are congruent using the definition of congruent triangles. If they are, write the congruence statement.

1. Suppose the two triangles below are congruent. Write a congruence statement for these triangles.
2. Explain how we know that if the two triangles are congruent, then BZ\begin{align*}\angle{B} \cong \angle{Z}\end{align*}.
3. If TBSFAM\begin{align*}\triangle TBS \cong \triangle FAM\end{align*}, what else do you know?
4. If PAMSTE\begin{align*}\triangle PAM \cong \triangle STE\end{align*}, what else do you know?
5. If INTWEB\begin{align*}\triangle INT \cong \triangle WEB\end{align*}, what else do you know?
6. If ADGBCE\begin{align*}\triangle ADG \cong \triangle BCE\end{align*}, what angle is congruent to G\begin{align*} \angle{G}\end{align*}?

To see the Review answers, open this PDF file and look for section 4.4.

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