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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

What if you were told that ABCXYZ\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}? How could you determine which side in XYZ\begin{align*}\triangle XYZ\end{align*} is congruent to BA¯¯¯¯¯¯¯¯\begin{align*}\overline{BA}\end{align*} and which angle is congruent to C\begin{align*}\angle{C}\end{align*}

### Congruence Statements

When stating that two triangles are congruent, use a congruence statement. The order of the letters is very important, as corresponding parts must be written in the same order. 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 several 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:

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

One congruence statement can always be written six ways. Any of the six ways above would be correct.

Watch the first part of this video.

#### Writing a Congruence Statement

Write a congruence statement for the two triangles below.

To write the congruence statement, you need to line up the corresponding parts 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, the triangles are RSTFED\begin{align*}\triangle RST \cong \triangle FED\end{align*}.

#### Making Conclusions from Congruence Statements

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

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

CDCA¯¯¯¯¯¯¯¯DO¯¯¯¯¯¯¯¯AOAT¯¯¯¯¯¯¯OG¯¯¯¯¯¯¯¯TGCT¯¯¯¯¯¯¯DG¯¯¯¯¯¯¯¯\begin{align*}& \angle C \cong \angle D && \angle A \cong \angle O && \angle T \cong \angle G\\ & \overline{CA} \cong \overline{DO} && \overline{AT} \cong \overline{OG} && \overline{CT} \cong \overline{DG}\end{align*}

#### Finding Congruent Angles

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.

#### Earlier Problem Revisited

If ABCXYZ\begin{align*} \triangle ABC \cong \triangle XYZ\end{align*}, then BA¯¯¯¯¯¯¯¯YX¯¯¯¯¯¯¯¯\begin{align*} \overline{BA} \cong \overline{YX}\end{align*} and CZ\begin{align*} \angle C \cong \angle Z\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

If EWNMAP\begin{align*}\triangle EWN \cong \triangle MAP\end{align*}, what else do you know?

From this congruence statement, we know three pairs of angles and three pairs of sides are congruent. EM,WA,NP\begin{align*} \angle{E} \cong \angle{M}, \angle{W} \cong \angle{A}, \angle{N} \cong \angle{P}\end{align*}, EW¯¯¯¯¯¯¯¯¯MA¯¯¯¯¯¯¯¯¯, WN¯¯¯¯¯¯¯¯¯¯AP¯¯¯¯¯¯¯¯, EN¯¯¯¯¯¯¯¯MP¯¯¯¯¯¯¯¯¯\begin{align*}\overline{EW} \cong \overline{MA}, \ \overline{WN} \cong \overline{AP}, \ \overline{EN} \cong \overline{MP}\end{align*}.

### 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 to the right 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*}.

Suppose TBSFAM\begin{align*}\triangle TBS \cong \triangle FAM\end{align*}.

1. What angle is congruent to B\begin{align*}\angle B\end{align*}?
2. What side is congruent to FM¯¯¯¯¯¯¯¯¯\begin{align*}\overline{FM}\end{align*}?
3. What side is congruent to SB¯¯¯¯¯¯¯\begin{align*}\overline{SB}\end{align*}?

Suppose INTWEB\begin{align*}\triangle INT \cong \triangle WEB\end{align*}.

1. What side is congruent to IT¯¯¯¯¯¯\begin{align*}\overline{IT}\end{align*}?
2. What angle is congruent to W\begin{align*}\angle W\end{align*}?
3. What angle is congruent to I\begin{align*}\angle I\end{align*}?

Suppose ADGBCE\begin{align*}\triangle ADG \cong \triangle BCE\end{align*}.

1. What side is congruent to CE¯¯¯¯¯¯¯¯\begin{align*}\overline{CE}\end{align*}?
2. What side is congruent to DA¯¯¯¯¯¯¯¯\begin{align*}\overline{DA}\end{align*}?
3. What angle is congruent to G\begin{align*}\angle G\end{align*}?

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