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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 \begin{align*}&\triangle ABC \cong \triangle XYZ\end{align*}? How could you determine which side in \begin{align*}\triangle XYZ\end{align*} is congruent to \begin{align*}\overline{BA}\end{align*} and which angle is congruent to \begin{align*}\angle{C}\end{align*}? After completing this Concept, you'll be able to use congruence statements to state which sides and angles are congruent in congruent triangles.

Watch This

Watch the first part of this video.

Guidance

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.

\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 \begin{align*}\triangle ABC \cong \triangle LMN\end{align*} as:

\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.

Example A

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: \begin{align*}\angle R \cong \angle F, \angle S \cong \angle E,\end{align*} and \begin{align*}\angle T \cong \angle D\end{align*}. Therefore, the triangles are \begin{align*}\triangle RST \cong \triangle FED\end{align*}.

Example B

If \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.

\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*}

Example C

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

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

Watch this video for help with the Examples above.

Concept Problem Revisited

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

Vocabulary

To be congruent means to be the same size and shape. Two triangles are congruent if their corresponding angles and sides are congruent. The symbol \begin{align*}\cong\end{align*} means congruent.

Guided Practice

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

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

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

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

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

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

Practice

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 \begin{align*}\angle{B} \cong \angle{Z}\end{align*}.

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

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

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

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

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

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