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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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Practice Congruence Statements
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Congruence Statements

What if you were told that $\triangle FGH \cong \triangle XYZ$ ? How could you determine which side in $\triangle XYZ$ is congruent to $\overline{GH}$ and which angle is congruent to $\angle{F}$ ? After completing this Concept, you'll be able 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, the corresponding parts must be written in the same order. For example, if we know that $\triangle ABC$ and $\triangle LMN$ are congruent then we know that:

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

$\overline{AB} \cong \overline{LM}, \ \overline{BC} \cong \overline{MN}, \ \overline{AC} \cong \overline{LN}$

We can also write this congruence statement five other ways, as long as the congruent angles match up. For example, we can also write $\triangle ABC \cong \triangle LMN$ as:

$&\triangle ACB \cong \triangle LNM && \triangle BCA \cong \triangle MNL && \triangle BAC \cong \triangle MLN\\&\triangle CBA \cong \triangle NML && \triangle CAB \cong \triangle NLM &&$

#### Example A

Write a congruence statement for the two triangles below.

Line up the corresponding angles in the triangles:

$\angle{R} \cong \angle{F}, \ \angle{S} \cong \angle{E}$ , and $\angle{T} \cong \angle{D}$ .

Therefore, one possible congruence statement is $\triangle RST \cong \angle{FED}$

#### Example B

If $\triangle CAT \cong \triangle DOG$ , what else do you know?

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

#### Example C

If $\triangle BUG \cong \triangle ANT$ , what angle is congruent to $\angle{N}$ ?

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

### Guided Practice

1. If $\triangle ABC \cong \triangle DEF$ , what else do you know?

2. If $\triangle KBP \cong \triangle MRS$ , what else do you know?

3. If $\triangle EWN \cong \triangle MAP$ , what else do you know?

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

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

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

### 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 $\angle{B} \cong \angle{Z}$ .
3. If $\triangle TBS \cong \triangle FAM$ , what else do you know?
4. If $\triangle PAM \cong \triangle STE$ , what else do you know?
5. If $\triangle INT \cong \triangle WEB$ , what else do you know?
6. If $\triangle ADG \cong \triangle BCE$ , what angle is congruent to $\angle{G}$ ?