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# Congruence Statements

## Corresponding angles and sides of congruent triangles are congruent.

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Congruence Statements

What if you were told that $\triangle ABC \cong \triangle XYZ$ ? How could you determine which side in $\triangle XYZ$ is congruent to $\overline{BA}$ and which angle is congruent to $\angle{C}$ ? 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.

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

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

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

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: $\angle R \cong \angle F, \angle S \cong \angle E,$ and $\angle T \cong \angle D$ . Therefore, the triangles are $\triangle RST \cong \triangle FED$ .

#### Example B

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

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

$& \angle C \cong \angle D && \angle A \cong \angle O && \angle T \cong \angle G\\& \overline{CA} \cong \overline{DO} && \overline{AT} \cong \overline{OG} && \overline{CT} \cong \overline{DG}$

#### 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.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

If $\triangle ABC \cong \triangle XYZ$ , then $\overline{BA} \cong \overline{YX}$ and $\angle C \cong \angle Z$ .

### 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 $\cong$ means congruent .

### 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 $\angle{B} \cong \angle{Z}$ .

Suppose $\triangle TBS \cong \triangle FAM$ .

1. What angle is congruent to $\angle B$ ?
2. What side is congruent to $\overline{FM}$ ?
3. What side is congruent to $\overline{SB}$ ?

Suppose $\triangle INT \cong \triangle WEB$ .

1. What side is congruent to $\overline{IT}$ ?
2. What angle is congruent to $\angle W$ ?
3. What angle is congruent to $\angle I$ ?

Suppose $\triangle ADG \cong \triangle BCE$ .

1. What side is congruent to $\overline{CE}$ ?
2. What side is congruent to $\overline{DA}$ ?
3. What angle is congruent to $\angle G$ ?

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