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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.

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CK-12 Creating Congruence Statements

Watch the first part of this video.

James Sousa: Introduction to Congruent Triangles

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.

CK-12 Creating Congruence Statements

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?

Answers:

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} ?

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