Triangle Congruence from Rigid Motions - Geometry
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What is the definition of congruent segments using rigid motions?
What is the definition of congruent segments using rigid motions?
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A rigid motion maps one segment onto the other. Congruence means one maps to the other.
A rigid motion maps one segment onto the other. Congruence means one maps to the other.
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What does it mean to say $AB \cong DE$ in terms of length?
What does it mean to say $AB \cong DE$ in terms of length?
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$AB = DE$. Congruent segments have equal lengths.
$AB = DE$. Congruent segments have equal lengths.
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What does “map $\triangle ABC$ onto $\triangle DEF$” mean precisely?
What does “map $\triangle ABC$ onto $\triangle DEF$” mean precisely?
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A rigid motion sends $A \to D$, $B \to E$, and $C \to F$. Each vertex goes to its corresponding vertex.
A rigid motion sends $A \to D$, $B \to E$, and $C \to F$. Each vertex goes to its corresponding vertex.
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Find and correct the false claim: “A dilation is a rigid motion because it preserves angles.”
Find and correct the false claim: “A dilation is a rigid motion because it preserves angles.”
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Correct: A dilation is not rigid; it does not preserve distances. Dilation changes size, so it's not rigid.
Correct: A dilation is not rigid; it does not preserve distances. Dilation changes size, so it's not rigid.
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Identify the correct conclusion: If $AB = DE$ but no rigid motion maps $A \to D$ and $B \to E$, are the triangles necessarily congruent?
Identify the correct conclusion: If $AB = DE$ but no rigid motion maps $A \to D$ and $B \to E$, are the triangles necessarily congruent?
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No, one side length alone is not sufficient. Need all six corresponding parts to match.
No, one side length alone is not sufficient. Need all six corresponding parts to match.
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Which statement is equivalent to “corresponding angles are congruent” for triangles?
Which statement is equivalent to “corresponding angles are congruent” for triangles?
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All three pairs of corresponding angle measures are equal. Angle congruence means equal measures.
All three pairs of corresponding angle measures are equal. Angle congruence means equal measures.
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Which statement is equivalent to “corresponding sides are congruent” for triangles?
Which statement is equivalent to “corresponding sides are congruent” for triangles?
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All three pairs of corresponding side lengths are equal. Side congruence means equal lengths.
All three pairs of corresponding side lengths are equal. Side congruence means equal lengths.
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Which statement completes the definition: Triangles are congruent if and only if there exists a ______ mapping one to the other.
Which statement completes the definition: Triangles are congruent if and only if there exists a ______ mapping one to the other.
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Rigid motion. This completes the if-and-only-if definition.
Rigid motion. This completes the if-and-only-if definition.
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Identify the final check in a rigid-motion proof once $A \to D$ and $B \to E$ are matched.
Identify the final check in a rigid-motion proof once $A \to D$ and $B \to E$ are matched.
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Verify that $C$ maps to $F$ using preserved distances/angles. Check that all distances/angles are preserved.
Verify that $C$ maps to $F$ using preserved distances/angles. Check that all distances/angles are preserved.
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Identify the needed second step after $A$ is mapped to $D$ to match side $AB$ with $DE$.
Identify the needed second step after $A$ is mapped to $D$ to match side $AB$ with $DE$.
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Rotate about $D$ to align $\overrightarrow{AB}$ with $\overrightarrow{DE}$. Use rotation to match ray directions.
Rotate about $D$ to align $\overrightarrow{AB}$ with $\overrightarrow{DE}$. Use rotation to match ray directions.
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Identify the needed first step in a rigid-motion proof to map $\triangle ABC$ to $\triangle DEF$ when $A \leftrightarrow D$.
Identify the needed first step in a rigid-motion proof to map $\triangle ABC$ to $\triangle DEF$ when $A \leftrightarrow D$.
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Translate so that $A$ maps to $D$. Start by matching first corresponding vertices.
Translate so that $A$ maps to $D$. Start by matching first corresponding vertices.
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Find the correct correspondence: If $\triangle JKL \cong \triangle LMN$, which angle corresponds to $\angle K$?
Find the correct correspondence: If $\triangle JKL \cong \triangle LMN$, which angle corresponds to $\angle K$?
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$\angle M$. Second letters: $K \leftrightarrow M$.
$\angle M$. Second letters: $K \leftrightarrow M$.
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Find the correct correspondence: If $\triangle JKL \cong \triangle LMN$, which side corresponds to $JK$?
Find the correct correspondence: If $\triangle JKL \cong \triangle LMN$, which side corresponds to $JK$?
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$LM$. First letters: $J \leftrightarrow L$, second: $K \leftrightarrow M$.
$LM$. First letters: $J \leftrightarrow L$, second: $K \leftrightarrow M$.
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Identify whether a rotation can change a triangle’s angle measures.
Identify whether a rotation can change a triangle’s angle measures.
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No, a rotation preserves angle measures. Rotation is a rigid motion.
No, a rotation preserves angle measures. Rotation is a rigid motion.
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Identify whether a translation can change a triangle’s size.
Identify whether a translation can change a triangle’s size.
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No, a translation preserves size. Translation is a rigid motion.
No, a translation preserves size. Translation is a rigid motion.
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Identify the rigid motion that can change a triangle’s orientation (clockwise to counterclockwise).
Identify the rigid motion that can change a triangle’s orientation (clockwise to counterclockwise).
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Reflection. Only reflection can flip orientation.
Reflection. Only reflection can flip orientation.
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Choose the correct statement: If $AB = DE$, $BC = EF$, $CA = FD$, and corresponding angles match, then $\triangle ABC$ and $\triangle DEF$ are what?
Choose the correct statement: If $AB = DE$, $BC = EF$, $CA = FD$, and corresponding angles match, then $\triangle ABC$ and $\triangle DEF$ are what?
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Congruent. All corresponding parts match means congruence.
Congruent. All corresponding parts match means congruence.
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Identify the correct conclusion: If a rigid motion maps $\triangle ABC$ onto $\triangle DEF$, then $m\angle A$ and $m\angle D$ are what?
Identify the correct conclusion: If a rigid motion maps $\triangle ABC$ onto $\triangle DEF$, then $m\angle A$ and $m\angle D$ are what?
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Equal. Rigid motions preserve angle measures.
Equal. Rigid motions preserve angle measures.
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Find the missing match: If $\triangle ABC \cong \triangle DEF$ and $\angle B \leftrightarrow \angle E$, then $\angle C \leftrightarrow$ what?
Find the missing match: If $\triangle ABC \cong \triangle DEF$ and $\angle B \leftrightarrow \angle E$, then $\angle C \leftrightarrow$ what?
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$\angle F$. Third letters correspond in the statement.
$\angle F$. Third letters correspond in the statement.
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Which congruence is guaranteed if $\triangle ABC \cong \triangle DEF$? Choose $AB$ or $AC$ for matching $DF$.
Which congruence is guaranteed if $\triangle ABC \cong \triangle DEF$? Choose $AB$ or $AC$ for matching $DF$.
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$AC \cong DF$. First and third letters: $A \leftrightarrow D$, $C \leftrightarrow F$.
$AC \cong DF$. First and third letters: $A \leftrightarrow D$, $C \leftrightarrow F$.
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Identify the angle in $\triangle ABC$ that corresponds to $\angle F$ if $\triangle ABC \cong \triangle DEF$.
Identify the angle in $\triangle ABC$ that corresponds to $\angle F$ if $\triangle ABC \cong \triangle DEF$.
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$\angle C$. Third letters correspond: $C \leftrightarrow F$.
$\angle C$. Third letters correspond: $C \leftrightarrow F$.
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Identify the side in $\triangle DEF$ that corresponds to side $BC$ if $\triangle ABC \cong \triangle DEF$.
Identify the side in $\triangle DEF$ that corresponds to side $BC$ if $\triangle ABC \cong \triangle DEF$.
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$EF$. Second letters correspond: $B \leftrightarrow E$, so $C \leftrightarrow F$.
$EF$. Second letters correspond: $B \leftrightarrow E$, so $C \leftrightarrow F$.
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Identify the correspondence implied by $\triangle PQR \cong \triangle RST$.
Identify the correspondence implied by $\triangle PQR \cong \triangle RST$.
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$P \leftrightarrow R$, $Q \leftrightarrow S$, $R \leftrightarrow T$. Order shows $P \to R$, $Q \to S$, $R \to T$.
$P \leftrightarrow R$, $Q \leftrightarrow S$, $R \leftrightarrow T$. Order shows $P \to R$, $Q \to S$, $R \to T$.
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Which statement is equivalent to $\triangle ABC \cong \triangle DEF$ by definition?
Which statement is equivalent to $\triangle ABC \cong \triangle DEF$ by definition?
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There exists a rigid motion sending $\triangle ABC$ onto $\triangle DEF$. This is the definition of triangle congruence.
There exists a rigid motion sending $\triangle ABC$ onto $\triangle DEF$. This is the definition of triangle congruence.
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If a rigid motion sends rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ to $\overrightarrow{ED}$ and $\overrightarrow{EF}$, what follows?
If a rigid motion sends rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ to $\overrightarrow{ED}$ and $\overrightarrow{EF}$, what follows?
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$\angle ABC \cong \angle DEF$. Rigid motions map angles to congruent angles.
$\angle ABC \cong \angle DEF$. Rigid motions map angles to congruent angles.
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If a rigid motion sends $A \to D$ and $B \to E$, what must happen to segment $AB$?
If a rigid motion sends $A \to D$ and $B \to E$, what must happen to segment $AB$?
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It maps to segment $DE$ with $AB = DE$. Rigid motions preserve segment length.
It maps to segment $DE$ with $AB = DE$. Rigid motions preserve segment length.
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Which fact justifies that a rigid motion preserves $m\angle A = m\angle B$ if it holds before the motion?
Which fact justifies that a rigid motion preserves $m\angle A = m\angle B$ if it holds before the motion?
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Rigid motions preserve angle measures. Angle measure is an invariant under rigid motions.
Rigid motions preserve angle measures. Angle measure is an invariant under rigid motions.
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Which fact justifies that a rigid motion preserves $AB = BC$ if it holds before the motion?
Which fact justifies that a rigid motion preserves $AB = BC$ if it holds before the motion?
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Rigid motions preserve distances. Distance is an invariant under rigid motions.
Rigid motions preserve distances. Distance is an invariant under rigid motions.
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Identify the correct correspondence if $\triangle ABC \cong \triangle A'B'C'$ is written in order.
Identify the correct correspondence if $\triangle ABC \cong \triangle A'B'C'$ is written in order.
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$A \leftrightarrow A'$, $B \leftrightarrow B'$, $C \leftrightarrow C'$. First letters match, second letters match, etc.
$A \leftrightarrow A'$, $B \leftrightarrow B'$, $C \leftrightarrow C'$. First letters match, second letters match, etc.
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What is the condition “corresponding sides and corresponding angles are congruent” called?
What is the condition “corresponding sides and corresponding angles are congruent” called?
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CPCTC conditions (all corresponding parts are congruent). All six parts must match for congruence.
CPCTC conditions (all corresponding parts are congruent). All six parts must match for congruence.
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