Theorems about Lines and Angles - Geometry
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What property of segment length is preserved by any rigid motion?
What property of segment length is preserved by any rigid motion?
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If $AB = CD$, then $A'B' = C'D'$. Distances remain unchanged under rigid motions.
If $AB = CD$, then $A'B' = C'D'$. Distances remain unchanged under rigid motions.
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Which congruence criterion uses two pairs of corresponding angles and the included side?
Which congruence criterion uses two pairs of corresponding angles and the included side?
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ASA. Uses two angles with their included side.
ASA. Uses two angles with their included side.
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Identify the correct conclusion: If a rigid motion maps $\triangle ABC$ onto $\triangle DEF$, then $AB$ and $DE$ are what?
Identify the correct conclusion: If a rigid motion maps $\triangle ABC$ onto $\triangle DEF$, then $AB$ and $DE$ are what?
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Congruent segments, so $AB = DE$. Rigid motions preserve all distances between points.
Congruent segments, so $AB = DE$. Rigid motions preserve all distances between points.
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Identify the correct conclusion: If a rigid motion maps $\angle A$ to $\angle D$, then their measures satisfy what equation?
Identify the correct conclusion: If a rigid motion maps $\angle A$ to $\angle D$, then their measures satisfy what equation?
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$m\angle A = m\angle D$. Rigid motions preserve all angle measures.
$m\angle A = m\angle D$. Rigid motions preserve all angle measures.
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What is the rigid-motion reason that SSS can produce at most two mirror-image triangles?
What is the rigid-motion reason that SSS can produce at most two mirror-image triangles?
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A reflection can swap the two circle-intersection positions. Two circles intersect at most at two points.
A reflection can swap the two circle-intersection positions. Two circles intersect at most at two points.
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In SAS, if two triangles share the same included angle and adjacent side lengths, what is fixed about the third vertex position?
In SAS, if two triangles share the same included angle and adjacent side lengths, what is fixed about the third vertex position?
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It is uniquely determined up to reflection. Only reflection can change orientation.
It is uniquely determined up to reflection. Only reflection can change orientation.
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In ASA, if $\overline{AB}$ is fixed and both endpoint angles are fixed, what is fixed about the third vertex?
In ASA, if $\overline{AB}$ is fixed and both endpoint angles are fixed, what is fixed about the third vertex?
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It is the intersection of the two determined rays. Two rays from fixed points have unique intersection.
It is the intersection of the two determined rays. Two rays from fixed points have unique intersection.
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Choose the correct included angle for sides $AB$ and $BC$ in $\triangle ABC$: $\angle A$, $\angle B$, or $\angle C$.
Choose the correct included angle for sides $AB$ and $BC$ in $\triangle ABC$: $\angle A$, $\angle B$, or $\angle C$.
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$\angle B$. Vertex $B$ is where sides $AB$ and $BC$ meet.
$\angle B$. Vertex $B$ is where sides $AB$ and $BC$ meet.
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Choose the correct included side for angles $\angle B$ and $\angle C$ in $\triangle ABC$: $\overline{AB}$, $\overline{BC}$, or $\overline{AC}$.
Choose the correct included side for angles $\angle B$ and $\angle C$ in $\triangle ABC$: $\overline{AB}$, $\overline{BC}$, or $\overline{AC}$.
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$\overline{BC}$. Side $BC$ connects angles $\angle B$ and $\angle C$.
$\overline{BC}$. Side $BC$ connects angles $\angle B$ and $\angle C$.
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What does SSS stand for in triangle congruence criteria?
What does SSS stand for in triangle congruence criteria?
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Side-Side-Side. All three pairs of corresponding sides are equal.
Side-Side-Side. All three pairs of corresponding sides are equal.
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What is the congruence criterion if $\angle A = \angle D$, $AC = DF$, and $\angle C = \angle F$?
What is the congruence criterion if $\angle A = \angle D$, $AC = DF$, and $\angle C = \angle F$?
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ASA (the included side is $\overline{AC}$). Side $AC$ is between the two given angles.
ASA (the included side is $\overline{AC}$). Side $AC$ is between the two given angles.
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What is the congruence criterion if $AB = DE$, $BC = EF$, and $\angle A = \angle D$?
What is the congruence criterion if $AB = DE$, $BC = EF$, and $\angle A = \angle D$?
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None; this is SSA and does not guarantee congruence. The angle is not between the two given sides.
None; this is SSA and does not guarantee congruence. The angle is not between the two given sides.
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What property of segment length is preserved by any rigid motion?
What property of segment length is preserved by any rigid motion?
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If $AB = CD$, then $A'B' = C'D'$. Distances remain unchanged under rigid motions.
If $AB = CD$, then $A'B' = C'D'$. Distances remain unchanged under rigid motions.
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What three transformations are typically used as rigid motions in Geometry?
What three transformations are typically used as rigid motions in Geometry?
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Translations, rotations, and reflections. These preserve distances and angles without distortion.
Translations, rotations, and reflections. These preserve distances and angles without distortion.
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What is the congruence criterion if $AB = DE$, $\angle B = \angle E$, and $BC = EF$?
What is the congruence criterion if $AB = DE$, $\angle B = \angle E$, and $BC = EF$?
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SAS (the included angle is $\angle B$). Angle $\angle B$ is between the two given sides.
SAS (the included angle is $\angle B$). Angle $\angle B$ is between the two given sides.
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What does SAS stand for in triangle congruence criteria?
What does SAS stand for in triangle congruence criteria?
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Side-Angle-Side. Two sides and the angle between them are equal.
Side-Angle-Side. Two sides and the angle between them are equal.
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What does ASA stand for in triangle congruence criteria?
What does ASA stand for in triangle congruence criteria?
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Angle-Side-Angle. Two angles and the side between them are equal.
Angle-Side-Angle. Two angles and the side between them are equal.
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Which angle is required in SAS: the included angle or a non-included angle?
Which angle is required in SAS: the included angle or a non-included angle?
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The included angle between the two sides. The angle must be between the two given sides.
The included angle between the two sides. The angle must be between the two given sides.
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Which side is required in ASA: the included side or a non-included side?
Which side is required in ASA: the included side or a non-included side?
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The included side between the two angles. The side must be between the two given angles.
The included side between the two angles. The side must be between the two given angles.
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What is the included angle between sides $AB$ and $AC$ in $\triangle ABC$?
What is the included angle between sides $AB$ and $AC$ in $\triangle ABC$?
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$\angle BAC$. The angle at vertex $A$ is between sides $AB$ and $AC$.
$\angle BAC$. The angle at vertex $A$ is between sides $AB$ and $AC$.
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What is the included side between angles $\angle A$ and $\angle B$ in $\triangle ABC$?
What is the included side between angles $\angle A$ and $\angle B$ in $\triangle ABC$?
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$\overline{AB}$. Side $AB$ connects vertices $A$ and $B$.
$\overline{AB}$. Side $AB$ connects vertices $A$ and $B$.
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In SAS, if $AB = DE$, $AC = DF$, and $\angle A = \angle D$, what is the conclusion?
In SAS, if $AB = DE$, $AC = DF$, and $\angle A = \angle D$, what is the conclusion?
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$\triangle ABC \cong \triangle DEF$ by SAS. Two sides and included angle establish congruence.
$\triangle ABC \cong \triangle DEF$ by SAS. Two sides and included angle establish congruence.
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In SSS, if $AB = DE$, $BC = EF$, and $AC = DF$, what is the conclusion?
In SSS, if $AB = DE$, $BC = EF$, and $AC = DF$, what is the conclusion?
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$\triangle ABC \cong \triangle DEF$ by SSS. All three corresponding sides are equal.
$\triangle ABC \cong \triangle DEF$ by SSS. All three corresponding sides are equal.
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In ASA, if $\angle A = \angle D$, $AB = DE$, and $\angle B = \angle E$, what is the conclusion?
In ASA, if $\angle A = \angle D$, $AB = DE$, and $\angle B = \angle E$, what is the conclusion?
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$\triangle ABC \cong \triangle DEF$ by ASA. Two angles and included side establish congruence.
$\triangle ABC \cong \triangle DEF$ by ASA. Two angles and included side establish congruence.
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Which congruence criterion uses three pairs of corresponding sides?
Which congruence criterion uses three pairs of corresponding sides?
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SSS. Uses all three pairs of corresponding sides.
SSS. Uses all three pairs of corresponding sides.
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Which congruence criterion uses two pairs of corresponding sides and the included angle?
Which congruence criterion uses two pairs of corresponding sides and the included angle?
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SAS. Uses two sides with their included angle.
SAS. Uses two sides with their included angle.
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What is the key rigid-motion idea behind proving SSS congruence?
What is the key rigid-motion idea behind proving SSS congruence?
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Match one side, then the third vertex is fixed by two distances. Two circle intersections determine the third vertex.
Match one side, then the third vertex is fixed by two distances. Two circle intersections determine the third vertex.
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What is the key rigid-motion idea behind proving SAS congruence?
What is the key rigid-motion idea behind proving SAS congruence?
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Match a side and included angle; the second side fixes the third vertex. The angle direction constrains the third vertex location.
Match a side and included angle; the second side fixes the third vertex. The angle direction constrains the third vertex location.
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What is the key rigid-motion idea behind proving ASA congruence?
What is the key rigid-motion idea behind proving ASA congruence?
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Match the included side; the two angles fix the rays to the third vertex. Two angle directions intersect at the third vertex.
Match the included side; the two angles fix the rays to the third vertex. Two angle directions intersect at the third vertex.
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What rigid motions can be used to map a segment $\overline{AB}$ onto $\overline{DE}$?
What rigid motions can be used to map a segment $\overline{AB}$ onto $\overline{DE}$?
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A translation and rotation, possibly followed by a reflection. These transformations preserve all distances.
A translation and rotation, possibly followed by a reflection. These transformations preserve all distances.
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