Formal Geometric Constructions - Geometry
Card 1 of 30
What angle congruence follows from $AB \parallel CD$ with transversal $AC$ in $ABCD$?
What angle congruence follows from $AB \parallel CD$ with transversal $AC$ in $ABCD$?
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$\angle BAC \cong \angle DCA$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
$\angle BAC \cong \angle DCA$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
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In parallelogram $ABCD$, diagonals meet at $E$. If $AE = 2x + 1$ and $CE = 11$, what is $x$?
In parallelogram $ABCD$, diagonals meet at $E$. If $AE = 2x + 1$ and $CE = 11$, what is $x$?
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$5$. Set $2x + 1 = 11$ and solve: $2x = 10$, so $x = 5$.
$5$. Set $2x + 1 = 11$ and solve: $2x = 10$, so $x = 5$.
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What is the theorem: if a parallelogram has congruent diagonals, then what type is it?
What is the theorem: if a parallelogram has congruent diagonals, then what type is it?
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It is a rectangle. Congruent diagonals characterize rectangles among parallelograms.
It is a rectangle. Congruent diagonals characterize rectangles among parallelograms.
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Identify the conclusion: if a parallelogram has $AC \cong BD$, what is the most specific type?
Identify the conclusion: if a parallelogram has $AC \cong BD$, what is the most specific type?
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Rectangle. Congruent diagonals characterize rectangles among parallelograms.
Rectangle. Congruent diagonals characterize rectangles among parallelograms.
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What is the converse theorem: if both pairs of opposite sides of a quadrilateral are congruent, then what?
What is the converse theorem: if both pairs of opposite sides of a quadrilateral are congruent, then what?
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The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
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What is the definition of a parallelogram in terms of parallel sides?
What is the definition of a parallelogram in terms of parallel sides?
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A quadrilateral with both pairs of opposite sides parallel. This is the basic definition requiring two pairs of parallel opposite sides.
A quadrilateral with both pairs of opposite sides parallel. This is the basic definition requiring two pairs of parallel opposite sides.
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What theorem states the relationship between opposite sides in any parallelogram $ABCD$?
What theorem states the relationship between opposite sides in any parallelogram $ABCD$?
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$AB \cong CD$ and $BC \cong AD$. Opposite sides in any parallelogram are always congruent.
$AB \cong CD$ and $BC \cong AD$. Opposite sides in any parallelogram are always congruent.
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What theorem states the relationship between opposite angles in any parallelogram $ABCD$?
What theorem states the relationship between opposite angles in any parallelogram $ABCD$?
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$\angle A \cong \angle C$ and $\angle B \cong \angle D$. Opposite angles in any parallelogram are always congruent.
$\angle A \cong \angle C$ and $\angle B \cong \angle D$. Opposite angles in any parallelogram are always congruent.
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What theorem describes how the diagonals behave in a parallelogram $ABCD$?
What theorem describes how the diagonals behave in a parallelogram $ABCD$?
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The diagonals bisect each other. The diagonals intersect at their midpoints.
The diagonals bisect each other. The diagonals intersect at their midpoints.
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What does it mean to say the diagonals of $ABCD$ bisect each other at $E$?
What does it mean to say the diagonals of $ABCD$ bisect each other at $E$?
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$AE \cong CE$ and $BE \cong DE$. Each diagonal is split into two equal segments at point $E$.
$AE \cong CE$ and $BE \cong DE$. Each diagonal is split into two equal segments at point $E$.
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What angle relationship is always true for consecutive angles in a parallelogram?
What angle relationship is always true for consecutive angles in a parallelogram?
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Consecutive angles are supplementary. Adjacent angles in a parallelogram always add to $180°$.
Consecutive angles are supplementary. Adjacent angles in a parallelogram always add to $180°$.
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In parallelogram $ABCD$, what is the equation relating consecutive angles $\angle A$ and $\angle B$?
In parallelogram $ABCD$, what is the equation relating consecutive angles $\angle A$ and $\angle B$?
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$m\angle A + m\angle B = 180^\circ$. Consecutive angles are supplementary in any parallelogram.
$m\angle A + m\angle B = 180^\circ$. Consecutive angles are supplementary in any parallelogram.
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In parallelogram $ABCD$, what is the equation relating consecutive angles $\angle B$ and $\angle C$?
In parallelogram $ABCD$, what is the equation relating consecutive angles $\angle B$ and $\angle C$?
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$m\angle B + m\angle C = 180^\circ$. Consecutive angles are supplementary in any parallelogram.
$m\angle B + m\angle C = 180^\circ$. Consecutive angles are supplementary in any parallelogram.
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What is a standard proof strategy to show opposite sides of a parallelogram are congruent?
What is a standard proof strategy to show opposite sides of a parallelogram are congruent?
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Prove two triangles are congruent using diagonal $AC$ or $BD$. Use a diagonal to create congruent triangles, then apply CPCTC.
Prove two triangles are congruent using diagonal $AC$ or $BD$. Use a diagonal to create congruent triangles, then apply CPCTC.
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What is a standard proof strategy to show diagonals bisect each other in parallelogram $ABCD$?
What is a standard proof strategy to show diagonals bisect each other in parallelogram $ABCD$?
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Prove $\triangle AEB \cong \triangle CED$ (or similar). Use vertical angles and parallel line properties to prove triangle congruence.
Prove $\triangle AEB \cong \triangle CED$ (or similar). Use vertical angles and parallel line properties to prove triangle congruence.
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What congruence reason lets you conclude parts are equal after proving triangles congruent?
What congruence reason lets you conclude parts are equal after proving triangles congruent?
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CPCTC. Corresponding Parts of Congruent Triangles are Congruent.
CPCTC. Corresponding Parts of Congruent Triangles are Congruent.
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Identify the parallel-line angle fact used when $AB \parallel CD$ and $AC$ is a transversal.
Identify the parallel-line angle fact used when $AB \parallel CD$ and $AC$ is a transversal.
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Alternate interior angles are congruent. When parallel lines are cut by a transversal, alternate interior angles are equal.
Alternate interior angles are congruent. When parallel lines are cut by a transversal, alternate interior angles are equal.
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What angle congruence follows from $AB \parallel CD$ with transversal $BD$ in $ABCD$?
What angle congruence follows from $AB \parallel CD$ with transversal $BD$ in $ABCD$?
Tap to reveal answer
$\angle ABD \cong \angle CDB$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
$\angle ABD \cong \angle CDB$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
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What angle congruence follows from $AD \parallel BC$ with transversal $BD$ in $ABCD$?
What angle congruence follows from $AD \parallel BC$ with transversal $BD$ in $ABCD$?
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$\angle ADB \cong \angle CBD$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
$\angle ADB \cong \angle CBD$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
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What angle congruence follows from $AB \parallel CD$ with transversal $AC$ in $ABCD$?
What angle congruence follows from $AB \parallel CD$ with transversal $AC$ in $ABCD$?
Tap to reveal answer
$\angle BAC \cong \angle DCA$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
$\angle BAC \cong \angle DCA$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
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What angle congruence follows from $AD \parallel BC$ with transversal $AC$ in $ABCD$?
What angle congruence follows from $AD \parallel BC$ with transversal $AC$ in $ABCD$?
Tap to reveal answer
$\angle BCA \cong \angle CAD$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
$\angle BCA \cong \angle CAD$. Alternate interior angles are congruent when parallel lines are cut by a transversal.
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In parallelogram $ABCD$, which diagonal is commonly drawn to prove $AB \cong CD$?
In parallelogram $ABCD$, which diagonal is commonly drawn to prove $AB \cong CD$?
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Draw diagonal $BD$ or $AC$. Either diagonal works to create the necessary congruent triangles.
Draw diagonal $BD$ or $AC$. Either diagonal works to create the necessary congruent triangles.
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What is the theorem: if a parallelogram has congruent diagonals, then what type is it?
What is the theorem: if a parallelogram has congruent diagonals, then what type is it?
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It is a rectangle. Congruent diagonals characterize rectangles among parallelograms.
It is a rectangle. Congruent diagonals characterize rectangles among parallelograms.
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What is the minimum angle information needed to prove a parallelogram is a rectangle?
What is the minimum angle information needed to prove a parallelogram is a rectangle?
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One right angle, for example $m\angle A = 90^\circ$. One right angle forces all angles to be $90°$ in a parallelogram.
One right angle, for example $m\angle A = 90^\circ$. One right angle forces all angles to be $90°$ in a parallelogram.
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In a parallelogram, what is the relationship between $\angle A$ and $\angle C$?
In a parallelogram, what is the relationship between $\angle A$ and $\angle C$?
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$\angle A \cong \angle C$. Opposite angles are always congruent in parallelograms.
$\angle A \cong \angle C$. Opposite angles are always congruent in parallelograms.
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What is the converse theorem: if one pair of opposite sides is both parallel and congruent, then what?
What is the converse theorem: if one pair of opposite sides is both parallel and congruent, then what?
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The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
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What is the converse theorem: if the diagonals of a quadrilateral bisect each other, then what?
What is the converse theorem: if the diagonals of a quadrilateral bisect each other, then what?
Tap to reveal answer
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
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What is the converse theorem: if both pairs of opposite angles of a quadrilateral are congruent, then what?
What is the converse theorem: if both pairs of opposite angles of a quadrilateral are congruent, then what?
Tap to reveal answer
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
The quadrilateral is a parallelogram. This is one of the standard converses for proving parallelograms.
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What is the definition of a rectangle in terms of angles?
What is the definition of a rectangle in terms of angles?
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A quadrilateral with four right angles. A rectangle is defined by having all four angles equal to $90°$.
A quadrilateral with four right angles. A rectangle is defined by having all four angles equal to $90°$.
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What statement connects rectangles and parallelograms in a standard classification hierarchy?
What statement connects rectangles and parallelograms in a standard classification hierarchy?
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Every rectangle is a parallelogram. Rectangles inherit all parallelogram properties plus additional ones.
Every rectangle is a parallelogram. Rectangles inherit all parallelogram properties plus additional ones.
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