Other Quadrilaterals - GMAT Quantitative
Card 0 of 296
Is parallelogram
a rectangle?
Statement 1: 
Statement 2: 
Is parallelogram a rectangle?
Statement 1:
Statement 2:
Any two consecutive angles of a parallelogram are supplementary, so if one angle has measure
, all angles can be proven to have measure
. This is the definition of a rectangle. Statement 1 therefore proves the parallelogram to be a rectangle.
Also, a parallelogram is a rectangle if and only if its diagonals are congruent, which is what Statement 2 asserts.
From either statement, it follows that parallelogram
is a rectangle.
Any two consecutive angles of a parallelogram are supplementary, so if one angle has measure , all angles can be proven to have measure
. This is the definition of a rectangle. Statement 1 therefore proves the parallelogram to be a rectangle.
Also, a parallelogram is a rectangle if and only if its diagonals are congruent, which is what Statement 2 asserts.
From either statement, it follows that parallelogram is a rectangle.
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Given a quadrilateral
, can a circle be circumscribed about it?
Statement 1: Quadrilateral
is not a rectangle.
Statement 2: 
Given a quadrilateral , can a circle be circumscribed about it?
Statement 1: Quadrilateral is not a rectangle.
Statement 2:
A circle can be circumscribed about a quadrilateral if and only if both pairs of opposite angles are supplementary. This is not proved or disproved by Statement 1 alone:
Case 1: 
This is not a rectangle, and opposite angles are supplementary, so a circle can be constructed to circumscribe the quadrilateral.
Case 2: 
This is not a rectangle, and opposite angles are not supplementary, so a circle cannot be constructed to circumscribe the quadrilateral.
From Statement 2, however, it follows that a two opposite angles are not a supplementary pair, so a circle cannot be circumscribed about it.
A circle can be circumscribed about a quadrilateral if and only if both pairs of opposite angles are supplementary. This is not proved or disproved by Statement 1 alone:
Case 1:
This is not a rectangle, and opposite angles are supplementary, so a circle can be constructed to circumscribe the quadrilateral.
Case 2:
This is not a rectangle, and opposite angles are not supplementary, so a circle cannot be constructed to circumscribe the quadrilateral.
From Statement 2, however, it follows that a two opposite angles are not a supplementary pair, so a circle cannot be circumscribed about it.
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Given a quadrilateral
, can a circle be circumscribed about it?
Statement 1: Quadrilateral
is an isosceles trapezoid.
Statement 2: 
Given a quadrilateral , can a circle be circumscribed about it?
Statement 1: Quadrilateral is an isosceles trapezoid.
Statement 2:
A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary.
An isosceles trapezoid has this characteristic. Assume without loss of generality that
and
are the pairs of base angles.
Then, since base angles are congruent,
and
. Since the bases of a trapezoid are parallel, from the Same-Side Interior Angles Theorem,
and
are supplementary, and, subsequently, so are
and
, as well as
and
.
If
, then
and
form a supplementary pair, as their measures total
; since the measures of the angles of a quadrilateral total
, the measures of
and
also total
, making them supplementary as well.
Therefore, it follows from either statement that both pairs of opposite angles are supplementary, and that a circle can be circumscribed about the quadrilateral.
A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary.
An isosceles trapezoid has this characteristic. Assume without loss of generality that and
are the pairs of base angles.
Then, since base angles are congruent, and
. Since the bases of a trapezoid are parallel, from the Same-Side Interior Angles Theorem,
and
are supplementary, and, subsequently, so are
and
, as well as
and
.
If , then
and
form a supplementary pair, as their measures total
; since the measures of the angles of a quadrilateral total
, the measures of
and
also total
, making them supplementary as well.
Therefore, it follows from either statement that both pairs of opposite angles are supplementary, and that a circle can be circumscribed about the quadrilateral.
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Given Parallelogram
.
True or false: 
Statement 1: 
Statement 2: 
Given Parallelogram .
True or false:
Statement 1:
Statement 2:
and
, the diagonals of Parallelogram
, are perpendicular if and only if Parallelogram
is also a rhombus.
Opposite sides of a parallelogram are congruent, so if Statement 1 is assumed,
. Parallelogram
a rhombus; subsequently,
.
The angle measures are irrelevant, so Statement 2 is unhelpful.
and
, the diagonals of Parallelogram
, are perpendicular if and only if Parallelogram
is also a rhombus.
Opposite sides of a parallelogram are congruent, so if Statement 1 is assumed, . Parallelogram
a rhombus; subsequently,
.
The angle measures are irrelevant, so Statement 2 is unhelpful.
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Are the diagonals of Quadrilateral
perpendicular?
(a) 
(b) 
Are the diagonals of Quadrilateral perpendicular?
(a)
(b)
For the diagonals of a quadrilateral to be perpendicular, the quadrilateral must be a kite or a rhombus - in either case, there must be two pairs of adjacent congruent sides. Neither statement alone proves this, but both statements together do.
For the diagonals of a quadrilateral to be perpendicular, the quadrilateral must be a kite or a rhombus - in either case, there must be two pairs of adjacent congruent sides. Neither statement alone proves this, but both statements together do.
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Quadrilateral
is inscribed in a circle.
What is
?
Statement 1: 
Statement 2: 
Quadrilateral is inscribed in a circle.
What is ?
Statement 1:
Statement 2:
From Statement 1 alone, we can calculate
, since two opposite angles of a quadrilateral inscribed inside a circle are supplementary:

From Statement 2 alone, we can calculate
, since the degree measure of an inscribed angle of a circle is half that of the arc it intersects:

From Statement 1 alone, we can calculate , since two opposite angles of a quadrilateral inscribed inside a circle are supplementary:
From Statement 2 alone, we can calculate , since the degree measure of an inscribed angle of a circle is half that of the arc it intersects:
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The above shows Parallelogram
. Is it a rectangle?
Statement 1: 
Statement 2: 
The above shows Parallelogram . Is it a rectangle?
Statement 1:
Statement 2:
To prove that Parallelogram
is also a rectangle, we need to prove that any one of its angles is a right angle.
If we assume Statement 1 alone, that
, then, since
and
form a linear pair,
is right.
If we assume Statement 2 alone, that
, it follows from the converse of the Pythagorean Theorem that
is a right triangle with right angle
.
Either way, we have proved that the parallelogram is a rectangle.
To prove that Parallelogram is also a rectangle, we need to prove that any one of its angles is a right angle.
If we assume Statement 1 alone, that , then, since
and
form a linear pair,
is right.
If we assume Statement 2 alone, that , it follows from the converse of the Pythagorean Theorem that
is a right triangle with right angle
.
Either way, we have proved that the parallelogram is a rectangle.
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Refer to the above figure. You are given that Polygon
is a parallelogram but not that it is a rectangle. Is it a rectangle?
Statement 1: 
Statement 2:
and
are complementary angles.
Refer to the above figure. You are given that Polygon is a parallelogram but not that it is a rectangle. Is it a rectangle?
Statement 1:
Statement 2: and
are complementary angles.
It is necessary and sufficient to prove that one of the angles of the parallelogram is a right angle.
Assume Statement 1 alone - that
.
and
are supplementary, since they are same-side interior angles of parallel lines. Since
,
is also supplementary to
. But as corresponding angles of parallel lines,
. Two angles that are conruent and supplementary are both right angles, so
is a right angle.
Assume Statement 2 alone - that
and
are complementary angles, or, equivalently,
. Since the angles of a triangle have measures that add up to
, the third angle of
, which is
, measures
, and is a right angle.
Either statement alone proves
a right angle and subsequently proves
a rectangle.
It is necessary and sufficient to prove that one of the angles of the parallelogram is a right angle.
Assume Statement 1 alone - that .
and
are supplementary, since they are same-side interior angles of parallel lines. Since
,
is also supplementary to
. But as corresponding angles of parallel lines,
. Two angles that are conruent and supplementary are both right angles, so
is a right angle.
Assume Statement 2 alone - that and
are complementary angles, or, equivalently,
. Since the angles of a triangle have measures that add up to
, the third angle of
, which is
, measures
, and is a right angle.
Either statement alone proves a right angle and subsequently proves
a rectangle.
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True or false: Quadrilateral
is a rectangle.
Statement 1: 
Statement 2: 
True or false: Quadrilateral is a rectangle.
Statement 1:
Statement 2:
Assume Statement 1 alone. By congruence,
and
, making Quadrilateral
a parallelogram. However, no clue is given to whether any angles are right or not, so whether the quadrilateral is a rectangle or not remains open.
Assume Statement 2 alone. By congruence, opposite sides
, but no clue is provided as to the lengths of opposite sides
and
. Also,
, but no clue is provided as to whether the angles are right. A rectangle would have both characteristics, but so would an isosceles trapezoid with legs
and
.
Assume both statements are true. Quadrilateral
is a parallelogram as a consequence of Statement 1. Since
and
are consecutive angles of the parallelogram, they are supplementary, but they are also congruent as a consequence of Statement 2. Therefore, they are right angles, and a parallelogram with right angles is a rectangle.
Assume Statement 1 alone. By congruence, and
, making Quadrilateral
a parallelogram. However, no clue is given to whether any angles are right or not, so whether the quadrilateral is a rectangle or not remains open.
Assume Statement 2 alone. By congruence, opposite sides , but no clue is provided as to the lengths of opposite sides
and
. Also,
, but no clue is provided as to whether the angles are right. A rectangle would have both characteristics, but so would an isosceles trapezoid with legs
and
.
Assume both statements are true. Quadrilateral is a parallelogram as a consequence of Statement 1. Since
and
are consecutive angles of the parallelogram, they are supplementary, but they are also congruent as a consequence of Statement 2. Therefore, they are right angles, and a parallelogram with right angles is a rectangle.
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True or false: Quadrilateral
is a rectangle.
Statement 1:
and
are right angles.
Statement 2: 
True or false: Quadrilateral is a rectangle.
Statement 1: and
are right angles.
Statement 2:
Statement 1 alone is insufficient to answer the question; A quadrilateral in which
and
are right angles,
, and
fits the statement, as well as a rectangle, which by defintion has four right angles.
Statement 2 alone is insufficient as well, as a parallelogram with acute and obtuse angles, as well as a rectangle, fits the description.
Assume both statements, and construct diagonal
to form two triangles
and
. By Statement 1, both triangles are right with congruent legs
, and congruent hypotenuses, both being the same segment
. By the Hypotenuse Leg Theorem,
. By congruence,
. The quadrilateral, having two sets of congruent opposite sides, is a parallelogram; a parallelogram with right angles is a rectangle.
Statement 1 alone is insufficient to answer the question; A quadrilateral in which and
are right angles,
, and
fits the statement, as well as a rectangle, which by defintion has four right angles.
Statement 2 alone is insufficient as well, as a parallelogram with acute and obtuse angles, as well as a rectangle, fits the description.
Assume both statements, and construct diagonal to form two triangles
and
. By Statement 1, both triangles are right with congruent legs
, and congruent hypotenuses, both being the same segment
. By the Hypotenuse Leg Theorem,
. By congruence,
. The quadrilateral, having two sets of congruent opposite sides, is a parallelogram; a parallelogram with right angles is a rectangle.
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What is the perimeter of Rhombus
?
Statement 1:
has perimeter
.
Statement 2:
is equilateral.
What is the perimeter of Rhombus ?
Statement 1: has perimeter
.
Statement 2: is equilateral.
Each diagonal divides Rhombus
into two triangles, both isosceles.
Statement 1 alone establishes the perimeter of one such triangle. However, it does not make it clear what equal side lengths
and
and diagonal length
are. For example,
fits the perimeter, but so does
.
Statement 2 alone gives no information about the actual lengths of the sides.
Assume both statements are true. Since
is equilateral,
. It follows that
, and
. Also, the diagonals of a rhombus bisect their angles and are each other's perpendicular bisectors, so the rhombus, with their diagonals, is given below.

has perimeter
, which means that




Since
is known to be a
triangle, the proportions of the side lengths are known; along with the above equation,
, and, subsequently, the perimeter, can be determined.
Each diagonal divides Rhombus into two triangles, both isosceles.
Statement 1 alone establishes the perimeter of one such triangle. However, it does not make it clear what equal side lengths and
and diagonal length
are. For example,
fits the perimeter, but so does
.
Statement 2 alone gives no information about the actual lengths of the sides.
Assume both statements are true. Since is equilateral,
. It follows that
, and
. Also, the diagonals of a rhombus bisect their angles and are each other's perpendicular bisectors, so the rhombus, with their diagonals, is given below.
has perimeter
, which means that
Since is known to be a
triangle, the proportions of the side lengths are known; along with the above equation,
, and, subsequently, the perimeter, can be determined.
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What is the perimeter of quadrilateral
?
(1) Diagonal
and
are perpendicular with midpoint
.
(2) 
What is the perimeter of quadrilateral ?
(1) Diagonal and
are perpendicular with midpoint
.
(2)
To find the perimeter of the quadrilateral, we need to know whether it is of a special type of quadrilaterals and we need to know the length of the sides.
Statement 1 tells us only that the quadrilateral is a rhombus. Indeed, a quadrilateral with perpendicular diagonals intersecting at their midpoint must be a rhombus. However we don't know any length of the sides.
Statement 2 says gives us the length us two consecutive sides. It could be tempting to answer that it is sufficient, however, we can't conclude that the quadrilateral has equal lengths. Therefore this statement alone is insufficient.
Both statements together are sufficient since we can conclude that the quadrilateral is a rhombus, and twice
will give us the perimeter.
To find the perimeter of the quadrilateral, we need to know whether it is of a special type of quadrilaterals and we need to know the length of the sides.
Statement 1 tells us only that the quadrilateral is a rhombus. Indeed, a quadrilateral with perpendicular diagonals intersecting at their midpoint must be a rhombus. However we don't know any length of the sides.
Statement 2 says gives us the length us two consecutive sides. It could be tempting to answer that it is sufficient, however, we can't conclude that the quadrilateral has equal lengths. Therefore this statement alone is insufficient.
Both statements together are sufficient since we can conclude that the quadrilateral is a rhombus, and twice will give us the perimeter.
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Consider rectangle
.
I) Side
is three fourths of side
.
II) Side
is
meters long.
What is the perimeter of
?
Consider rectangle .
I) Side is three fourths of side
.
II) Side is
meters long.
What is the perimeter of ?
To find perimeter, we need to find the length of all the sides. Recall that rectangles are made up of two pairs of equal sides.
I) Relates one side to another non-equivalent side.
II) Gives us side
, which must be equivalent to
.
Use II) and I) to find all the side lengths, then add them up. Both are needed.
Recap:
Consider rectangle CONT
I) Side CO is three fourths of side ON
II) Side NT is 15.7 meters long
What is the perimeter of CONT?
Because we are dealing with a rectangle, we know the following:

Find perimeter with:

Use I) and II) to write the following equation:

So:

And finally:

To find perimeter, we need to find the length of all the sides. Recall that rectangles are made up of two pairs of equal sides.
I) Relates one side to another non-equivalent side.
II) Gives us side , which must be equivalent to
.
Use II) and I) to find all the side lengths, then add them up. Both are needed.
Recap:
Consider rectangle CONT
I) Side CO is three fourths of side ON
II) Side NT is 15.7 meters long
What is the perimeter of CONT?
Because we are dealing with a rectangle, we know the following:
Find perimeter with:
Use I) and II) to write the following equation:
So:
And finally:
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Find the perimeter of the rectangle.
Statement 1: The area of the rectangle is 24.
Statement 2: The diagonal of the rectangle is 5.
Find the perimeter of the rectangle.
Statement 1: The area of the rectangle is 24.
Statement 2: The diagonal of the rectangle is 5.
Statement 1): The area of the rectangle is 24.
Write the area for a rectangle and substitute the value of the area.


The length and width of the rectangle are unknown, and each set of dimensions will provide a different perimeter. This statement is insufficient to find the perimeter of the rectangle.
Statement 2): The diagonal of the rectangle is 5.
Given the diagonal of the rectangle, the Pythagorean Theorem can be used to solve for the diagonal. Express the equation in terms of length and width.

Similar to the case in Statement 1), both the length and width are unknown, and the equation by itself is insufficient to solve for the perimeter of the rectangle.
Attempting to use both equations:
and
to solve for length and width will yield complex numbers as part of the solution.
Therefore:

Statement 1): The area of the rectangle is 24.
Write the area for a rectangle and substitute the value of the area.
The length and width of the rectangle are unknown, and each set of dimensions will provide a different perimeter. This statement is insufficient to find the perimeter of the rectangle.
Statement 2): The diagonal of the rectangle is 5.
Given the diagonal of the rectangle, the Pythagorean Theorem can be used to solve for the diagonal. Express the equation in terms of length and width.
Similar to the case in Statement 1), both the length and width are unknown, and the equation by itself is insufficient to solve for the perimeter of the rectangle.
Attempting to use both equations: and
to solve for length and width will yield complex numbers as part of the solution.
Therefore:
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Consider rectangle
.
I) Side
is
fathoms long.
II) Side
is three fourths of
.
What is the length segment
, the diagonal of
?
Consider rectangle .
I) Side is
fathoms long.
II) Side is three fourths of
.
What is the length segment , the diagonal of
?
To find the diagonal of a rectangle, we need the length of two sides.
I) Gives us the length of one side.
II) Lets us find the length of the next side.
Use I) and II) with Pythagorean Theorem to find the diagonal.


Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.
To find the diagonal of a rectangle, we need the length of two sides.
I) Gives us the length of one side.
II) Lets us find the length of the next side.
Use I) and II) with Pythagorean Theorem to find the diagonal.
Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.
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Find the diagonal of a square.
Statement 1: The perimeter of the square is known.
Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.
Find the diagonal of a square.
Statement 1: The perimeter of the square is known.
Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.
Statement 1: The perimeter of the square is known.
A square has four equal sides. Write the perimeter formula for squares.

The side length of a square is a fourth of the perimeter.
Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.
If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge. Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square. Write the formula for the area of a square.

The diagonal of the secondary square can be solved by using the Pythagorean Theorem.
The side length of the secondary square also must equal the diagonal of the primary square.
Therefore:

Statement 1: The perimeter of the square is known.
A square has four equal sides. Write the perimeter formula for squares.
The side length of a square is a fourth of the perimeter.
Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.
If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge. Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square. Write the formula for the area of a square.
The diagonal of the secondary square can be solved by using the Pythagorean Theorem.
The side length of the secondary square also must equal the diagonal of the primary square.
Therefore:
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What are the lengths of the diagonals in the parallelogram
?
-

-

What are the lengths of the diagonals in the parallelogram ?
Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:

Since this is a parellelogram, knowing one angle allows us to know all the angles.
Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:
Since this is a parellelogram, knowing one angle allows us to know all the angles.
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For the parellogram
, the longest diagonal is
. Is
?
-
The area of
is 
-
The perimeter of
is 
For the parellogram , the longest diagonal is
. Is
?
-
The area of
is
-
The perimeter of
is
Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of
and a longer side of
. The diagonal would be well above 
However, with the perimeter, the smaller and larger sides must add up to one half of it,
.
The longer diagonal reaches its maximum with the larger internal angle widens towards
degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:


.
At its max, the diagonal could be no greater than 
Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of and a longer side of
. The diagonal would be well above
However, with the perimeter, the smaller and larger sides must add up to one half of it, .
The longer diagonal reaches its maximum with the larger internal angle widens towards degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:
.
At its max, the diagonal could be no greater than
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NOTE: Figure NOT drawn to scale.
Is the above figure a parallelogram?
Statement 1: 
Statement 2: 
NOTE: Figure NOT drawn to scale.
Is the above figure a parallelogram?
Statement 1:
Statement 2:
Knowing one pair of sides of a quadrilateral to be congruent is not alone sufficient to prove that figure to be a parallelogram, nor is knowing one pair of sides of a quadrilateral to be parallel. But knowing both about the same pair of sides is sufficient by a theorem of parallelograms.
Knowing one pair of sides of a quadrilateral to be congruent is not alone sufficient to prove that figure to be a parallelogram, nor is knowing one pair of sides of a quadrilateral to be parallel. But knowing both about the same pair of sides is sufficient by a theorem of parallelograms.
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Notes:
refers to the length of the entire dashed line. Figure not drawn to scale.
Calculate
, the height of the large trapezoid.
Statement 1: 
Statement 2: The area of the trapezoid is 7,000.
Notes: refers to the length of the entire dashed line. Figure not drawn to scale.
Calculate , the height of the large trapezoid.
Statement 1:
Statement 2: The area of the trapezoid is 7,000.
Consider the area formula for a trapezoid:

is the _midsegment_of the trapezoid, which, by the Trapezoid Midsegment Theorem, has length equal to the mean of the bases - in other words,
. The area formula can be expressed, after substitution, as

So, if you know both the area and
- but not just one - you can find the height by dividing.
Consider the area formula for a trapezoid:
is the _midsegment_of the trapezoid, which, by the Trapezoid Midsegment Theorem, has length equal to the mean of the bases - in other words,
. The area formula can be expressed, after substitution, as
So, if you know both the area and - but not just one - you can find the height by dividing.
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