Quadrilaterals - ACT Math
Card 0 of 2088
If the diagonals of the quadrilateral above were drawn in the figure, they would form four 90 degree angles at the center. In degrees, what is the value of 
?
If the diagonals of the quadrilateral above were drawn in the figure, they would form four 90 degree angles at the center. In degrees, what is the value of
A quadrilateral is considered a kite when one of the following is true:
(1) it has two disjoint pairs of sides are equal in length or
(2) one diagonal is the perpendicular bisector of the other diagonal. Given the information in the question, we know (2) is definitely true.
To find 
we must first find the values of all of the angles.
The sum of angles within any quadrilateral is 360 degrees.
Therefore 
.
To find 
:
A quadrilateral is considered a kite when one of the following is true:
(1) it has two disjoint pairs of sides are equal in length or
(2) one diagonal is the perpendicular bisector of the other diagonal. Given the information in the question, we know (2) is definitely true.
To find
The sum of angles within any quadrilateral is 360 degrees.
Therefore
To find
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A kite has two adjacent sides both with a measurement of 
. The perimeter of the kite is 
. Find the length of one of the remaining two sides.
A kite has two adjacent sides both with a measurement of
A kite must have two sets of congruent adjacent sides. This question provides the length for one set of congruent adjacent sides, thus the two remaining sides must be congruent to each other. Since, this problem also provides the perimeter measurement of the kite, find the difference between the perimeter and the sum of the congruent adjacent sides provided. Then divide that remaining difference in half, because each of the two sides must have the same length.
The solution is:
, where 
one of the two missing sides.
A kite must have two sets of congruent adjacent sides. This question provides the length for one set of congruent adjacent sides, thus the two remaining sides must be congruent to each other. Since, this problem also provides the perimeter measurement of the kite, find the difference between the perimeter and the sum of the congruent adjacent sides provided. Then divide that remaining difference in half, because each of the two sides must have the same length.
The solution is:
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A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where 
is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 
degrees 
degrees 
degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between 
degrees and the non-congruent opposite angles sum by 
:
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
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Using the kite shown above, find the sum of the two remaining congruent interior angles.
Using the kite shown above, find the sum of the two remaining congruent interior angles.
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
degrees
Thus, 
degrees is the sum of the remaining two opposite angles.
Check:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
Thus,
Check:
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A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between degrees and the non-congruent opposite angles sum by :
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
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Using the kite shown above, find the sum of the two remaining congruent interior angles.
Using the kite shown above, find the sum of the two remaining congruent interior angles.
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
degrees
Thus, 
degrees is the sum of the remaining two opposite angles.
The sum of the interior angles of any polygon can be found by applying the formula:
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
Thus,
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Find the area of a rectangle whose width is 
and length is 
.
Find the area of a rectangle whose width is
To solve, simply multiply width and length. Thus,
To solve, simply multiply width and length. Thus,
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A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between degrees and the non-congruent opposite angles sum by :
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement of the sum of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
degrees
This means that 
degrees is the sum of the remaining two opposite angles and that each have an individual measurement of 
degrees.
Check:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
This means that
Check:
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between degrees and the non-congruent opposite angles sum by :
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement of the sum of the two remaining interior angles.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
degrees
This means that 
degrees is the sum of the remaining two opposite angles.
Check:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
This means that
Check:
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between degrees and the non-congruent opposite angles sum by :
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
Compare your answer with the correct one above
Using the kite shown above, find the sum of the two remaining congruent interior angles.
Using the kite shown above, find the sum of the two remaining congruent interior angles.
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
degrees
degrees
Thus, 
degrees is the sum of the remaining two opposite angles.
The sum of the interior angles of any polygon can be found by applying the formula:
A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
Thus,
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement for one of the two remaining interior angles in this kite.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between degrees and the non-congruent opposite angles sum by :
This means that 
is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between
This means that
Compare your answer with the correct one above
A kite has one set of opposite interior angles where the two angles measure 
and 
, respectively. Find the measurement of the sum of the two remaining interior angles.
A kite has one set of opposite interior angles where the two angles measure
The sum of the interior angles of any polygon can be found by applying the formula:
degrees, where is the number of sides in the polygon.
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.
To find the sum of the remaining two angles, determine the difference between degrees and the sum of the non-congruent opposite angles.
The solution is:
This means that 
is the sum of the remaining two opposite angles.
The sum of the interior angles of any polygon can be found by applying the formula:
By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:
To find the sum of the remaining two angles, determine the difference between
The solution is:
This means that
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In a particular kite, one angle that lies between congruent sides measures 
, and one angle that lies between non-congruent sides measures 
. What is the measure of the angle opposite the 
angle?
In a particular kite, one angle that lies between congruent sides measures
One of the rules governing kites is that the angles which lie between non-congruent sides are congruent to each other. Thus, we know one of the missing angles is also 
. Since all angles in a quadrilateral must sum to 
, we know that the other missing angle is
One of the rules governing kites is that the angles which lie between non-congruent sides are congruent to each other. Thus, we know one of the missing angles is also
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Find the area of a kite if one diagonal is 
long, and the other diagonal is 
long.
Find the area of a kite if one diagonal is
The formula for the area of a kite is
Plug in the values for each of the diagonals and solve.
The formula for the area of a kite is
Plug in the values for each of the diagonals and solve.
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Find the area of a kite with the diagonal lengths of 
and 
.
Find the area of a kite with the diagonal lengths of
Write the formula to find the area of a kite. Substitute the diagonals and solve.
Write the formula to find the area of a kite. Substitute the diagonals and solve.
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Find the area of a kite with diagonal lengths of 
and 
.
Find the area of a kite with diagonal lengths of
Write the formula for the area of a kite.
Plug in the given diagonals.
Pull out a common factor of two in 
and simplify.
Use the FOIL method to simplify.
Write the formula for the area of a kite.
Plug in the given diagonals.
Pull out a common factor of two in
Use the FOIL method to simplify.
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A kite has two perpendicular interior diagonals. One diagonal is twice the length of the other diagonal. The total area of the kite is 
. Find the length of each interior diagonal.
A kite has two perpendicular interior diagonals. One diagonal is twice the length of the other diagonal. The total area of the kite is
To solve this problem, apply the formula for finding the area of a kite:
However, in this problem the question only provides information regarding the exact area. The lengths of the diagonals are represented as a ratio, where
Therefore, it is necessary to plug the provided information into the area formula. Diagonal 
is represented by 
and diagonal 
.
The solution is:
Thus, if 
, then diagonal 
must equal 
To solve this problem, apply the formula for finding the area of a kite:
However, in this problem the question only provides information regarding the exact area. The lengths of the diagonals are represented as a ratio, where
Therefore, it is necessary to plug the provided information into the area formula. Diagonal
The solution is:
Thus, if
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