Quadrilaterals - ACT Math

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Question

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 ?

Answer

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 $\angle D=\angle B$ .

To find $\angle A$ :

$x=\frac{130-8}{2}=\frac{122}{2}=61$

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Question

A kite has one set of opposite interior angles where the two angles measure $28$^{\circ}$$ and $84$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{248}{2}=124$

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Question

Using the kite shown above, find the sum of the two remaining congruent interior angles.

Answer

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.

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Question

A kite has one set of opposite interior angles where the two angles measure $118$^{\circ}$$ and $51$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{191}{2}=95.5$

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Question

Using the kite shown above, find the sum of the two remaining congruent interior angles.

Answer

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.

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Question

A kite has one set of opposite interior angles where the two angles measure $91$^{\circ}$$ and $99$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{170}{2}=85$

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Question

A kite has one set of opposite interior angles where the two angles measure $82$^{\circ}$$ and $58$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles in this kite.

Answer

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.

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Question

A kite has one set of opposite interior angles where the two angles measure $204$^{\circ}$$ and $36$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{120}{2}=60$

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Question

A kite has one set of opposite interior angles where the two angles measure $112$^{\circ}$$ and $101$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles.

Answer

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:

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Question

A kite has one set of opposite interior angles where the two angles measure $16$^{\circ}$$ and $187$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{157}{2}=78.5$

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Question

Using the kite shown above, find the sum of the two remaining congruent interior angles.

Answer

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.

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Question

A kite has one set of opposite interior angles where the two angles measure $44$^{\circ}$$ and $59$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Answer

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:

$\frac{257}{2}=128.5$

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Question

A kite has one set of opposite interior angles where the two angles measure $58$^{\circ}$$ and $67$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles.

Answer

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:

$360-125=235$^{\circ}$$

This means that $235$^{\circ}$$ is the sum of the remaining two opposite angles.

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Question

In a particular kite, one angle that lies between congruent sides measures $50^{\circ}$ , and one angle that lies between non-congruent sides measures $25^{\circ}$ . What is the measure of the angle opposite the $50^{\circ}$ angle?

Answer

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 $25^{\circ}$ . Since all angles in a quadrilateral must sum to $360^{\circ}$ , we know that the other missing angle is

$\angle x = 360^{\circ} - (25^{\circ} + 25^{\circ} + 50^{\circ}) = 260^{\circ}$

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Question

In the parallellogram, what is the value of ?

Answer

Opposite angles are equal, and adjacent angles must sum to 180.

Therefore, we can set up an equation to solve for z:

(z – 15) + 2z = 180

3z - 15 = 180

3z = 195

z = 65

Now solve for x:

2_z_ = x = 130°

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Question

In parallelogram , $\measuredangle A = 45^{\circ}$ . What is $\measuredangle B?$

Answer

In the above parallelogram, $\measuredangle A$ and $\measuredangle B$ are consecutive angles (i.e. next to each other). In a parallelogram, consecutive angles are supplementary, meaning they add to $180^{\circ}$ .

$\measuredangle A + \measuredangle B = 180^{\circ}$

$45^{\circ} + \measuredangle B = 180^{\circ}$

$\measuredangle B = 135^{\circ}$

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Question

In parallelogram , $\measuredangle B = 117^{\circ}$ . What is $\measuredangle D$ ?

Answer

In parallelogram , $\measuredangle B$ and $\measuredangle D$ are opposite angles. In a parallelogram, opposite angles are congruent. This means these two angles are equal.

$\measuredangle D = \measuredangle B$

$\measuredangle D = 117^{\circ}$

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Question

In parallelogram , $\overline{AB} = 15$ and the height is . What is $\measuredangle A$ ?

Answer

We can start this problem by drawing the height and labeling the lengths with the given values.

When we do this, we can see that we have drawn a triangle inside the paralellogram including $\measuredangle A$ . Because we know the lengths of two sides of this triangle, we can use trigonometry to find $\measuredangle A$ .

With respect to $\measuredangle A$ , we know the values of the opposite and hypotenuse sides of the triangle. Thus, we can use the sine function to solve for $\measuredangle A$ .

$\sin \left (\measuredangle A\right ) = \frac{\textup{opposite}}{\textup{hypotenuse}}$

$\sin \left (\measuredangle A\right ) = \frac{12}{15}= 0.8$

$\measuredangle A = \arcsin\left ( 0.8\right ) = 53.1^{\circ}$

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Question

In parallelogram , what is the sum of $\measuredangle A$ and $\measuredangle D$ ?

Answer

In a parallelogram, consecutive angles are supplementary. $\measuredangle A$ and $\measuredangle D$ are consecutive, so their sum is $180^{\circ}$ .

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Question

In parallelogram , $\measuredangle B = \left (2x + 15 \right )^{\circ}$ and $\measuredangle C = x^{\circ}$ . Find .

Answer

In a parallelogram, consecutive angles are supplementary. Thus,

$\measuredangle B + \measuredangle C = 180^{\circ}$

$\left ( 2x + 15\right ) + x = 180$

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