Understanding Quadrilaterals

Identify the number of sides in the polygon shown.

What is the sum of the measures of the exterior angles of the polygon shown?

In the figure, which vertex pair forms an adjacent side with $AB$?

What is the name of a polygon with 8 sides as shown in the figure?

In the triangle shown, find the value of the exterior angle $x$ if the opposite interior angles are $60^\circ$ and $70^\circ$.

The figure shows a regular polygon. If the sum of its interior angles is $1080^\circ$, what is the measure of each interior angle?

If the ratio of the interior angles of a quadrilateral is $1:2:3:4$, what is the measure of the largest angle?

In a certain $n$-sided convex polygon, one interior angle is $140^\circ$ and all the other interior angles are $160^\circ$ each. What is the value of $n$?

The number of sides of two regular polygons are in the ratio $5:4$ and the difference between their interior angles is $9^\circ$. Find the number of sides of these polygons.

A regular polygon is such that the sum of its interior angles is three times the sum of its exterior angles. If a triangle is formed by connecting three consecutive vertices of this polygon, what are the angles of that triangle?

In the given figure, what is the measure of the unknown angle marked $z$?

The angles of a quadrilateral are $x, x+10^{\circ}, x+20^{\circ}$ and $x+30^{\circ}$. Find the value of $x$.

Find the missing angle $a$ in the quadrilateral.

In a quadrilateral, if the ratio of angles is $1:1:1:1$, the figure is most likely a:

In a quadrilateral $ABCD$, $\angle A + \angle C = 180^{\circ}$. What is the value of $\angle B + \angle D$?

One angle of a quadrilateral is $108^{\circ}$ and the remaining three angles are equal. Find the measure of each equal angle.

In the figure, find the value of $a+b+c+d$.

A designer creates a quadrilateral logo where the measure of one angle is $x^\circ$, and the other three angles are $(x-20)^\circ$, $(x+40)^\circ$, and $(2x-60)^\circ$. What is the measure of the largest interior angle of this logo?

In a quadrilateral $ABCD$, the bisectors of the four interior angles form a smaller quadrilateral $PQRS$. If the original quadrilateral $ABCD$ has angles in the ratio $1:2:3:4$, what is the sum of the interior angles of the newly formed quadrilateral $PQRS$?

A metal bracket is bent into a quadrilateral $ABCD$. The exterior angles at $A$ and $C$ are $80^\circ$ and $100^\circ$ respectively. If the interior angle at $B$ is double the interior angle at $D$, find the interior angle $\angle D$.

In the given figure, $PQRS$ is a quadrilateral with $PQ = QR = RS = SP$ and $\angle P = \angle Q = \angle R = \angle S = 90^\circ$. What is the specific name given to this type of quadrilateral?

A quadrilateral with all sides equal and all angles equal is a:

The perimeter of a rhombus is $52$ cm. What is the length of each side?

What is the name of a quadrilateral that has exactly one pair of parallel sides?

In the given parallelogram, find the value of $x + y$ if opposite sides are $3x$ and $18$, and $2y-1$ and $11$.

Which of these quadrilaterals has diagonals that are perpendicular and bisect each other, but the adjacent sides are not necessarily equal?

Find the value of $y$ in the following parallelogram if the diagonals are $2y + 4$ and $12$ (bisected part is $6$).

A decorative window is in the shape of a regular hexagon, which can be seen as composed of two congruent isosceles trapeziums joined at their longer base. If the perimeter of the regular hexagon is $72$ cm, and the distance between two opposite parallel sides is $12\sqrt{3}$ cm, find the area of one such trapezium.

In a parallelogram $MNOP$, the bisectors of $\angle M$ and $\angle N$ meet at point $X$. What is the measure of $\angle MXN$?

A rectangular photo frame $PQRS$ has a perimeter of $60$ cm. If the ratio of the diagonal $PR$ to the side $PQ$ is $13:12$, find the area of the frame.

In the given parallelogram $ABCD$, the measure of $\angle A$ is $70^{\circ}$. Find the measure of its adjacent angle $\angle B$.

In parallelogram $ABCD$, if the diagonal $AC = 12\text{ cm}$ and $BD = 16\text{ cm}$, they bisect each other at $O$. What is the length of $BO$?

What is the sum of any two adjacent angles in a parallelogram?

In the parallelogram $WXYZ$, $\angle W = (x + 20)^\circ$ and $\angle Y = 80^\circ$. Find the value of $x$.

In parallelogram $ABCD$, $\angle A$ and $\angle C$ are opposite angles. If $\angle A = (4x - 5)^\circ$ and $\angle C = (3x + 15)^\circ$, find the measure of $\angle B$.

If the perimeter of a parallelogram is $50\text{ cm}$ and one side is $15\text{ cm}$, what is the length of the adjacent side?

In a parallelogram, if the sum of two opposite angles is $160^\circ$, find the measure of each of the other two opposite angles.

In a parallelogram $ABCD$, if the diagonal $AC$ also bisects $\angle A$, then what kind of special parallelogram is $ABCD$ most specifically, and what is the relationship between sides $AB$ and $BC$?

A metal frame is in the shape of a parallelogram $PQRS$. If the side $PQ = (5y - 7)$ cm and $RS = (2y + 8)$ cm, and side $QR = (3x - 4)$ cm and $PS = (x + 6)$ cm, find the values of $x$ and $y$.

In a parallelogram $ABCD$, the measure of $\angle A$ is $30^\circ$ less than twice the measure of $\angle B$. Find the measure of the largest angle of the parallelogram.

If one angle of a parallelogram is a right angle and all sides are equal, then it is a:

In a square $PQRS$, the diagonals intersect at $O$. What is the measure of $\angle POQ$?

Which of the following properties is true for a rhombus?

In the rhombus $ABCD$, side $AB = 6\text{ cm}$. What is the perimeter of the rhombus?

If the diagonals of a rhombus are equal, then the rhombus is a:

In a rectangle $ABCD$, diagonal $AC$ is inclined to side $AB$ at $35^{\circ}$. Find the acute angle between the diagonals.

ABCD is a rhombus such that $\angle ACB = 40^{\circ}$. Find the measure of $\angle ADB$.

A square $PQRS$ is such that the sum of its diagonals is $20\sqrt{2}$ cm. What is the perimeter of the square?

The diagonal of a rectangle is inclined to one side at $30^{\circ}$. If the length of the diagonal is $12$ cm, what is the area of the rectangle?

In a rhombus $ABCD$, the diagonals intersect at $O$. If $OA = x + y$, $OC = 12$, $OB = 2x - y$, and $OD = 9$, determine the values of $x$ and $y$.