Understanding Quadrilaterals

Polygons can be classified as concave or convex. A concave polygon is one where at least one interior angle is reflex (greater than $180^{\circ}$). Looking at the polygon $PQRS$ in the diagram, how is it classified?

A polygon is classified based on the number of its sides. Consider the polygon shown in the figure. Based on the number of vertices, what is the specific name of this polygon?

Which of the following figures is a simple closed curve?

In a regular hexagon, find the measure of each interior angle.

A polygon has 27 diagonals. How many sides does it have?

Find the value of $x$ in the figure where five exterior angles of a pentagon are given as $x, 2x, 3x, 4x$, and $5x$.

Two angles of a hexagon are $120^\circ$ and $160^\circ$. If the remaining four angles are equal, find the measure of each equal angle.

A convex polygon has $n$ sides. The sum of $n-1$ of its interior angles is $2190^{\circ}$. Find the measure of the $n$-th interior angle and the value of $n$.

A regular polygon 'A' has $n$ sides and a regular polygon 'B' has $2n$ sides. If the ratio of an interior angle of A to an interior angle of B is $3:4$, find the value of $n$.

In a quadrilateral $ABCD$, the bisectors of $\angle A$ and $\angle B$ meet at point $O$. If $\angle C = 70^{\circ}$ and $\angle D = 110^{\circ}$, calculate the measure of $\angle AOB$.

In the given figure, find the value of $x+y+z+w$.

What is the minimum interior angle possible for a regular polygon?

In the given diagram, find the sum $a + b + c + d$.

If three angles of a quadrilateral are equal and the fourth angle is $60^{\circ}$, find the measure of each equal angle.

Find the measure of $x$ in the figure where the exterior angles of a pentagon are shown.

Find the number of sides of a regular polygon whose each interior angle is $165^{\circ}$.

Two adjacent angles of a parallelogram are $(2x+10)^{\circ}$ and $(3x-40)^{\circ}$. Find the value of $x$.

A polygon has $n$ sides. If the sum of its interior angles is $1800^{\circ}$, and it is a regular polygon, what is the measure of each exterior angle?

The difference between an interior angle and its corresponding exterior angle of a regular polygon is $108^{\circ}$. How many sides does this polygon have?

In a convex octagon, seven of the interior angles are $145^{\circ}$ each. What is the measure of the eighth interior angle?

In the given figure, $ABCD$ is a parallelogram where $\angle DAB = 75^{\circ}$. Find the measure of $\angle ABC$.

The sum of all interior angles of a trapezium is:

In the given diagram, which shape is a quadrilateral with two pairs of equal adjacent sides but unequal opposite sides?

If two adjacent angles of a parallelogram are in the ratio $2:3$, find the smaller angle.

One angle of a parallelogram is $30^{\circ}$ less than twice the smallest angle. Find the measure of the largest angle.

In a trapezium $ABCD$, $AB \parallel DC$ and $AB = 2 DC$. If the area is $60\text{ cm}^2$ and the height is $8\text{ cm}$, find the length of $AB$.

In the figure, $ABCD$ is a parallelogram. If $\angle DAB = 60^{\circ}$ and $\angle DBC = 80^{\circ}$, find $\angle BDC$.

A kite $ABCD$ has diagonals $AC$ and $BD$ intersecting at $O$. If $AO = 4$ cm, $OC = 9$ cm, and $OB = 6$ cm, find the perimeter of the kite.

A mechanical linkage is in the shape of a parallelogram $PQRS$. If the diagonals $PR$ and $QS$ intersect at $O$ such that $PO = 3x - 5$, $OR = x + 7$, $QO = 2y + 1$, and $OS = 3y - 2$, find the sum of $x + y$.

A decorative window pane is a trapezium $WXYZ$ with $WX \parallel ZY$. The length of $WX$ is $x + 5$, $ZY$ is $3x - 1$, and the median (line segment joining midpoints of non-parallel sides) length is $14$ cm. Find the value of $x$.

Which of the following properties is true for a rectangle but NOT necessarily true for a rhombus?

In parallelogram $ABCD$, $\angle A = (3x - 20)^{\circ}$ and $\angle C = (x + 40)^{\circ}$. Find the value of $x$.

If the diagonals of a quadrilateral bisect each other at right angles, and all sides are equal, which of these is it NOT necessarily?

A quadrilateral has all four sides equal and its diagonals are not equal. What is the most specific name for this shape?

If the diagonals of a rhombus are $10$ cm and $24$ cm, find the perimeter of the rhombus.

In a parallelogram $ABCD$, if $\angle A = (2x+10)^{\circ}$ and $\angle B = (3x-40)^{\circ}$, find the measure of $\angle A$.

In the square $ABCD$, the diagonal $BD$ is drawn. What is the measure of $\angle ABD$?

A plot is in the shape of a parallelogram $ABCD$. A fence is built such that $\angle DAB = 110^{\circ}$. If an internal pillar is placed at $P$ such that $\triangle PDC$ is an isosceles triangle with $PD = PC$ and $\angle PDC = 30^{\circ}$, calculate $\angle BCP$.

In a square $STUV$, the diagonals intersect at $W$. If the length of $SW$ is $5\sqrt{2}$ units, find the perimeter of the square.

A rectangle $ABCD$ has side $AB = 8$ cm and $BC = 6$ cm. If a point $P$ is the midpoint of $AD$, find the length of the line segment $CP$.