Quadrilaterals

In the given figure of a rectangle $ABCD$, if $\angle ABD = 35^{\circ}$, find $\angle BDC$.

Which of the following statement is true about the diagonals of a rectangle?

In square $ABCD$, the diagonals intersect at $O$. What is the measure of $\angle AOB$?

If all sides of a rectangle are equal, the rectangle is called a:

In rectangle $ABCD$, point $P$ is on $BC$ such that $AP = CD + BP$. If $\angle BAP = 25^{\circ}$, find $\angle DAP$.

If the length of a diagonal of a square is $12 \text{ cm}$, what is its area?

The diagonals of a rectangle $ABCD$ intersect at $O$. If $\angle BOC = 70^{\circ}$, then $\angle ODA$ is:

A square sheet of paper with side $20\text{ cm}$ has four identical small squares of side $x\text{ cm}$ cut out from its corners. If the remaining area is $336\text{ cm}^2$, find the side of the small squares.

In a rectangle $PQRS$, $PQ = 20\text{ cm}$ and $\angle PRQ = 60^\circ$. Calculate the length of the diagonal $PR$ and the side $QR$.

Two adjacent angles of a quadrilateral are $130^\circ$ and $50^\circ$. If the other two angles are in the ratio $5:8$ and the quadrilateral is a rectangle, what must be the values of all four angles?

In the given quadrilateral $ABCD$, the measures of three angles are $\angle A = 110^{\circ}$, $\angle B = 70^{\circ}$, and $\angle C = 80^{\circ}$. Find the measure of the fourth angle $\angle D$.

In the isosceles trapezoid $ABCD$, $\angle A = 70^{\circ}$. Find the measure of $\angle B$ (where $AB || CD$ and $AD = BC$).

Three angles of a quadrilateral are equal and the fourth angle is $120^{\circ}$. Find the measure of each equal angle.

In a rhombus $ABCD$, the diagonals intersect at $O$. What is the measure of $\angle AOB$?

In the figure, $ABCD$ is a parallelogram. Find $x$.

The exterior angle of a regular polygon is $72^{\circ}$. What is the sum of its interior angles?

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

A quadrilateral has two adjacent angles of $80^\circ$ and $120^\circ$. The other two angles are in the ratio $3:5$. Find the measure of the largest angle of the quadrilateral.

In a quadrilateral $ABCD$, $\angle A = 2\angle B = 3\angle C = 4\angle D$. What is the measure of $\angle C$ (rounded to the nearest degree)?

The measures of the angles of a quadrilateral are in the ratio $3:4:5:6$. Find the difference between the largest and the smallest angle.

The perimeter of a rhombus is $40$ cm. What is the length of each side?

In a rectangle $ABCD$, if $\angle ABD = 35^\circ$, find $\angle DBC$.

In the parallelogram $ABCD$, the diagonals $AC$ and $BD$ intersect at $O$. If $AC = 12$ cm, find $OC$.

Identify the figure $ABCD$ where $AB || DC$ and $AD || BC$, and all sides are equal but angles are not $90^\circ$.

In a square $ABCD$, $M$ is the midpoint of $AB$. If the area of the square is $100$ cm$^2$, what is the length of $MC$ (rounded to one decimal)?

In the figure, $ABCD$ is a rhombus. If $\angle CAD = 35^\circ$, find $\angle BCD$.

In a parallelogram $PQRS$, $\angle P$ is $30^\circ$ more than twice $\angle Q$. Find the measure of $\angle P$.

The diagonals of a quadrilateral $ABCD$ bisect each other at right angles. If $AB = 3x - 4$ and $BC = x + 6$, find the perimeter of the quadrilateral.

In a kite $ABCD$ where $AB = AD$ and $CB = CD$, if $\angle B = 110^\circ$ and $\angle A = 50^\circ$, find the measure of $\angle C$.

A rectangle has a length of $x+2$ and a width of $x-1$. If its diagonal is $\sqrt{25}$, find the value of $x$.

In the given figure, $PQRS$ is a quadrilateral where all sides are equal ($PQ = QR = RS = SP$). Additionally, the diagonals $PR$ and $QS$ are equal in length. Which specific type of quadrilateral is $PQRS$?

True or False: The diagonals of a square are equal in length.

If one side of a rhombus is $6\text{ cm}$, what is the sum of all its sides?

Identify the property that a rhombus possesses but a general parallelogram does not.

If the diagonals of a rhombus are $10\text{ cm}$ and $24\text{ cm}$, what is its area?

The angle between a side and a diagonal of a square is:

In a rhombus, the diagonals are in the ratio $3:4$. If the perimeter is $40\text{ cm}$, find the length of the longer diagonal.

In square $ABCD$, points $E, F, G, H$ are the midpoints of sides $AB, BC, CD, DA$ respectively. If the area of the inner quadrilateral $EFGH$ is $50\text{ cm}^2$, what is the length of the side of the original square $ABCD$?

A rhombus-shaped field has a side length of $25\text{ m}$ and one diagonal of $30\text{ m}$. A cow is tied to a post at one of the vertices representing an obtuse angle. If the cow can graze the entire area of the field, what is the area of the field?

A wire in the shape of a circle of radius $14\text{ cm}$ is bent into the shape of a square. What is the difference between the area of the circle and the area of the square? (Use $\pi = \frac{22}{7}$)

The measure of one angle of a parallelogram is $50^\circ$. Find the measure of its opposite angle.

In the parallelogram $ABCD$, if the base $AB = 10$ cm and the corresponding height is $6$ cm, what is the area?

Identify the figure where all sides are equal and all angles are $90^\circ$.

In the isosceles trapezoid $ABCD$, $AB$ is parallel to $CD$. If $\angle A = 75^\circ$, find $\angle B$.

In the given parallelogram $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. If $OA = (2x + 3)$ cm and $OC = (3x - 1)$ cm, and $OB = (y + 5)$ cm and $OD = (2y - 2)$ cm, find the value of $(x + y)$.

A floral design is made using 4 identical rhombi as shown. If each side of a rhombus is $5$ cm and one angle is $60^\circ$, find the length of the shorter diagonal of one rhombus.

In the given diagram of a square $ABCD$, the diagonal $AC$ makes an angle $\angle CAD = y$. Find the value of $y$.

In the figure, $ABCD$ is a parallelogram where $\angle DAB = 60^{\circ}$ and $AD = 6$ cm. If the bisector of $\angle DAB$ meets $DC$ at $E$, find the length of $DE$.

A quadrilateral has three acute angles, each measuring $75^{\circ}$. What is the measure of the fourth angle, and what kind of quadrilateral can it NOT be?

In a rectangle $ABCD$, the diagonals $AC$ and $BD$ intersect at $O$. If $\angle BOC = 70^{\circ}$, calculate the measure of $\angle ODA$.

In the given trapezium $PQRS$, $PQ$ is parallel to $SR$. If $\angle P = 120^{\circ}$ and $\angle Q = 110^{\circ}$, find the measure of $\angle S$.

In the given diagram of kite $ABCD$, where $AB = AD$ and $CB = CD$, if the angle $\angle ABC = 110^{\circ}$ and $\angle BCD = 40^{\circ}$, find the measure of $\angle ADC$.

In a kite, the diagonals intersect at what angle?

Which quadrilateral has exactly one pair of parallel sides?

In the trapezium shown, the height is $h$ and the parallel sides are $a$ and $b$. If $a=5$, $b=9$, and $h=4$, what is the area?

An isosceles trapezium has parallel sides of $10$ cm and $20$ cm. If the non-parallel sides are $13$ cm each, find the height of the trapezium.

In the kite below, if $\angle 1 = 35^{\circ}$, what is the measure of $\angle 2$ (angle between a diagonal and a side)?

In a kite $EFGH$, the diagonals $EG$ and $FH$ intersect at $O$. If $EO = 3$ cm, $OG = 7$ cm, and $FH = 8$ cm, find the difference between the areas of $\triangle EGH$ and $\triangle EFH$.

The cross-section of a canal is a trapezium. The canal is $10$ m wide at the top and $6$ m wide at the bottom. If the area of the cross-section is $72$ m$^2$, find the depth of the canal.

A kite $ABCD$ has its diagonal $AC$ of length $24$ cm. The other diagonal $BD$ is bisected by $AC$ at $O$. If $AB = 13$ cm and $BC = 20$ cm, find the length of the diagonal $BD$.