Practice Questions

State the sum of the measures of the exterior angles of any convex polygon.

Define a regular polygon.

Recall the definition and properties of a rhombus. List at least three distinct properties.

Justify the statement: "Every square is a rhombus, but not every rhombus is a square."

In a rectangle ABCD, the length of the diagonal AC is 10 cm. What is the length of the other diagonal BD?

What is the name of a regular polygon with 6 sides?

Calculate the number of sides of a regular polygon whose each exterior angle measures $40^\circ$.

A quadrilateral has three of its angles measuring $80^\circ, 100^\circ,$ and $110^\circ$. Calculate the measure of the fourth angle.

In a parallelogram PQRS, if the measure of $\angle P$ is $75^\circ$, calculate the measure of its adjacent angle, $\angle Q$.

Explain the relationship between a parallelogram, a rhombus, a rectangle, and a square. Describe how each is a special case of another.

In a trapezium ABCD with $\overline{AB} \| \overline{DC}$, the ratio of $\angle D$ to $\angle A$ is $5:4$. Calculate the measure of $\angle A$ and $\angle D$.

Design a quadrilateral whose diagonals are unequal, bisect each other, but are not perpendicular. Name the most general type of quadrilateral you have designed and justify your choice.

Identify the quadrilateral that has exactly one pair of parallel sides.

List two properties of the diagonals of a square.

Describe the main properties of a parallelogram concerning its sides and angles.

Explain why a square is also considered a special type of rectangle.

Define the following terms. For each term, also state one key property. (i) Polygon (ii) Convex Quadrilateral (iii) Kite (iv) Rectangle (v) Trapezium

Explain the difference between a convex polygon and a concave polygon. You may use a simple sketch to illustrate.

Summarize the key features of a kite. Mention its properties related to sides, angles, and diagonals.

Describe how a trapezium is different from a parallelogram.

The ratio of two adjacent sides of a parallelogram is $3:5$ and its perimeter is 48 cm. Calculate the lengths of the sides of the parallelogram.

In parallelogram ABCD, the diagonals intersect at point O. If $AO = x+8$, $OC = 17$, $DO = 2y-3$, and $OB=15$, solve for $x$ and $y$.

In a regular polygon, the measure of an exterior angle is one-fifth of its interior angle. Calculate the number of sides in the polygon.

In a kite ABCD, the sides AB and AD are equal, and sides CB and CD are equal. If $\angle A = 100^\circ$ and $\angle C = 60^\circ$, calculate the measure of $\angle B$.

The perimeter of a kite is 108 cm. If one of its sides is 12 cm longer than another side, calculate the length of each side of the kite.

A student claims it is possible to create a regular polygon whose interior angle is $155^\circ$. Critique this claim and justify your conclusion.

Evaluate whether a regular polygon can have an exterior angle of $23^\circ$. Provide a mathematical justification for your answer.

Formulate a rule using the concept of diagonals to definitively distinguish a convex polygon from a concave polygon.

In a parallelogram ABCD, the bisectors of adjacent angles $\angle A$ and $\angle B$ intersect at a point P. Justify why $\angle APB = 90^\circ$.

In rectangle ABCD, diagonal AC is drawn. If $\angle BAC = 32^\circ$, find the measure of $\angle DBC$. Justify each step of your calculation using the properties of a rectangle.

A carpenter builds a four-sided frame PQRS. He measures the sides and finds $PQ = RS = 50$ cm and $PS = QR = 80$ cm. He concludes it must be a parallelogram. Is his conclusion sufficient? Evaluate what additional measurement he must take to guarantee the frame is a perfect rectangle, and justify your reasoning.

The diagonals of a rhombus are 6 cm and 8 cm. Calculate the length of a side of the rhombus.

In parallelogram ABCD, the angle bisector of $\angle A$ meets the side DC at point P. If $\angle B = 110^\circ$, calculate all the angles of $\triangle APD$.

In a rhombus PQRS, the diagonal PR is equal in length to its side PQ. Calculate all the interior angles of the rhombus.

Analyze if a quadrilateral with interior angles in the ratio $1:3:7:9$ can be a trapezium. Justify your answer.

The diagonals of a rhombus are in the ratio $3:4$. If its perimeter is $40$ cm, formulate a method and find the lengths of the diagonals.

In the figure provided, ABCD is a square and CDE is an equilateral triangle constructed on the side CD outside the square. Calculate the measure of $\angle BEC$.

A quadrilateral has opposite sides that are parallel and equal. Identify this quadrilateral and explain its other two main properties related to angles and diagonals.

Propose a method to find the area of a kite given the lengths of its diagonals, $d_1$ and $d_2$. Justify your proposed formula using geometric properties.

Prove that the quadrilateral formed by joining the mid-points of the sides of a parallelogram is also a parallelogram.

List the four main properties of a parallelogram. Then, explain which of these properties are always true for a kite and a trapezium.