Quadrilateral

Introduction

The word "quadrilateral" originates from Latin words, with "quadra" meaning four and "latus" meaning sides. This accurately reflects the defining characteristic of a quadrilateral: it's a polygon with four sides.

A quadrilateral is a closed shape and a type of polygon with four sides and four vertices. It's a broad category that includes various shapes like squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each of these shapes has its own unique properties and characteristics, but they all share the common feature of having four sides. Quadrilaterals play a significant role in geometry and mathematics, often used in calculations involving area, perimeter, angles, and more.

What Is a Quadrilateral?

A quadrilateral is defined as a polygon with `4` sides, `4` angles, and `4` vertices. The interior angles of a quadrilateral add up to `360°`.

When naming a quadrilateral, it's essential to maintain the order of the vertices. For example, if the vertices of a quadrilateral are `A, B, C`, and `D`, it can be named `ABCD, BCDA, ADCB`, or `DCBA`. Other combinations, such as `ACBD` or `DBAC`, change the order of the vertices and are not valid names for the quadrilateral.

In addition to four sides `(AB, BC, CD, DA)`, the quadrilateral `ABCD` has two diagonals (`AC` and `BD`).

Maintaining consistency in naming conventions and understanding the properties of quadrilaterals helps in clear communication and accurate geometry calculations.

It's also important to note that while quadrilaterals have four sides, those sides may or may not be equal in length. This property distinguishes various types of quadrilaterals from one another. Different types of quadrilaterals or example quadrilateral , such as squares, rectangles, parallelograms, rhombuses, trapezoids, and kites, are identified based on their unique properties, including the lengths of their sides, the measures of their angles, and the relationships between their sides and angles. These distinctions play a significant role in geometry and mathematics, allowing for precise classification and analysis of geometric shapes.

Types of Quadrilateral:

• Convex and concave quadrilaterals
• Regular and irregular quadrilaterals

What Is a Convex Quadrilateral?

A convex quadrilateral is a four-sided polygon where all interior angles measure less than `180` degrees. This means that no angle within the quadrilateral "turns inwards" or is greater than a straight angle. Additionally, both diagonals of a convex quadrilateral lie entirely within the closed figure formed by its sides. Examples of convex quadrilaterals include familiar shapes like squares, rectangles, parallelograms, rhombuses, trapezoids, and kites.

A convex quadrilateral `ABCD` is a four-sided shape where all its internal angles, namely `∠ABC, ∠BCD, ∠CDA`, and `∠DAB`, are smaller than `180` degrees. This means that no angle inside the shape "bends inward" or exceeds the measure of a straight angle. Moreover, both diagonals `AC` and `DB` of the quadrilateral lie entirely within the boundaries of the closed figure formed by its sides.

Example: In a quadrilateral `ABCD`, having `∠ABC = 80°, ∠BCD = 70°, ∠CDA = 110°`. Calculate the measure of the `∠DAB` and state whether it is a convex quadrilateral.

Solution:

To find the measure of angle `∠DAB` and determine whether the quadrilateral is convex, we can use the property that the sum of the interior angles of a quadrilateral is always `360` degrees.

Given:

`∠ABC = 80°`

`∠BCD = 70°`

`∠CDA = 110°`

Let's denote `∠DAB` as `x`.

We know that the sum of the interior angles of a quadrilateral is `360` degrees:

`∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°`

Substituting the given angle measures:

`80° + 70° + 110° + x = 360°`

Now, let's solve for `x`:

`260° + x = 360°`

`x = 360° - 260°`

`x = 100°`

So, `∠DAB` measures `100` degrees.

To determine whether the quadrilateral is convex, we need to check if all the interior angles are less than `180` degrees. Since all the given angles (`∠ABC, ∠BCD, ∠CDA`, and `∠DAB`) are less than `180` degrees, the quadrilateral is indeed convex.

What Is a Concave Quadrilateral?

A concave quadrilateral is a four-sided polygon where at least one of its interior angles measures more than `180` degrees. This means that within the shape, there is at least one angle that "turns inwards" or exceeds the measure of a straight angle. Additionally, in a concave quadrilateral, one of the diagonals lies partially or entirely outside the closed figure formed by its sides. In simpler terms, any quadrilateral shape that is not convex is classified as a concave quadrilateral. Examples include shapes like a dart or an arrowhead, where the interior angles exceed `180` degrees and the diagonals extend outside the shape.

In the provided concave quadrilateral `ABCD`, the angle `∠BCD` measures more than `180` degrees, indicating that it "turns inwards" beyond the measure of a straight angle. Additionally, the diagonal `BD` extends outside the closed figure formed by the sides of the quadrilateral.

Example: In a quadrilateral `ABCD`, having `∠ABC = 30°, ∠DAB = 80°`, and `∠CDA = 25°`. Calculate the measure of the `∠BCD` and state whether it is a concave quadrilateral.

Solution:

To find the measure of angle `∠BCD` and determine whether the quadrilateral is concave, we can use the property that the sum of the interior angles of a quadrilateral is always `360` degrees.

Given:

`∠ABC = 30°`

`∠DAB = 80°`

`∠CDA = 25°`

Let's denote `∠BCD` as `x`.

We know that the sum of the interior angles of a quadrilateral is `360` degrees:

`∠ABC + ∠DAB + ∠CDA + ∠BCD = 360°`

Substituting the given angle measures:

`30° + 80° + 25° + x = 360°`

Now, let's solve for `x`:

`135° + x = 360°`

`x = 360° - 135°`

`x = 225°`

So, `∠BCD` measures `225` degrees.

To determine whether the quadrilateral is concave, we need to check if any of its interior angles are greater than `180` degrees. Since `∠BCD` measures `225` degrees, which is greater than `180` degrees, the quadrilateral is indeed concave.

What Is a Regular Quadrilateral?

A regular quadrilateral or an equilateral quadrilateral, often known simply as a square, is a type of quadrilateral with four sides of equal length and four angles of equal measure. In simpler terms, all sides are of equal length, and all angles are congruent, each measuring `90` degrees. Therefore, a regular quadrilateral has both the properties of a square and is sometimes referred to as such.

What Is an Irregular Quadrilateral?

An irregular quadrilateral definition states that it is a type of quadrilateral that does not have all sides of equal length or all angles of equal measure. In simpler terms, it has one or more sides of unequal length and one or more angles of unequal measure.

Irregular quadrilateral examples include shapes like rectangles, parallelograms, rhombuses, trapezoids, and kites. These shapes have sides of different lengths and/or angles of different measures, making them irregular compared to regular quadrilaterals like squares.

Please note that though a rhombus has all its sides of equal measure, its four angles do not measure the same. Hence, the rhombus falls under irregular quadrilaterals.

Here's a table summarizing the different types of quadrilaterals, along with their defining properties:

Properties of Quadrilateral

The properties of quadrilaterals encompass various aspects including the lengths of sides, measures of angles, types of symmetry, and relationships between diagonals. Here are some key properties of quadrilaterals:

`1`. Number of Sides, Angles, and Vertices: A quadrilateral has four sides, four angles, and four vertices.

`2`. Sum of Interior Angles: The sum of the interior angles of a quadrilateral is always `360` degrees.

`3`. Types of Angles:

• Right angles: In rectangles and squares, all interior angles are right angles (`90` degrees).
   
• Supplementary angles: The sum of adjacent angles equals `180` degrees.
   
• Opposite angles: In parallelograms, opposite angles are equal.

`4`. Types of Symmetry:

• Diagonal Symmetry: Some quadrilaterals have a line of symmetry through one or both diagonals (e.g., rhombus, square).
   
• Rotational Symmetry: A square has rotational symmetry of order four, meaning it looks the same after a rotation of `90, 180`, or `270` degrees.

`5`. Diagonals:

• Length of Diagonals: Diagonals bisect each other at their intersection point in parallelograms and rectangles.
   
• Perpendicular Bisector: Diagonals of rectangles and squares are perpendicular to each other, bisecting each other.

Square

A square is a special type of quadrilateral with several unique properties:

`1`. Equal Sides: In a square, all four sides are of equal length. This means that `AB = BC = CD = DA`, where `AB`, `BC`, `CD`, and `DA` represent the lengths of the sides of the square.

`2`. Parallel Sides: Opposite sides of a rectangle are parallel to each other.

`3`. Right Angles: Each interior angle of a square measures `90` degrees. This means that all four angles in a square are right angles (`∠A = ∠B = ∠C = ∠D = 90^\circ`).

`4`. Diagonals: The diagonals of a square are equal in length and bisect each other at right angles. This means that `AC = BD`, and the point where the diagonals intersect divides each diagonal into two equal segments, forming right angles.

`5`. Area: The area of a square can be calculated using the formula: ` \text{Area} = (\text{side length})^2`.

`6`. Diagonal Symmetry: A square has two lines of symmetry, passing through the midpoints of opposite sides and intersecting at right angles at the center of the square.

`7`. Rotational Symmetry: A square has rotational symmetry of order four, meaning it looks the same after a rotation of `90, 180`, or `270` degrees.

Rectangle

A rectangle is a type of quadrilateral with several distinctive properties:

`1`. Opposite Sides: In a rectangle, opposite sides are equal in length. This means that `AB = CD` and `BC = DA`, where `AB` and `CD` represent the lengths of the rectangle while `BC`, and `AD` represent the widths of the rectangle.

`2`. Right Angles: Each interior angle of a rectangle measures `90` degrees. This means that all four angles in a rectangle are right angles (`∠A = ∠B = ∠C = ∠D = 90^\circ`).

`3`. Diagonals: The diagonals of a rectangle are equal in length. This means that `AC = BD`, where `AC` and `BD` represent the lengths of the diagonals.

`4`. Diagonal Symmetry: A rectangle has two lines of symmetry, passing through the midpoints of opposite sides. These lines intersect each other at the center of the rectangle, dividing it into four congruent right triangles.

`5`. Area: The area of a rectangle can be calculated using the formula: ` \text{Area} = \text{length} \times \text{width}`.

`6`. Parallel Sides: Opposite sides of a rectangle are parallel to each other.

`7`. Adjacent Angles: Adjacent angles in a rectangle are supplementary, meaning they add up to `180` degrees.

Parallelogram

A parallelogram is a type of quadrilateral with several distinct properties:

`1`. Opposite sides: In a parallelogram, opposite sides are equal in length. This means that `PQ = RT` and `PR = QT`, where `PQ`, `RT`, `PR`, and `QT` represent the lengths of the sides of the parallelogram.

`2`. Parallel sides: Opposite sides of a parallelogram are parallel to each other. This property distinguishes a parallelogram from other quadrilaterals.

`3`. Opposite angles: Opposite angles in a parallelogram are equal. This means that `∠P = ∠T` and `∠Q = ∠R`.

`4`. Consecutive angles: Consecutive angles in a parallelogram are supplementary, meaning they add up to `180` degrees. For example, `∠P + ∠R = 180^\circ` and `∠Q + ∠T = 180^\circ`.

`5`. Diagonals: The diagonals of a parallelogram bisect each other. This means that the point where the diagonals intersect divides each diagonal into two equal segments.

`6`. Area: The area of a parallelogram can be calculated using the formula: ` \text{Area} = \text{base} \times \text{height}`, where the base is any of the parallel sides, and the height is the perpendicular distance between the parallel sides.

Trapezium

The term "trapezium" can have different meanings depending on the geographical region. In some countries, such as the United Kingdom, a trapezium refers to a quadrilateral with no parallel sides, while in others, such as the United States, it refers to a quadrilateral with at least one pair of parallel sides. 

For consistency, let's discuss both interpretations:

`1`. United Kingdom (No parallel sides):

• A trapezium in this context is a quadrilateral with no parallel sides. It may also be referred to as a "trapezoid" in the United States. In this type of trapezium, the lengths of all four sides can vary, and the angles may also vary.

`2`. United States (At least one pair of parallel sides):

• In the United States, a trapezium is a quadrilateral with at least one pair of parallel sides. This type of trapezium has two non-parallel sides called the legs and two parallel sides called the bases. The distance between the bases is the height of the trapezium.

In both cases, the following properties can be observed:

`1`. Sum of Angles: The sum of the interior angles of a trapezium is always `360` degrees.

`2`. Diagonals: The diagonals of a trapezium intersect each other, dividing the trapezium into four triangles. The lengths of the diagonals and their intersection point may vary depending on the specific properties of the trapezium.

`3`. Area: The area of a trapezoid can be calculated using the formula: ` \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}`, where the "height" is the perpendicular distance between the parallel sides.

Rhombus

A rhombus is a special type of quadrilateral with several distinct properties:

Let's consider the rhombus `EFGH`:

`1`. Equal Sides: In a rhombus, all four sides are of equal length. So, in rhombus `EFGH`, `EF = FG = GH = HE`.

`2`. Opposite Angles: Opposite angles in a rhombus are equal. Therefore, `∠E = ∠G` and `∠F = ∠H`.

`3`. Diagonals: The diagonals of a rhombus bisect each other at right angles. Thus, the point where the diagonals intersect divides each diagonal into two equal segments, forming right angles. 

`4`. Diagonal Symmetry: Rhombus `EFGH` has two lines of symmetry. These lines pass through the opposite angles.

`5`. Area: The area of rhombus `EFGH` can be calculated using the formula: ` \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2`, where `\text{diagonal}_1` and `\text{diagonal}_2` are the lengths of the diagonals `EG` and `FH`.

Kite

A kite is a special type of quadrilateral with the following properties:

`1`. Adjacent Sides: In a kite, two pairs of adjacent sides are congruent. So, in kite `ABCD`, `AB = BC` and `CD = DA`.

`2`. Diagonals: The diagonals of a kite intersect at a right angle and bisect each other. Let the diagonals intersect at point `O`. So, `AC` and `BD` are diagonals of kite `ABCD`, and they intersect at right angles at point `O`.

`3`. Diagonal Symmetry: A kite has one line of symmetry. This line passes through `B` and `D` dividing the kite into two congruent triangles.

`4`. Opposite Obtuse Angles: A kite has one pair of opposite angles that are obtuse and equal in measure. Here `∠A = ∠C`.

`5`. Area: The area of rhombus `EFGH` can be calculated using the formula: ` \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2`, where `\text{diagonal}_1` and `\text{diagonal}_2` are the lengths of the diagonals `AC` and `BD`.

Area of Quadrilaterals

The area of a quadrilateral can be calculated using different methods, depending on the given information about the quadrilateral. Here is the summary of common methods for finding the area of different types of quadrilaterals:

Real-Life Applications

Quadrilaterals have numerous real-life applications across various fields. Here are some examples:

`1`. Architecture and Construction: Quadrilaterals are fundamental shapes used in architectural design and construction. Buildings, rooms, doors, and windows often have rectangular or square shapes. Parallelograms are also used in designing roofs and floors, while trapezoids are used in constructing ramps, stairs, and awnings.

`2`. Urban Planning: City planners use quadrilaterals extensively to layout streets, blocks, and neighborhoods. City blocks are often rectangular or square, while roads and intersections may form parallelograms and trapezoids.

`3`. Geometry and Engineering: Engineers use quadrilaterals in various applications, such as designing structures, bridges, and roadways. Understanding the properties of quadrilaterals is crucial for optimizing designs and ensuring structural integrity.

`4`. Manufacturing and Carpentry: Quadrilaterals play a vital role in manufacturing processes, such as cutting materials into specific shapes. Carpentry involves working with various quadrilaterals to build furniture, cabinets, and other wooden structures.

`5`. Graphics and Design: In graphic design and digital modeling, quadrilaterals are used to create shapes, patterns, and layouts. Computer-aided design (CAD) software utilizes quadrilaterals extensively for drafting and modeling.

`6`. Land Surveying: Surveyors use quadrilaterals to measure and divide land plots. The boundaries of land parcels are often represented by quadrilateral shapes, and surveying techniques involve calculating areas and distances using quadrilateral formulas.

`7`. Navigation and Mapping: Maps and navigation systems represent geographical features using quadrilateral shapes. Quadrilaterals are used to define land boundaries, bodies of water, and other landmarks on maps.

Solved Examples

Example `1`: In quadrilateral `ABCD, ∠A = 70°, ∠B = 110°`, and `∠C = 100°`. Find the measure of `∠D`.

Solution:

We know that the sum of the interior angles of a quadrilateral is always `360°`. Therefore, we can use this property to find the measure of `∠D`.

First, let's find the sum of the angles given:

Sum of interior angles of quadrilateral `ABCD = ∠A + ∠B + ∠C + ∠D = 360°`

Substituting the given angle measures:

`70^\circ + 110^\circ + 100^\circ + \text{∠D} = 360^\circ`

`280^\circ + \text{∠D} = 360^\circ`

Now, let's solve for `∠D`:

`\text{∠D} = 360^\circ - 280^\circ`

`\text{∠D} = 80^\circ`

So, the measure of `∠D` in quadrilateral `ABCD` is `80^\circ`.

Example `2`: The perimeter of a square is `24` meters. Find its area.

Solution:

Given that the perimeter of the square is `24` meters, we know that the perimeter of a square is equal to four times the length of one side. 

So, let ` s ` be the length of one side of the square.

The perimeter formula for a square is:

\( \text{Perimeter} = 4s \)

Given that the perimeter is `24` meters, we can write:

\( 24 = 4s \)

Solving for ` s `:

`s = \frac{24}{4} = 6 \text{ meters}`

Now that we know the side length of the square is `6` meters, we can use this to find its area.

The formula for the area of a square is:

\( \text{Area} = s^2 \)

Substituting ` s = 6 ` into the formula:

\( \text{Area} = 6^2 = 36 \text{ square meters} \)

So, the area of the square is ` 36 \text{ square meters} `.

Example `3`: The perimeter of quadrilateral `PQRS` is `56` units. `PQ = 2x + 5`, `QR = 3x - 4`, `RS = 4x – 3`, and `SP = 5x – 2`. Find the length of the shortest side of the quadrilateral.

Solution:

Let's say we have a quadrilateral `PQRS` with sides defined as follows:

• ` PQ = 2x + 5 `
• ` QR = 3x - 4 `
• ` RS = 4x + 3 `
• ` SP = 5x - 2 `

And the perimeter of the quadrilateral is `56` units.

To find the length of the shortest side, we follow this approach:

\(PQ + QR + RS + SP = 56\)

Substituting the expressions for each side:

\((2x + 5) + (3x - 4) + (4x + 3) + (5x - 2) = 56\)

Combine like terms:

\(2x + 5 + 3x - 4 + 4x + 3 + 5x - 2 = 56\)

\( 14x + 2 = 56 \)

\( 14x = 54 \)

`x = 54/14`

`x = 27/7`

Now, find the lengths of each side:

• ` PQ = 2(27/7) + 5 = 64/7 `
• ` QR = 3(27/7) - 4 = 81/7 - 4 = 65/7 `
• ` RS = 4(27/7) + 3 = 108/7 + 3 = 129/7 `
• ` SP = 5(27/7) - 2 = 135/7 - 2 = 121/7 `

The shortest side is ` QR ` with a length of ` 65/7 ` units.

Example `4`: Find the area of a parallelogram with a base of `10` meters and a height of `6` meters.In a quadrilateral ` UVWX `, having ` \angle UVX = 60^\circ `, ` \angle VWX = 100^\circ `, and ` \angle UVX = 70^\circ `. Calculate the measure of ` \angle UWX ` and state whether it is a concave quadrilateral.

Solution:

Certainly, here's another similar example:

In quadrilateral `UVWX`, we have:

• `∠UVX = 60°`
• `∠VWX = 100°`
• `∠UWX = 70°`

We want to find the measure of angle `UWX` and determine whether it is a concave quadrilateral.

Using the fact that the sum of the angles in a quadrilateral is always `360` degrees:

\( ∠UVX + ∠VWX + ∠UWX + ∠UVX = 360 \)

Substituting the given angle measures:

\( 60° + 100° + ∠UWX + 70° = 360 \)

Solving for `∠UWX`:

\(230° + ∠UWX = 360\)

\( ∠UWX = 360 - 230 \)

\(∠UWX = 130°\)

So, the measure of `∠UWX` is `130` degrees.

To determine if the quadrilateral is concave, we check if any of its angles are greater than `180` degrees. Since the measure of `∠UWX` is `130` degrees, which is not greater than `180` degrees, quadrilateral `UVWX` is not concave.

Example `5`: Given a trapezoid `PQRS` `(PQ` `||` `RS)` with median `MN`. `PQ = 4x + 3, RS = 3x - 1`, and `MN = 5x - 2`. We want to find the value of `MN`.

Solution:

Using the property of trapezoids where the length of the median is equal to the average of the lengths of the bases:

`MN = \frac{PQ + RS}{2}`

Substitute the given values:

`MN = \frac{4x + 3 + 3x - 1}{2}`

Simplify the expression:

`MN = \frac{4x + 3x + 3 - 1}{2}`

`MN = \frac{7x + 2}{2}`

Now, we know that ` MN ` is equal to ` 5x - 2 `. So, we can set up an equation:

`\frac{7x + 2}{2} = 5x - 2`

Now, solve for ` x `:

\( 7x + 2 = 10x - 4 \)

Subtract ` 7x ` from both sides:

\( 2 = 3x - 4 \)

Add `4` to both sides:

\( 6 = 3x \)

Divide both sides by `3`:

\( x = 2 \)

Now that we've found the value of ` x `, we can substitute it back into the expression for ` MN `:

\( MN = 5x - 2 \)

\( MN = 5(2) - 2 \)

\( MN = 10 - 2 \)

\( MN = 8 \)

So, the value of ` MN ` is `8`.

Example `6`: Find the area of a kite with diagonals of lengths `16` meters and `12` meters.

Solution:

Using the formula for the area of a kite, which is ` \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 `, we can substitute the given values:

`\text{Area} = \frac{1}{2} \times 16 \ \text{m} \times 12 \ \text{m}`

`\text{Area} = \frac{1}{2} \times 192 \ \text{m}^2`

\( \text{Area} = 96 \, \text{m}^2 \)

So, the area of the kite is \( 96 \, \text{m}^2\).

Therefore, the area of a kite with diagonals of lengths `16` meters and `12` meters is \(96 \, \text{m}^2\).

Practice Problems

Q`1`. In quadrilateral `PQRS, ∠P = 90°, ∠Q = 120°`, and `∠R = 70°`. Find the measure of `∠S`.

1. `80^\circ`
2. `90^\circ`
3. `180^\circ`
4. `120^\circ`

Answer: a

Q`2`. The perimeter of quadrilateral `ABCD` is `46` units. `AB = x + 7, BC = 2x + 3, CD = 3x – 8`, and `DA = 4x – 6`. Find the length of the shortest side of the quadrilateral.

1. Shortest side of the quadrilateral is ` BC ` with a length of `7` units
2. Shortest side of the quadrilateral is ` CD ` with a length of `8` units
3. Shortest side of the quadrilateral is ` CD ` with a length of `7` units
4. None

Answer: c

Q`3`. Find the area of a parallelogram with a base of `12` centimeters and a height of `8` centimeters.

1. ` 20 \ \text{cm}^2 `
2. ` 16 \ \text{cm}^2 `
3. ` 69 \ \text{cm}^2 `
4. ` 96 \ \text{cm}^2 `

Answer: d

Q`4`. In a trapezoid `ABCD` `(AB` `||` `DC)` with median `EF`. `AB = 3x – 5, CD = 2x -1` and `EF = 2x + 1`. Find the value of `EF`.

1. `17` units
2. `18` units
3. `15` units
4. `7` units

Answer: a

Q`5`. In a quadrilateral ` PQRS `, having ` \angle PQR = 40^\circ `, ` \angle QSP = 70^\circ `, and ` \angle SRP = 110^\circ `. Calculate the measure of ` \angle QRS ` and state whether it is a concave quadrilateral.

1. ` 140^\circ ` and quadrilateral ` PQRS ` is concave
2. ` 140^\circ ` and quadrilateral ` PQRS ` is not concave
3. Both a and b
4. None

Answer: b

Q`6`. Find the area of a kite with diagonals of lengths `10` meters and `6` meters.

1. `60 \ \text{m}^2`
2. `20 \ \text{m}^2`
3. `30 \ \text{m}^2`
4. `16 \ \text{m}^2`

Answer: c

Frequently Asked Questions

Q`1`. What is a quadrilateral?

Answer: A quadrilateral is a polygon with four sides, four angles, and four vertices.

Q`2`. What are the properties of a quadrilateral?

Answer: The properties of a quadrilateral include:

• Four sides
• Four angles
• Sum of interior angles equals `360` degrees
• Diagonals intersect each other inside the quadrilateral
• Diagonals divide the quadrilateral into four triangles with equal areas (if the diagonals bisect each other)
• Depending on the type of quadrilateral, additional properties may apply.

Q`3`. How do you classify quadrilaterals?

Answer: Quadrilaterals can be classified based on various criteria, including:

• The length of their sides
• The measure of their angles
• Whether their opposite sides are parallel
• Whether they have equal sides or angles

Q`4`. What are some common types of quadrilaterals?

Answer: Some common types of quadrilaterals include:

• Square
• Rectangle
• Parallelogram
• Rhombus
• Trapezium (Trapezoid in American English)
• Kite

Q`5`. What is the sum of interior angles in a quadrilateral?

Answer: The sum of interior angles in a quadrilateral is always `360` degrees.

Q`6`. What is the perimeter of a quadrilateral?

Answer: The perimeter of a quadrilateral is the sum of the lengths of its four sides.

Q`7`. Can a quadrilateral have equal sides but unequal angles?

Answer: Yes, a quadrilateral can have equal sides but unequal angles. An example of such a quadrilateral is a rhombus.