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20 June,2023

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The term quadrilateral is derived from the Latin terms ‘Quadra’ which means four and ‘Latus’ which means ‘sides’. A quadrilateral always does not have equal lengths on all of its four sides. As a result, depending on the sides and angles, we can have several sorts of quadrilaterals.

From house designs to windows, doors, swimming pools, football fields, boxes, and paper – the quadrilateral is the shape** **that is most frequently used in our daily lives. Other quadrilateral examples are the technical devices include televisions, laptops, computers, and mobile phones. Books, chart paper, copies, and other office supplies. So, we find it impossible to imagine a world without quadrilaterals.

Egyptians divided a rectangle into two congruent triangles using the principle of congruence, which they employed to build the Great Pyramids of Giza.

Leonardo Da Vinci also used the idea of triangular congruence, which comes from the diagonal of quadrilaterals, to create the renowned Mona Lisa.

Let us now look at some more fascinating facts about quadrilateral.

A quadrilateral is a closed form of polygon with 4 sides, 4 vertices, and 4 angles. 4 non-collinear points are joined together to draw a quadrilateral. The sum of total angles of a quadrilateral is 360 degrees.

The quadrilateral has angles at each of its four vertices, or corners. The angles formed by the vertices of a quadrilateral named ABCD are A, B, C, and D. A quadrilateral has the following sides: AB, BC, CD, and DA.

The diagonals are obtained by joining the opposing vertices of the quadrilateral. The diagonals of the quadrilateral ABCD are shown in the figure below as AC and BD.** **

A quadrilateral has four inner angles, which together total 360°. Using a quadrilateral’s angle sum attribute, this number was calculated. In line with the polygon’s angle sum attribute, the internal angle sum can be calculated using the number of triangles that can be formed in the polygon.

The proof is as below:

Consider a quadrilateral ABCD.

To prove, ∠A+∠B+∠C+∠D=360∘.

In this method, we join A and C first, i.e., we draw the diagonal AC.

In △ABC,

∠CAB+∠ABC+∠BCA=180∘ [Sum of all angles of a triangle is 180∘]

In △ACD,

∠CAD+∠ADC+∠DCA=180∘ [Sum of all angle of a triangle is 180∘]

Adding (1) and (2), we get,

(∠CAB+∠ABC+∠BCA) + (∠CAD+∠ADC+∠DCA) = 180∘+180∘

⟹ ∠ABC+∠ADC + (CAB+CAD) + (BCA+DCA) = 360∘

⟹ ∠ABC+∠ADC+∠BAD+∠BCD = 360∘

∴∠A+∠B+∠C+∠D=360∘

**Sum of Exterior Angles:**

The external angles of a quadrilateral are those that are created between a side and a line that extends from a neighbouring side. The exterior and interior angles converge to produce a straight line in the aforementioned example, which is the reason they form a linear pair. As a result, if we know the value of one internal angle in a quadrilateral, we may calculate its equivalent exterior angle.

Exterior angle = 180° – Interior angle.

When a quadrilateral’s inner angle, you can use this formula to ascertain the value of the matching exterior angle. Since both the angles make up a linear pair, they are complementary, which means that their sum is always 180°.

**Diagonals:**

A diagonal connects two solid or polygonal vertices whose vertices are not located on the same edge. In general, a diagonal is a line that connects the vertices of a quadrilateral by drawing slanting lines.

The line segment connecting the quadrilateral’s opposing, non-adjacent vertices or corners is known as the diagonal.

Since a quadrilateral has four sides, it also has two diagonals. A quadrilateral’s diagonals are parallel to and intersect one another. A quadrilateral is divided into two congruent right triangles if a diagonal bisects it in half.

**PARALLELOGRAM:**

A parallelogram is a type of quadrilateral which is having two sets of parallel sides. A parallelogram has opposing sides that are the same length and angles that are the same size. Moreover, the interior angles on the same side that is further than the transversal. The interior angles’ total sum is 360°.

**Properties**

- The opposite sides are parallel and congruent.
- Congruent angles are opposite angles.
- Adjacent angles are helpful.
- Each diagonal splits the parallelogram into two congruent triangles and bisects the other diagonal.
- If a parallelogram has a right angle, all other angles are also right, and the parallelogram turns into a rectangle.

- Area = L * H
- Perimeter = 2(L+B)
- RECTANGLE:

A rectangle is a four-sided polygon with opposite sides that are parallel and equal to each other. It belongs to a category of quadrilaterals where each of the four angles is a straight angle or measures 90 degrees. A particular kind of parallelogram with all equal angles forms a rectangle.

- The opposite sides are parallel and congruent.
- Every angle is correct.
- The diagonals divide each other equally and are congruent with one another.
- At the intersection of diagonals, opposite angles are formed that are congruent.
- A particular kind of parallelogram with the right angles is a rectangle.

- Length of a diagonal = (L2 + B2); where L = length and B = breadth.
- •Area = L * B
- •Perimeter = 2(L+B)
- SQUARE
**:**

A square is a common quadrilateral which has 4 equal sides and 4 equal angles. Angles in a square are either straight or 90 degrees. The square’s diagonals are also equal and intersect at a 90-degree angle.

- All sides and angles line up.
- The two sides are directly opposite one another.
- The diagonals are aligned.
- The diagonals are divided in half and parallel.
- A parallelogram with equal angles and sides is referred to as a square.
- When a parallelogram’s diagonals and their right bisectors are equal, it likewise becomes a square.

- Diagonal of square length = L √2; where L = length
- Area = L2.
- Perimeter = 4L
- RHOMBUS:

A rhombus is a subset of a parallelogram. The rhombus has 4 opposite sides and angles which are parallel and equal. A rhombus also has equal-length sides on each side, and its diagonals meet at right angles to form its shape. The rhombus is also referred to as a diamond or rhombus.

- The sides all congruent.
- Congruent angles are opposite angles.
- The diagonals are parallel to and split in half.
- Adjacent angles are supplementary (A + B = 180°, for instance).
- A parallelogram whose diagonals are perpendicular to one another is a rhombus.

If a and b are the diagonal length of a rhombus, then

- Area = (a* b) / 2
- Perimeter = 4L
**TRAPEZOID**:

A polygon with only one set of parallel sides is called a trapezium. The parallel bases of a trapezium are another name for these parallel sides. Trapezoids have 2 extra sides that are not parallel and are generally known as their legs.

**Properties**

- The bases of a trapezium are parallel to one another.
- In a trapezium there are no congruent sides, angles, or diagonals.
- A trapezium is closed, 4-sided shapes that has a perimeter and cover a particular region.
- The bases of the trapezium are the sides that are parallel to one another.
- Legs or lateral sides refer to the non-parallel sides.
- The height is the separation between the parallel sides.
**KITE**:

A quadrilateral called a kite has two adjacent pairs of congruent sides that are each the same length and is next to one another.

Properties:

- Where the uneven sides meet, the two angles are equal.
- It can be thought of as two congruent triangles sharing a base.
- It has 2 diagonals which intersect at right-angle with one another.
- The other diagonal is divided by the longer or main diagonal.
- A kite’s primary diagonal is symmetrical.
- The kite is divided into two different sizes of isosceles triangles divided by two different size diagonals.

IV. **OTHER IMPORTANT QUADRILATERALS:**

**ISOSCELES TRAPEZOID**:

A trapezium with non-parallel sides and equal-sized base angles is known as an isosceles trapezium. In other terms, a trapezoid is an isosceles trapezoid if its two opposite sides (or bases) are parallel and its two non-parallel sides are of equal length.

**Properties**

- It has a symmetry axis. It only has one line of symmetry connecting the middle of the parallel sides and no rotational symmetry.
- Only the two sides of bases are parallel in isosceles trapezoid.
- All of the other sides, save the base, are non-parallel and equal in length.
- The base angles are equal; the diagonals are of equal length.
- The complement of opposite angles is 180°.
- A line segment perpendicular to the bases connects the midpoints of the parallel sides.

**Formula for Isosceles Trapezoid**

Area = (sum of parallel sides ÷ 2) × h

Perimeter = sum of all sides

B. **CYCLIC QUADRILATERAL**:

Cyclic quadrilateral refers to a quadrilateral whose four vertices are all located on a circle. Another name for it is an inscribed quadrilateral. The circumscribed circle is the circle that has every vertex of any polygon along its perimeter.

**Properties**

- The sum of two opposed angles in a cyclic quadrilateral is 1800(supplementary).
- It is a cyclic quadrilateral if the product of two opposed angles is additional.
- The circle’s perimeter is where a cyclic quadrilateral’s four vertices are located.
- Simply connect the midpoints of the four sides in descending order to create a rectangle or a parallelogram.
- When a cyclic quadrilateral is built, an external angle is created that is equal to the interior angle on the other side.
- The perpendicular bisectors must be contemporaneous if it is a cyclic quadrilateral.
- The centre O of a cyclic quadrilateral is where the four perpendicular bisectors of the specified four sides meet.

**Formula for Cyclic Quadrilateral**

Radius

Area

**Endnote:**

The quadrilateral is a fundamental topic in mathematics. Since degree courses will involve the concepts of area and perimeter of quadrilaterals, students studying for specific exams should get familiar with quadrilaterals.

There are many different sizes and shapes of quadrilaterals, some of which are more elusive in the actual world. So, it is quite normal to feel confused when working on mathematical concepts revolving around quadrilaterals.

One of the common uses of quadrilaterals in daily life is architecture. Quadrilaterals and triangles can both be used to make a variety of shapes. For instance, the architectural layout of some homes might be entirely dependent on the utilisation of various quadrilateral shapes.

Quadrilaterals are also used in a variety of everyday objects, including picture frames, table tops, doors, and books. In computer programming, logo design, visual art, sculpting, packaging, and web design, quadrilaterals are frequently used. Nearly all facets of daily life involve quadrilaterals.

While we hope this blog was helpful in clearing out the queries regarding the basics, feel free to reach out to us for more customised guidance.

A closed quadrilateral has 4 sides, 4 vertices, and 4 angles. It is a form of polygon. You can join four non-collinear points to draw a quadrilateral. If you add all of the interior angles, the sum total becomes 360 degrees.

There are six types of quadrilateral. These are:

- Parallelogram
- Trapezium
- Rhombus
- Rectangle
- Square
- Kite

A square is a two-dimensional shape in geometry with four equal sides and four angles that are each 90 degrees.

Yes a quadrilateral can have all four sides equal with all equal angles. This type of quadrilateral is known as square.

Can a quadrilateral have all sides equal but not all angles?

Yes rhombus is a type of quadrilateral that has all equal sides but not equal angles.

A quadrilateral can have two opposite sides parallel, which is known as parallelogram.