Exploring Trapezoid and its Theorems – Properties, examples, and Application

A Trapezoid is a quadrilateral, which is a polygon with four sides. It has two parallel sides, called the bases, and two non-parallel sides, called the legs. The bases are typically different lengths, and the legs can be different lengths as well. The height of the trapezoid is the perpendicular distance between the bases.trapezoid and its theorems

Trapezoids can have a variety of shapes and sizes, as long as they meet the definition of having two parallel sides and two non-parallel sides.

trapezoid

Let's go through each of the properties of a trapezoid in detail, with examples.

Theorems of Trapezoid

1.   A trapezoid is isosceles if and only if the base angles are congruent.

2.   A trapezoid is an isosceles trapezoid if and only if the non-parallel sides are congruent.

3.   A trapezoid is isosceles if and only if the diagonals are congruent.

4.   If a trapezoid is isosceles, the opposite angles are supplementary.

5.   The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

6.   Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

Trapezoid is a quadrilateral with at least one pair of parallel sides. AB || CD. (if there are two pairs of parallel lines then it is a parallelogram) . When non-parallel sides in trapezoid are equal then it is known as isosceles trapezoid.trapezoid and its theorems

Theorem 1: A trapezoid is isosceles if and only if the base angles are congruent.trapezoid and its theorems

trapezoid

Given : ABCD is an isosceles trapezoid. AD = BC and AB || CD.

Prove that : ∠C = ∠D

trapezoid statements & reasons

Example: In a trapezoid PQRS, PQ||RS and PS = QR. If ∠S = 600 then find the remaining angles.

trapezoid angles

Solution :
PQ||RS and PS = QR, so trapezoid PQRS is an isosceles trapezoid.
In isosceles trapezoid, base angles are equal.(trapezoid and its theorems)
∠S = ∠R and ∠P = ∠Q
But ∠S = 600
∴ ∠R = 600
Let ∠P = ∠Q = x
Sum of all the angles in a quadrilateral is 360.
∴ ∠P + ∠Q + ∠S + ∠R = 360
x + x + 60 + 60 = 360
2x +120 = 360
2x = 360 -120
2x = 240
∴ x = 240/2
x = 120
∠P = ∠Q = 1200 trapezoid and its theorems

The parallel sides are called the bases of the trapezoid

In the trapezoid below, the parallel sides AB and CD are the bases.

parallel sides of trapezoid

The non-parallel sides are called the legs of the trapezoid

In the trapezoid below, AD and BC are the legs.

The height of the trapezoid is the perpendicular distance between the bases

In the trapezoid below, h is the height.

height of trapezoid

The sum of the angles of a trapezoid is 360 degrees

In the trapezoid below, the sum of angles A, B, C, and D is 360 degrees.

The two angles between each leg and a base are supplementary (add up to 180 degrees).

In the trapezoid below, angles A and D are supplementary, and angles B and C are supplementary.     

The mid-segment of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. Its length is the average of the lengths of the bases.

In the trapezoid below, E is the midpoint of AD, F is the midpoint of BC, and EF is the mid-segment.

trapezium midpoint

The length of EF is (AB + CD)/2.

The area of a trapezoid can be found using the formula: (1/2) x (sum of the bases) x height.

In the trapezoid below, the area is A = (1/2) x (AB + CD) x h.

area of trapezoid

For example, if AB = 4, CD = 6, and h = 3, then the area is

A = (1/2) x (4 + 6) x 3 = 15.

Some examples of trapezoids include:

In this example, AB and DC are the parallel sides, and AD and BC are the legs.

example of trapezoid

The trapezoid is also a rectangle, which is a special case of a trapezoid where the legs are perpendicular to the bases.

Trapezoids are commonly found in geometry and are used in many real-world applications, such as in architecture, engineering, and construction.

Practice

1)   In a trapezoid ABCD,AB|| CD and BC = AD. If m∠C=650 then find m∠D.

trapezium

2) PQRS is a trapezium in which PQ || RS. If ∠P = ∠Q = 40, find the measures of the other two angles.
3) In trapezoid ABCD, ∠B= 120° Find m∠C.
4) In a quadrilateral HELP, if EP = LH then what type of quadrilateral it is?
5) In a quadrilateral, the angles are in the ratio of 4:5:3:6.Find the measures of each angle.
6) If three angles in the trapezoid are 130°,120°,50° , and 2x°. Find x and the 4th angle.
7) Draw a isosceles trapezoid named PQRS, PS||QR and PQ = SR.

There are several types of trapezoids, based on the specific characteristics of their sides and angles. Let's discuss some of the most common types in detail, along with examples and diagrams:

Isosceles Trapezoid: An isosceles trapezoid is a trapezoid in which the two non-parallel sides are equal in length. The two parallel sides may also be of equal length.

Right Trapezoid: A right trapezoid is a trapezoid in which one of the angles formed between the two non-parallel sides is a right angle (90 degrees).

In the diagram below, PQRS is a right trapezoid, with sides PQ and RS parallel, and angle PQR a right angle.

right trapezoid

The area of a right trapezoid can be calculated using the formula

A = [(b1 + b2) / 2] * h, where b1 and b2 are the lengths of the parallel sides, and h is the height of the trapezoid.

scalene trapezoid

Scalene Trapezoid: A scalene trapezoid is a trapezoid in which all four sides have different lengths.

In the diagram below, ABCD is a scalene trapezoid, with sides AB and CD parallel and sides AD and BC of different lengths.

scalene trapezoid 1

The formula for the area of a scalene trapezoid is

A = [(AB + CD) / 2] * h, where AB and CD are the lengths of the parallel sides, and h is the height of the trapezoid.

Trapezium: In the United States, however, the term trapezium is not commonly used, and the term trapezoid is used to refer specifically to a quadrilateral with one pair of parallel sides.

Parallelogram: A parallelogram is a type of trapezoid in which both pairs of opposite sides are parallel. A parallelogram can also be thought of as a special case of a trapezoid in which the two parallel sides are equal in length.

In summary, the different types of trapezoids have varying properties and characteristics, and understanding them is important for working with trapezoids in geometry and other mathematical applications.

The importance of trapezoids lies in their various practical applications in different fields, such as:

Geometry: Trapezoids are an important concept in geometry and are often used as an example of a polygon with parallel sides. They also play a role in the study of area and perimeter of polygons.

Architecture: Trapezoidal shapes are often used in architecture for buildings, bridges, and other structures. This is because the trapezoidal shape provides stability and structural support, making it a popular choice for architects.

Engineering: Trapezoids are commonly used in engineering calculations, such as in the design of roads, runways, and canals. In these applications, the trapezoidal shape is used to create a slope or gradient that allows for the smooth flow of liquids or vehicles.

Mathematics: Trapezoids are used in various mathematical calculations, such as the trapezoidal rule in calculus, which is used to approximate the area under a curve.

In summary, trapezoids are an important shape with practical applications in various fields such as geometry, architecture, engineering, and mathematics. Their unique properties make them a versatile and useful tool for solving problems and creating structures that require stability and support.

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