Quadrilaterals room a special kind of polygon. Just like triangles and also other polygons, quadrilaterals have actually special properties and can it is in classified by attributes of their angles and sides. Expertise the nature of various quadrilaterals can help you in solving troubles that indicate this form of polygon.

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Picking personal the name “quadrilateral” helps you understand what that refers to. The prefix “quad-” means “four,” and also “lateral” is derived from the Latin word because that “side.” therefore a square is a four-sided polygon.

Since it is a polygon, you know that the is a two-dimensional figure consisted of of right sides. A quadrilateral likewise has four angles formed by its four sides. Below are some instances of quadrilaterals. An alert that each figure has four straight sides and also four angles.

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The sum of the interior angles of any type of quadrilateral is 360°. Consider the two examples below.

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You can draw plenty of quadrilaterals such as these and carefully measure up the four angles. Friend would find that for every quadrilateral, the sum of the interior angles will constantly be 360°.

You can likewise use your expertise of triangles together a method to understand why the amount of the internal angles of any kind of quadrilateral is 360°. Any type of quadrilateral have the right to be split into 2 triangles as shown in the photos below.

In the very first image, the quadrilaterals have each been separated into two triangles. The angle measurements of one triangle are presented for each.

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These measurements include up to 180º. Now look in ~ the measurements for the various other triangles—they also include up to 180º!

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Since the amount of the internal angles of any triangle is 180° and also there space two triangles in a quadrilateral, the sum of the angle for each quadrilateral is 360°.


Specific species of Quadrilaterals


Let’s start by assessing the team of square that have two bag of parallel sides. These quadrilaterals are called parallelograms They take it a range of shapes, however one standard example is displayed below.

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Imagine prolonging the pairs of opposite sides. They would never intersect because they room parallel. Notice, also, the the opposite angles of a parallelogram room congruent, as are the the opposite sides. (Remember that “congruent” means “the exact same size.”) The geometric symbol for congruent is , so you deserve to write

*
 and
*
. The parallel sides are additionally the exact same length:
*
 and
*
. This relationships are true for every parallelograms.

There room two special instances of parallelograms that will be acquainted to you from your earliest experiences v geometric shapes. The very first special case is referred to as a rectangle. By definition, a rectangle is a parallelogram because its pairs of opposite sides are parallel. A rectangle additionally has the one-of-a-kind characteristic that every one of its angles are appropriate angles; all four of that is angles space congruent.

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The various other special instance of a parallelogram is a special kind of rectangle, a square. A square is among the most basic geometric shapes. It is a special instance of a parallel that has 4 congruent sides and also four right angles.

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A square is additionally a rectangle due to the fact that it has actually two sets of parallel sides and also four right angles. A square is likewise a parallelogram since its opposite sides are parallel. So, a square can be divide in any kind of of these 3 ways, with “parallelogram” being the least specific description and “square,” the many descriptive.

Another quadrilateral that you could see is referred to as a rhombus. All four sides of a rhombus space congruent. The properties encompass that every pair of opposite political parties is parallel, additionally making the a parallelogram.

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In summary, every squares space rectangles, yet not every rectangles room squares. Every rectangles room parallelograms, yet not all parallelograms room rectangles. And all of these shapes are quadrilaterals.

The diagram listed below illustrates the relationship between the different species of quadrilaterals.

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You can use the properties of parallelograms to fix problems. Consider the example that follows.


Example

Problem

Determine the measures of

*
 and
*
.

*

 is the opposite

 is the opposite

Identify opposite angles.

A home of parallelograms is the opposite angles room congruent.

*
 = 60°, for this reason
*
 = 60°

*
 = 120°, so
*
 = 120°

Use the offered angle measurements to determine measures of opposite angles.

Answer

*
 = 60° and
*
 = 120°


There is an additional special type of quadrilateral. This quadrilateral has actually the property of having actually only one pair of opposite political parties that room parallel. Right here is one instance of a trapezoid.

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Notice the

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, and also that  and  are not parallel. Girlfriend can easily imagine that if you prolonged sides and , they would intersect above the figure.

If the non-parallel sides of a trapezoid space congruent, the trapezoid is referred to as an isosceles trapezoid. Choose the likewise named triangle that has two sides of same length, the isosceles trapezoid has a pair of opposite sides of same length. The other pair of opposite political parties is parallel. Below is an instance of one isosceles trapezoid.

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In this trapezoid ABCD,

*
 and
*
.

Which the the complying with statements is true?

A) some trapezoids are parallelograms.

B) every trapezoids room quadrilaterals.

C) all rectangles space squares.

D) A shape cannot it is in a parallelogram and also a quadrilateral.


A) some trapezoids are parallelograms.

Incorrect. Trapezoids have actually only one pair that parallel sides; parallelograms have two bag of parallel sides. A trapezoid have the right to never be a parallelogram. The correct answer is the all trapezoids are quadrilaterals.

B) every trapezoids space quadrilaterals.

Correct. Trapezoids space four-sided polygons, so they are all quadrilaterals.

C) all rectangles space squares.

Incorrect. Part rectangles may be squares, yet not all rectangles have four congruent sides. Every squares are rectangles however. The correct answer is the all trapezoids space quadrilaterals.

D) A form cannot it is in a parallelogram and a quadrilateral.

Incorrect. All parallelograms room quadrilaterals, for this reason if it is a parallelogram, it is additionally a quadrilateral. The correct answer is the all trapezoids are quadrilaterals.

You deserve to use the nature of quadrilaterals come solve problems involving trapezoids. Think about the example below.


Example

Problem

Find the measure of

*
.

*

*
 = 360°

The amount of the steps of the inner angles of a quadrilateral is 360°.

*
 = 90°

*
 = 90°

The square symbol shows a best angle.

60° +  + 90° + 90° = 360°

Since 3 of the 4 angle steps are given, you can uncover the 4th angle measurement.

 + 240° = 360°

 = 120°

Calculate the measure of

*
.

From the image, you deserve to see that it is an obtuse angle, so its measure have to be greater than 90°.

Answer

*
 = 120°


Name the Quadrilateral

Quadrilateral

Description

Parallelogram

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2 pairs of parallel sides.

Opposite sides and opposite angles space congruent.

Rectangle

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2 pairs of parallel sides.

4 best angles (90°).

Opposite sides are parallel and congruent.

All angles space congruent.

Square

*

4 congruent sides.

4 appropriate angles (90°).

Opposite sides room parallel.

All angles room congruent.

Trapezoid

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Only one pair that opposite political parties is parallel.

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A square is a mathematical surname for a four-sided polygon. Parallelograms, squares, rectangles, and also trapezoids room all examples of quadrilaterals. This quadrilaterals knife their difference based on your properties, consisting of the number of pairs that parallel sides they have and also their angle and also side measurements.