Quadrilaterals are a special type of polygon. Similar to triangles and also other polygons, quadrilaterals have special properties and also can it is in classified by features of their angles and also sides. Expertise the nature of different quadrilaterals can aid you in solving troubles that indicate this kind of polygon.

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Picking personal the surname “quadrilateral” helps you recognize what the refers to. The prefix “quad-” means “four,” and “lateral” is acquired from the Latin word for “side.” therefore a quadrilateral is a four-sided polygon.

Since that is a polygon, you recognize that the is a two-dimensional figure comprised of directly sides. A quadrilateral additionally has 4 angles formed by its four sides. Listed below are some instances of quadrilaterals. Notification that each figure has 4 straight sides and also four angles.

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The sum of the inner angles of any type of quadrilateral is 360°. Think about the two examples below.

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You could draw plenty of quadrilaterals such as these and also carefully measure up the 4 angles. Girlfriend would find that because that every quadrilateral, the amount of the inner angles will always be 360°.

You can likewise use your expertise of triangles together a method to recognize why the sum of the internal angles of any kind of quadrilateral is 360°. Any type of quadrilateral can be split into two triangles as displayed in the images below.

In the an initial image, the quadrilaterals have each been split into 2 triangles. The angle measurements of one triangle are shown for each.

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

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Since the amount of the internal angles of any kind of triangle is 180° and also there room two triangle in a quadrilateral, the amount of the angles for each square is 360°.


Specific types of Quadrilaterals


Let’s start by assessing the group of square that have actually two pairs of parallel sides. These quadrilaterals are dubbed parallelograms They take it a range of shapes, yet one classic example is presented below.

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Imagine expanding the bag of opposite sides. They would never ever intersect since they space parallel. Notice, also, the the opposite angle of a parallelogram room congruent, as are the opposite sides. (Remember the “congruent” method “the exact same size.”) The geometric symbol because that congruent is , therefore you deserve to write

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 and
*
. The parallel sides are also the very same length:
*
 and
*
. These relationships room true for every parallelograms.

There are two special instances of parallelograms that will certainly be familiar to you from your earliest experiences with geometric shapes. The an initial special instance is referred to as a rectangle. By definition, a rectangle is a parallelogram because its pairs of opposite sides room parallel. A rectangle also has the distinct characteristic that all of its angles are best angles; all four of the angles space congruent.

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The other special case of a parallel is a special type of rectangle, a square. A square is among the most basic geometric shapes. It is a special case of a parallel that has four congruent sides and four ideal angles.

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A square is also a rectangle since it has two set of parallel sides and four best angles. A square is likewise a parallelogram since its opposite sides are parallel. So, a square have the right to be share in any type of of these 3 ways, through “parallelogram” being the least certain description and “square,” the many descriptive.

Another quadrilateral that you can see is referred to as a rhombus. All four sides that a rhombus room congruent. The properties encompass that each pair of opposite political parties is parallel, also making that a parallelogram.

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In summary, every squares room rectangles, but not every rectangles space squares. Every rectangles room parallelograms, yet not all parallelograms room rectangles. And every one of these forms 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 deal with problems. Take into consideration the example that follows.


Example

Problem

Determine the steps of

*
 and
*
.

*

 is the contrary

 is the opposite

Identify the contrary angles.

A residential or commercial property of parallelograms is the opposite angles room congruent.

*
 = 60°, therefore
*
 = 60°

*
 = 120°, therefore
*
 = 120°

Use the provided angle measurements to identify measures of opposite angles.

Answer

*
 = 60° and also
*
 = 120°


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

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Notice that

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, and that  and  are no parallel. Friend can easily imagine the if you expanded sides and , they would certainly intersect above the figure.

If the non-parallel sides of a trapezoid space congruent, the trapezoid is called an isosceles trapezoid. Favor the likewise named triangle that has two political parties of same length, the isosceles trapezoid has actually a pair of opposite political parties of equal length. The various other pair of opposite sides is parallel. Below is an example of an isosceles trapezoid.

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In this trapezoid ABCD,

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 and
*
.

Which of the complying with statements is true?

A) part trapezoids room parallelograms.

B) every trapezoids room quadrilaterals.

C) every rectangles room squares.

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


A) some trapezoids space parallelograms.

Incorrect. Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid deserve to never be a parallelogram. The correct answer is the all trapezoids space quadrilaterals.

B) all trapezoids are quadrilaterals.

Correct. Trapezoids space four-sided polygons, for this reason they are all quadrilaterals.

C) all rectangles are squares.

Incorrect. Part rectangles might be squares, but not all rectangles have 4 congruent sides. All squares space rectangles however. The correct answer is that all trapezoids are quadrilaterals.

D) A shape cannot be a parallelogram and a quadrilateral.

Incorrect. Every parallelograms room quadrilaterals, for this reason if it is a parallelogram, the is likewise a quadrilateral. The exactly answer is that all trapezoids space quadrilaterals.

You deserve to use the nature of quadrilaterals come solve difficulties involving trapezoids. Consider the instance below.


Example

Problem

Find the measure up of

*
.

*

*
 = 360°

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

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 = 90°

*
 = 90°

The square symbol indicates a ideal angle.

60° +  + 90° + 90° = 360°

Since 3 of the four angle procedures are given, girlfriend can discover the fourth angle measurement.

 + 240° = 360°

 = 120°

Calculate the measure of

*
.

From the image, you can see the it is an obtuse angle, so its measure need to be better than 90°.

Answer

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 = 120°


Name of Quadrilateral

Quadrilateral

Description

Parallelogram

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2 pairs of parallel sides.

Opposite sides and also opposite angles are congruent.

Rectangle

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2 pairs of parallel sides.

4 best angles (90°).

Opposite sides room parallel and congruent.

All angles space congruent.

Square

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4 congruent sides.

4 best angles (90°).

Opposite sides space parallel.

All angles are congruent.

Trapezoid

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Only one pair of opposite political parties is parallel.

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A square is a mathematical surname for a four-sided polygon. Parallelograms, squares, rectangles, and trapezoids are all examples of quadrilaterals. This quadrilaterals earn their difference based on your properties, including the number of pairs that parallel political parties they have and their angle and side measurements.