Quadrilaterals are a special kind of polygon. As with triangles and also other polygons, quadrilaterals have actually special properties and can it is in classified by attributes of their angles and also sides. Knowledge the nature of various quadrilaterals can assist you in solving difficulties that show off this form of polygon.

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Picking personally the surname “quadrilateral” helps you recognize what that refers to. The prefix “quad-” method “four,” and also “lateral” is acquired from the Latin word because that “side.” therefore a square is a four-sided polygon.

Since the is a polygon, you understand that the is a two-dimensional figure consisted of of right sides. A quadrilateral also has four angles created by its four sides. Listed below are some examples of quadrilaterals. Notice that each figure has 4 straight sides and also four angles.

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The amount 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 together these and carefully measure the 4 angles. Friend would discover that for every quadrilateral, the sum of the internal angles will constantly be 360°.

You can additionally use your knowledge of triangles as a means to understand why the sum of the interior angles of any type of quadrilateral is 360°. Any quadrilateral deserve to be separated into 2 triangles as shown in the pictures below.

In the very first image, the quadrilaterals have actually each been divided into 2 triangles. The angle measurements of one triangle are presented for each.

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These measurements add up come 180º. Now look at the measurements for the other triangles—they also include up to 180º!

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


Specific types of Quadrilaterals


Let’s start by examining the group of quadrilaterals that have actually two pairs of parallel sides. These quadrilaterals are called parallelograms They take it a range of shapes, however one standard example is shown below.

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Imagine expanding the bag of opposite sides. They would never ever intersect due to the fact that they are parallel. Notice, also, the the opposite angles of a parallelogram are congruent, as are the the opposite sides. (Remember the “congruent” method “the very same size.”) The geometric symbol because that congruent is , for this reason you deserve to write

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 and
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. The parallel sides are additionally the same length:
*
 and
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. These relationships are true for all parallelograms.

There room two special cases of parallelograms that will be acquainted to friend from her earliest experiences with geometric shapes. The very first special situation is dubbed a rectangle. By definition, a rectangle is a parallelogram due to the fact that its bag of the contrary sides are parallel. A rectangle also has the unique characteristic that all of its angle are appropriate angles; all 4 of the angles room congruent.

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The other special situation of a parallel is a special form of rectangle, a square. A square is just one of the most simple geometric shapes. That is a special instance of a parallel that has 4 congruent sides and four right angles.

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A square is additionally a rectangle because it has two set of parallel sides and four ideal angles. A square is additionally a parallelogram because its the opposite sides space parallel. So, a square have the right to be share in any of these three ways, with “parallelogram” being the least specific description and “square,” the most descriptive.

Another quadrilateral that you might see is dubbed a rhombus. All 4 sides of a rhombus room congruent. That properties encompass that each pair the opposite sides is parallel, likewise making that a parallelogram.

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

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

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You have the right to use the properties of parallelograms to solve problems. Think about the instance that follows.


Example

Problem

Determine the steps of

*
 and
*
.

*

 is opposite

 is the contrary

Identify opposite angles.

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

*
 = 60°, therefore
*
 = 60°

*
 = 120°, so
*
 = 120°

Use the given angle dimensions to identify measures of opposite angles.

Answer

*
 = 60° and
*
 = 120°


There is one more special form of quadrilateral. This quadrilateral has the residential property of having actually only one pair that opposite political parties that are parallel. Here is one example of a trapezoid.

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Notice the

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

If the non-parallel sides of a trapezoid are congruent, the trapezoid is referred to as an isosceles trapezoid. Favor the likewise named triangle that has actually two political parties of equal length, the isosceles trapezoid has actually a pair of opposite political parties of equal length. The other pair of opposite political parties 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 adhering to statements is true?

A) some trapezoids space parallelograms.

B) every trapezoids are quadrilaterals.

C) every rectangles space squares.

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


A) part trapezoids room parallelograms.

Incorrect. Trapezoids have only one pair of parallel sides; parallelograms have actually two pairs of parallel sides. A trapezoid can never it is in a parallelogram. The correct answer is that all trapezoids are quadrilaterals.

B) all trapezoids space quadrilaterals.

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

C) every rectangles are squares.

Incorrect. Part rectangles may be squares, yet not every rectangles have four congruent sides. All squares are rectangles however. The exactly answer is the all trapezoids room quadrilaterals.

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

Incorrect. All parallelograms space quadrilaterals, therefore if it is a parallelogram, that is additionally a quadrilateral. The correct answer is that all trapezoids are quadrilaterals.

You deserve to use the properties of quadrilaterals come solve troubles involving trapezoids. Take into consideration the instance below.


Example

Problem

Find the measure up of

*
.

*

*
 = 360°

The sum of the measures of the inner angles of a square is 360°.

*
 = 90°

*
 = 90°

The square symbol suggests a ideal angle.

60° +  + 90° + 90° = 360°

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

 + 240° = 360°

 = 120°

Calculate the measurement of

*
.

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

Answer

*
 = 120°


Name of Quadrilateral

Quadrilateral

Description

Parallelogram

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2 bag of parallel sides.

Opposite sides and also opposite angles room congruent.

Rectangle

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2 pairs of parallel sides.

4 right angles (90°).

Opposite sides space parallel and congruent.

All angles space congruent.

Square

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4 congruent sides.

4 best angles (90°).

Opposite sides room parallel.

All angles room congruent.

Trapezoid

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Only one pair the opposite sides is parallel.

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A quadrilateral is a mathematical name for a four-sided polygon. Parallelograms, squares, rectangles, and also trapezoids space all examples of quadrilaterals. This quadrilaterals knife their distinction based on their properties, including the variety of pairs of parallel political parties they have and also their angle and also side measurements.