Quadrilaterals room a special kind of polygon. Similar to triangles and other polygons, quadrilaterals have actually special properties and can be classified by features of their angles and sides. Knowledge the properties of different quadrilaterals can aid you in solving troubles that indicate this form of polygon.

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Picking apart the name “quadrilateral” helps you know what it refers to. The prefix “quad-” means “four,” and also “lateral” is derived from the Latin word for “side.” so a quadrilateral is a four-sided polygon.

Since the is a polygon, you know that the is a two-dimensional figure made up of straight sides. A quadrilateral also has four angles created by its 4 sides. Listed below are some instances of quadrilaterals. An alert that each number has four straight sides and also four angles.

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The amount of the interior angles of any type of quadrilateral is 360°. Take into consideration the two examples below.

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You might draw plenty of quadrilaterals such together these and carefully measure the four angles. Girlfriend would uncover that for every quadrilateral, the sum of the inner angles will constantly be 360°.

You can additionally use your knowledge of triangles as a means to understand why the sum of the internal angles of any kind of quadrilateral is 360°. Any quadrilateral can be divided into 2 triangles as presented in the photos below.

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

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

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


Specific types of Quadrilaterals


Let’s begin by assessing the group of quadrilateral that have actually two bag of parallel sides. This quadrilaterals are referred to as parallelograms They take a selection of shapes, but one classic example is presented below.

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

*
 and
*
. The parallel political parties are likewise the exact same length:
*
 and
*
. This relationships space true for every parallelograms.

There are two special situations of parallelograms that will certainly be familiar to friend from her earliest experiences v geometric shapes. The an initial special case is referred to as a rectangle. Through definition, a rectangle is a parallelogram due to the fact that its bag of the opposite sides room parallel. A rectangle likewise has the special characteristic that every one of its angle are ideal angles; all 4 of that is angles room congruent.

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

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A square is additionally a rectangle because it has two sets of parallel sides and four appropriate angles. A square is likewise a parallelogram due to the fact that its the contrary sides room parallel. So, a square have the right to be classified in any of these three ways, with “parallelogram” being the least specific description and “square,” the many descriptive.

Another quadrilateral that you could see is dubbed a rhombus. All four sides that a rhombus space congruent. That properties encompass that each pair of opposite political parties is parallel, additionally making it a parallelogram.

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

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

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You have the right to use the nature of parallelograms to solve problems. Take into consideration the instance that follows.


Example

Problem

Determine the measures of

*
 and
*
.

*

 is the opposite

 is opposite

Identify the contrary angles.

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

*
 = 60°, for this reason
*
 = 60°

*
 = 120°, for this reason
*
 = 120°

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

Answer

*
 = 60° and also
*
 = 120°


There is an additional special kind of quadrilateral. This quadrilateral has actually the home of having actually only one pair of opposite political parties that are parallel. Here is one example of a trapezoid.

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Notice that

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, and that  and  are not parallel. Friend can quickly imagine the if you extended sides and , they would intersect over the figure.

If the non-parallel political parties of a trapezoid are congruent, the trapezoid is dubbed an isosceles trapezoid. Like the similarly named triangle that has actually two political parties of same length, the isosceles trapezoid has actually a pair the opposite political parties of same length. The various other pair the opposite sides is parallel. Listed below is an instance of one isosceles trapezoid.

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In this trapezoid ABCD,

*
 and
*
.

Which that the adhering to statements is true?

A) some trapezoids are parallelograms.

B) all trapezoids space quadrilaterals.

C) every rectangles space squares.

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


A) some trapezoids are parallelograms.

Incorrect. Trapezoids have actually only one pair the parallel sides; parallelograms have two bag of parallel sides. A trapezoid deserve to never it is in a parallelogram. The exactly answer is the all trapezoids space quadrilaterals.

B) all trapezoids room quadrilaterals.

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

C) every rectangles space squares.

Incorrect. Part rectangles might be squares, but not every rectangles have four congruent sides. All squares room 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, therefore if the is a parallelogram, that is additionally a quadrilateral. The correct answer is the all trapezoids room quadrilaterals.

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


Example

Problem

Find the measure of

*
.

*

*
 = 360°

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

*
 = 90°

*
 = 90°

The square symbol suggests a best angle.

60° +  + 90° + 90° = 360°

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

 + 240° = 360°

 = 120°

Calculate the measurement of

*
.

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

Answer

*
 = 120°


Name the Quadrilateral

Quadrilateral

Description

Parallelogram

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2 bag of parallel sides.

Opposite sides and also opposite angles are congruent.

Rectangle

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2 bag of parallel sides.

4 appropriate angles (90°).

Opposite sides space parallel and also congruent.

All angles room congruent.

Square

*

4 congruent sides.

4 right angles (90°).

Opposite sides are parallel.

All angles room congruent.

Trapezoid

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Only one pair that opposite sides is parallel.

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