Quadrilaterals space a special kind of polygon. As with triangles and also other polygons, quadrilaterals have actually special properties and also can it is in classified by characteristics of your angles and also sides. Knowledge the properties of different quadrilaterals can aid you in solving problems that involve this type of polygon.

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Picking personal the surname “quadrilateral” helps you recognize what the refers to. The prefix “quad-” method “four,” and also “lateral” is derived from the Latin word for “side.” so a square is a four-sided polygon.

Since the is a polygon, you know that it is a two-dimensional figure comprised of directly sides. A quadrilateral additionally has 4 angles developed by its four sides. Below are some examples of quadrilaterals. Notice that each figure has 4 straight sides and also four angles.

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

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

You can additionally use your knowledge of triangles as a way to understand why the amount of the inner angles of any kind of quadrilateral is 360°. Any type of quadrilateral have the right to be divided into two triangles as shown in the images below.

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

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

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


Specific species of Quadrilaterals


Let’s start by examining the team of quadrilateral that have actually two bag of parallel sides. These quadrilaterals are dubbed parallelograms They take it a variety of shapes, but one classic example is presented below.

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

*
 and
*
. The parallel political parties are also the very same length:
*
 and
*
. These relationships room true for all parallelograms.

There space two special cases of parallelograms that will be acquainted to you from your earliest experiences through geometric shapes. The very first special instance is called a rectangle. By definition, a rectangle is a parallelogram since its pairs of the contrary sides space parallel. A rectangle also has the distinct characteristic that every one of its angles are best angles; all 4 of that is angles room congruent.

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

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A square is also a rectangle since it has two to adjust of parallel sides and also four ideal angles. A square is additionally a parallelogram due to the fact that its the contrary sides space parallel. So, a square have the right to be share in any type of of these 3 ways, v “parallelogram” gift the least details description and “square,” the most descriptive.

Another quadrilateral that you might see is dubbed a rhombus. All 4 sides that a rhombus are congruent. Its properties include that each pair the opposite political parties is parallel, likewise making it a parallelogram.

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

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

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You deserve to use the properties of parallelograms to deal with problems. Think about the example that follows.


Example

Problem

Determine the steps of

*
 and
*
.

*

 is opposite

 is the contrary

Identify opposite angles.

A property of parallelograms is the opposite angles room congruent.

*
 = 60°, so
*
 = 60°

*
 = 120°, so
*
 = 120°

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

Answer

*
 = 60° and
*
 = 120°


There is one more special type of quadrilateral. This quadrilateral has actually the home of having 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 no parallel. Girlfriend can conveniently imagine that if you expanded sides and , they would intersect above the figure.

If the non-parallel sides of a trapezoid room congruent, the trapezoid is referred to as an isosceles trapezoid. Prefer the similarly named triangle that has actually two political parties of equal length, the isosceles trapezoid has a pair that opposite political parties of same length. The other pair the opposite sides is parallel. Below is an instance of an isosceles trapezoid.

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In this trapezoid ABCD,

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 and
*
.

Which the the adhering to statements is true?

A) part trapezoids room parallelograms.

B) all trapezoids space quadrilaterals.

C) every rectangles space squares.

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


A) part trapezoids are parallelograms.

Incorrect. Trapezoids have actually only one pair the 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 room quadrilaterals.

B) every trapezoids space quadrilaterals.

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

C) all rectangles space squares.

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

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

Incorrect. Every parallelograms room quadrilaterals, so if that is a parallelogram, that is additionally a quadrilateral. The exactly answer is that all trapezoids space quadrilaterals.

You have the right to use the properties of quadrilaterals to solve difficulties involving trapezoids. Take into consideration the example below.


Example

Problem

Find the measure of

*
.

*

*
 = 360°

The amount of the procedures of the interior angles the a quadrilateral is 360°.

*
 = 90°

*
 = 90°

The square symbol indicates a right angle.

60° +  + 90° + 90° = 360°

Since three of the four angle procedures are given, you can find the 4th angle measurement.

 + 240° = 360°

 = 120°

Calculate the measurement of

*
.

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

Answer

*
 = 120°


Name of 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 bag of parallel sides.

4 best angles (90°).

Opposite sides room parallel and also congruent.

All angles room congruent.

Square

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4 congruent sides.

4 ideal 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 space all instances of quadrilaterals. This quadrilaterals knife their difference based on your properties, including the number of pairs the parallel sides they have and also their angle and also side measurements.