Quadrilaterals space a special kind of polygon. Similar to triangles and also other polygons, quadrilaterals have actually special properties and also can be classified by qualities of your angles and also sides. Knowledge the nature of various quadrilaterals can aid you in solving problems that indicate this type of polygon.

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Picking personal the name “quadrilateral” helps you recognize what the refers to. The prefix “quad-” way “four,” and “lateral” is acquired from the Latin word because that “side.” so a quadrilateral 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 created by its four sides. Below are some examples of quadrilaterals. An alert that each figure has 4 straight sides and also four angles.

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The sum of the inner angles of any quadrilateral is 360°. Consider the two instances below.

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You can draw numerous quadrilaterals such as these and also carefully measure the four angles. Friend would discover that for every quadrilateral, the amount of the interior angles will always be 360°.

You can likewise use your understanding of triangles together a way to recognize why the sum of the internal angles of any type of quadrilateral is 360°. Any quadrilateral have the right to be divided into 2 triangles as displayed in the pictures below.

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

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

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


Specific types of Quadrilaterals


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

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Imagine prolonging the pairs of the opposite sides. They would never intersect due to the fact that they space parallel. Notice, also, the the opposite angle of a parallelogram room congruent, as room the the opposite sides. (Remember that “congruent” way “the very same size.”) The geometric symbol for congruent is , so you can write

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

There space two special instances of parallelograms that will be acquainted to girlfriend from her earliest experiences with geometric shapes. The very first special instance is called a rectangle. By definition, a rectangle is a parallelogram since its pairs of opposite sides space parallel. A rectangle also has the one-of-a-kind characteristic that every one of its angles are best angles; all 4 of the angles space congruent.

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

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

Another quadrilateral that you could see is referred to as a rhombus. All 4 sides the a rhombus space congruent. That properties incorporate that each pair of opposite political parties is parallel, also making that a parallelogram.

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

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

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You can use the nature of parallelograms to resolve problems. Think about the instance that follows.


Example

Problem

Determine the measures of

*
 and
*
.

*

 is the contrary

 is the contrary

Identify the contrary angles.

A residential property of parallelograms is the opposite angles space congruent.

*
 = 60°, so
*
 = 60°

*
 = 120°, for this reason
*
 = 120°

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

Answer

*
 = 60° and
*
 = 120°


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

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Notice the

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

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

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In this trapezoid ABCD,

*
 and
*
.

Which that the complying with 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 are parallelograms.

Incorrect. Trapezoids have only one pair that parallel sides; parallelograms have actually 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 are quadrilaterals.

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

C) every rectangles room squares.

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

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

Incorrect. Every parallelograms room quadrilaterals, therefore if that is a parallelogram, that is also a quadrilateral. The exactly answer is the all trapezoids are quadrilaterals.

You have the right to use the properties of quadrilaterals come solve problems involving trapezoids. Consider the example below.


Example

Problem

Find the measure of

*
.

*

*
 = 360°

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

*
 = 90°

*
 = 90°

The square symbol shows a right angle.

60° +  + 90° + 90° = 360°

Since 3 of the 4 angle actions are given, friend can discover the 4th angle measurement.

 + 240° = 360°

 = 120°

Calculate the measure up of

*
.

From the image, you can see the it is one obtuse angle, so its measure need 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 are congruent.

Rectangle

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2 pairs of parallel sides.

4 ideal angles (90°).

Opposite sides are parallel and also congruent.

All angles are congruent.

Square

*

4 congruent sides.

4 ideal angles (90°).

Opposite sides space parallel.

All angles room congruent.

Trapezoid

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Only one pair the opposite sides is parallel.

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A square is a mathematical surname 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, consisting of the number of pairs of parallel political parties they have and their angle and also side measurements.