Quadrilaterals are a special kind of polygon. Similar to triangles and also other polygons, quadrilaterals have special properties and can it is in classified by characteristics of your angles and also sides. Expertise the properties of different quadrilaterals can assist you in solving troubles that show off this kind of polygon.

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Picking personal the name “quadrilateral” help you understand what that refers to. The prefix “quad-” way “four,” and “lateral” is obtained from the Latin word because that “side.” for this reason a quadrilateral is a four-sided polygon.

Since it is a polygon, you know that that is a two-dimensional figure consisted of of right sides. A quadrilateral additionally has four angles formed by its 4 sides. Listed below are some examples of quadrilaterals. Notice that each number has 4 straight sides and four angles.

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The amount of the internal angles of any type of quadrilateral is 360°. Consider the two examples below.

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You can draw plenty of quadrilaterals such as these and carefully measure the 4 angles. Girlfriend would discover that for every quadrilateral, the amount of the interior angles will always be 360°.

You can also use your understanding of triangles as a way to know why the amount of the interior angles of any type of quadrilateral is 360°. Any type of quadrilateral can be split into two triangles as displayed in the photos below.

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

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

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


Specific types of Quadrilaterals


Let’s start by examining the group of quadrilaterals that have two bag of parallel sides. This quadrilaterals are dubbed parallelograms They take a variety of shapes, but one standard example is displayed below.

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Imagine expanding the bag of opposite sides. Castle would never ever intersect due to the fact that they space parallel. Notice, also, the the opposite angles of a parallelogram room congruent, as are the the contrary sides. (Remember that “congruent” means “the exact same size.”) The geometric symbol because that congruent is , so you have the right to write

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

There are two special situations of parallelograms that will be familiar to friend from your earliest experiences with geometric shapes. The first special instance is dubbed a rectangle. By definition, a rectangle is a parallelogram because its pairs of the opposite sides room parallel. A rectangle also has the special characteristic that every one of its angle are appropriate angles; all four of that is angles are congruent.

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

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A square is also a rectangle since it has two sets of parallel sides and also four right angles. A square is additionally a parallelogram since its the contrary sides room parallel. So, a square have the right to be share in any type of of these three ways, with “parallelogram” being the least specific description and “square,” the most descriptive.

Another quadrilateral that you could see is referred to as a rhombus. All 4 sides of a rhombus are congruent. The properties incorporate that every pair that opposite sides is parallel, also making that a parallelogram.

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

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

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You have the right to use the properties of parallelograms to settle problems. Consider the example that follows.


Example

Problem

Determine the procedures of

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 and
*
.

*

 is the opposite

 is the opposite

Identify opposite angles.

A building of parallelograms is the opposite angles room congruent.

*
 = 60°, therefore
*
 = 60°

*
 = 120°, so
*
 = 120°

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

Answer

*
 = 60° and
*
 = 120°


There is another special type of quadrilateral. This quadrilateral has actually the building of having actually only one pair of opposite sides that room parallel. Below is one example of a trapezoid.

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Notice that

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

If the non-parallel sides of a trapezoid room congruent, the trapezoid is dubbed an isosceles trapezoid. Like the an in similar way named triangle that has two political parties of equal length, the isosceles trapezoid has a pair of opposite sides of same length. The various 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
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.

Which the the adhering to statements is true?

A) some trapezoids space parallelograms.

B) every trapezoids space 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 two pairs of parallel sides. A trapezoid can never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals.

B) every trapezoids room quadrilaterals.

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

C) every rectangles room squares.

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

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

Incorrect. Every parallelograms space quadrilaterals, therefore if the is a parallelogram, the is likewise a quadrilateral. The correct answer is that all trapezoids room quadrilaterals.

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


Example

Problem

Find the measure up of

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.

*

*
 = 360°

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

*
 = 90°

*
 = 90°

The square symbol shows a ideal angle.

60° +  + 90° + 90° = 360°

Since 3 of the 4 angle steps are given, you can find the fourth angle measurement.

 + 240° = 360°

 = 120°

Calculate the measure up of

*
.

From the image, you have the right to see the it is an obtuse angle, so its measure should 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 opposite angles space congruent.

Rectangle

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2 pairs of parallel sides.

4 ideal angles (90°).

Opposite sides space parallel and congruent.

All angles are congruent.

Square

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4 congruent sides.

4 appropriate angles (90°).

Opposite sides space parallel.

All angles room congruent.

Trapezoid

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Only one pair that 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 room all instances of quadrilaterals. This quadrilaterals earn their distinction based on your properties, including the variety of pairs the parallel political parties they have and their angle and also side measurements.