Quadrilaterals are a special kind of polygon. Just like triangles and also other polygons, quadrilaterals have actually special properties and also can be classified by features of your angles and sides. Knowledge the properties of various quadrilaterals can aid you in solving troubles that involve this form of polygon.

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Picking apart the name “quadrilateral” helps you understand what the refers to. The prefix “quad-” way “four,” and “lateral” is acquired from the Latin word for “side.” so a square is a four-sided polygon.

Since it is a polygon, you recognize that that is a two-dimensional figure made up of straight sides. A quadrilateral additionally has four angles developed by its four sides. Listed below are some examples of quadrilaterals. Notice that each number has four straight sides and four angles.

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

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You might draw plenty of quadrilaterals such together these and carefully measure up the 4 angles. Friend would uncover that because that every quadrilateral, the amount of the inner angles will constantly be 360°.

You can likewise use your knowledge of triangles together a way to understand why the sum of the interior angles of any quadrilateral is 360°. Any type of quadrilateral have the right to be separated into two triangles as displayed in the pictures below.

In the first 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. Now look in ~ the dimensions for the various other triangles—they also include up come 180º!

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


Specific varieties of Quadrilaterals


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

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Imagine expanding the bag of opposite sides. Lock would never intersect since they room parallel. Notice, also, that the opposite angles of a parallelogram are congruent, as space the the opposite sides. (Remember that “congruent” way “the very same size.”) The geometric symbol for congruent is , for this reason you deserve to write

*
 and
*
. The parallel sides are likewise the very same length:
*
 and
*
. This relationships space true for all parallelograms.

There are two special instances of parallelograms that will certainly be familiar to friend from your earliest experiences through geometric shapes. The first special case is referred to as a rectangle. By definition, a rectangle is a parallelogram since its pairs of the contrary sides room parallel. A rectangle additionally has the unique characteristic that every one of its angle are right angles; all four of its angles room congruent.

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The various other special instance of a parallel is a special type 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 appropriate angles.

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A square is additionally a rectangle since it has actually two to adjust of parallel sides and also four ideal angles. A square is also a parallelogram due to the fact that its opposite sides space parallel. So, a square deserve to be divide in any kind of of these three ways, v “parallelogram” being the least certain description and “square,” the most descriptive.

Another quadrilateral the you could see is dubbed a rhombus. All four sides that a rhombus are congruent. That is properties encompass that each pair that opposite sides is parallel, also making that a parallelogram.

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

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

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You have the right to use the nature of parallelograms to deal with problems. Consider the instance that follows.


Example

Problem

Determine the measures of

*
 and
*
.

*

 is the contrary

 is opposite

Identify opposite angles.

A building of parallelograms is that opposite angles are congruent.

*
 = 60°, therefore
*
 = 60°

*
 = 120°, so
*
 = 120°

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

Answer

*
 = 60° and also
*
 = 120°


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

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Notice that

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, and also that  and  are no parallel. You can quickly imagine the 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 actually two political parties of same length, the isosceles trapezoid has a pair of opposite sides of same length. The other pair of opposite sides is parallel. Listed below is an example of one isosceles trapezoid.

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In this trapezoid ABCD,

*
 and
*
.

Which of the following statements is true?

A) part trapezoids space parallelograms.

B) all trapezoids are quadrilaterals.

C) every rectangles room squares.

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


A) some trapezoids room parallelograms.

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

B) every trapezoids are quadrilaterals.

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

C) every rectangles are squares.

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

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

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

You deserve to use the nature of quadrilaterals to solve troubles involving trapezoids. Consider the example below.


Example

Problem

Find the measure up of

*
.

*

*
 = 360°

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

*
 = 90°

*
 = 90°

The square symbol suggests a best angle.

60° +  + 90° + 90° = 360°

Since three of the four angle procedures are given, friend can discover the fourth angle measurement.

 + 240° = 360°

 = 120°

Calculate the measurement of

*
.

From the image, you can see that it is an obtuse angle, so its measure should be greater 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 space congruent.

Rectangle

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2 pairs of parallel sides.

4 appropriate angles (90°).

Opposite sides room parallel and also congruent.

All angles room congruent.

Square

*

4 congruent sides.

4 right angles (90°).

Opposite sides room parallel.

All angles room congruent.

Trapezoid

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Only one pair of opposite sides is parallel.

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A square is a mathematical name for a four-sided polygon. Parallelograms, squares, rectangles, and also trapezoids room all instances of quadrilaterals. This quadrilaterals earn their distinction based on your properties, consisting of the number of pairs the parallel political parties they have and their angle and also side measurements.