Quadrilaterals room a special type of polygon. Just like triangles and also other polygons, quadrilaterals have actually special properties and also can be classified by qualities of their angles and sides. Knowledge the properties of various quadrilaterals can help you in solving troubles that indicate this kind of polygon.
You are watching: A quadrilateral with four congruent sides
Picking personal the surname “quadrilateral” help you understand what that refers to. The prefix “quad” way “four,” and also “lateral” is obtained from the Latin word because that “side.” therefore a quadrilateral is a foursided polygon.
Since that is a polygon, you understand that it is a twodimensional figure consisted of of right sides. A quadrilateral also has 4 angles created by its 4 sides. Below are some examples of quadrilaterals. An alert that each number has 4 straight sides and also four angles.
The amount of the interior angles of any quadrilateral is 360°. Take into consideration the two examples below.
You could draw countless quadrilaterals such as these and also carefully measure up the 4 angles. You would find that because that every quadrilateral, the amount of the inner angles will always be 360°.
You can also use your expertise of triangles together a way to know why the sum of the interior angles of any type of quadrilateral is 360°. Any type of quadrilateral can be divided into 2 triangles as shown in the images below.
In the very first image, the quadrilaterals have each been divided into two triangles. The angle dimensions of one triangle are shown for each.
These measurements include up come 180º. Currently look at the measurements for the other triangles—they also include up to 180º!
Since the sum of the inner 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 varieties of Quadrilaterals
Let’s start by analyzing the group of quadrilaterals that have actually two bag of parallel sides. This quadrilaterals are dubbed parallelograms They take it a range of shapes, yet one classic example is displayed below.
Imagine extending the bag of the opposite sides. Lock would never ever intersect since they space parallel. Notice, also, that the opposite angles of a parallelogram are congruent, as are the opposite sides. (Remember that “congruent” way “the same size.”) The geometric symbol because that congruent is , for this reason you deserve to write
and . The parallel political parties are additionally the exact same length: and . These relationships are true for every parallelograms.There are two special instances of parallelograms that will certainly be familiar to friend from her earliest experiences through geometric shapes. The first special case is called a rectangle. By definition, a rectangle is a parallelogram since its pairs of opposite sides are parallel. A rectangle likewise has the special characteristic that every one of its angles are ideal angles; all 4 of its angles space congruent.
The various other special case of a parallelogram is a special kind of rectangle, a square. A square is just one of the most straightforward geometric shapes. That is a special situation of a parallel that has 4 congruent sides and four appropriate angles.
A square is likewise a rectangle since it has two to adjust of parallel sides and also four ideal angles. A square is likewise a parallelogram because its the opposite sides are parallel. So, a square can be share in any type of of these three ways, with “parallelogram” gift the least specific description and also “square,” the most descriptive.
Another quadrilateral that you might see is dubbed a rhombus. All 4 sides the a rhombus room congruent. Its properties incorporate that each pair the opposite sides is parallel, also making that a parallelogram.
In summary, all squares are rectangles, however not all rectangles room squares. Every rectangles room parallelograms, however not all parallelograms room rectangles. And all of these forms are quadrilaterals.
The diagram below illustrates the relationship between the different types of quadrilaterals.
You have the right to use the nature of parallelograms to fix problems. Consider the example that follows.
Example  
Problem  Determine the actions of and .
 
 is the contrary is opposite  Identify opposite angles. 
A building of parallelograms is the opposite angles are congruent.  
= 60°, therefore = 60° = 120°, therefore = 120°  Use the provided angle dimensions to determine measures of the contrary angles.  
Answer  = 60° and also = 120° 
There is another special type of quadrilateral. This quadrilateral has actually the residential or commercial property of having only one pair of opposite political parties that space parallel. Below is one instance of a trapezoid.
Notice that
, and also that and are not parallel. You can conveniently imagine the if you extended sides and also , they would certainly intersect over the figure.If the nonparallel political parties of a trapezoid are congruent, the trapezoid is referred to as an isosceles trapezoid. Choose the likewise named triangle that has two political parties of same length, the isosceles trapezoid has actually a pair the opposite sides of same length. The various other pair that opposite sides is parallel. Listed below is an example of an isosceles trapezoid.
In this trapezoid ABCD,
and . Which the the following statements is true? A) some trapezoids space parallelograms. B) every trapezoids space quadrilaterals. C) every rectangles room squares. D) A shape cannot it is in a parallelogram and a quadrilateral. A) part trapezoids are parallelograms. Incorrect. Trapezoids have actually only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid deserve to never be a parallelogram. The correct answer is the all trapezoids are quadrilaterals. B) all trapezoids are quadrilaterals. Correct. Trapezoids room foursided polygons, therefore they room all quadrilaterals. C) every rectangles space squares. Incorrect. Some rectangles might be squares, however not every rectangles have 4 congruent sides. All squares are rectangles however. The correct answer is that all trapezoids space quadrilaterals. D) A shape cannot it is in a parallelogram and a quadrilateral. Incorrect. Every parallelograms are quadrilaterals, so if that is a parallelogram, it is additionally a quadrilateral. The correct answer is the all trapezoids room quadrilaterals. 
You deserve to use the properties of quadrilaterals come solve problems involving trapezoids. Take into consideration the example below.
Example  
Problem  Find the measure of .

 
 = 360°  The sum of the procedures of the inner angles that a quadrilateral is 360°.  
 = 90° = 90°  The square symbol suggests a appropriate angle.  
60° + + 90° + 90° = 360°  Since three of the 4 angle steps are given, girlfriend can find the 4th angle measurement.  
+ 240° = 360° = 120°  Calculate the measurement of .From the image, you have the right to see the it is one obtuse angle, so its measure need to be greater than 90°.  
Answer  = 120°  
Name the Quadrilateral  Quadrilateral  Description 
Parallelogram  2 bag of parallel sides. Opposite sides and opposite angles space congruent.  
Rectangle  2 bag of parallel sides. 4 ideal angles (90°). Opposite sides room parallel and congruent. All angles room congruent.  
Square  4 congruent sides. 4 appropriate angles (90°). Opposite sides space parallel. All angles are congruent.  
Trapezoid  Only one pair of opposite sides is parallel. 
A square is a mathematical surname for a foursided polygon. Parallelograms, squares, rectangles, and also trapezoids are all examples of quadrilaterals. This quadrilaterals knife their difference based on their properties, consisting of the number of pairs that parallel political parties they have and also their angle and side measurements.