Quadrilaterals are a special kind of polygon. Similar to triangles and also other polygons, quadrilaterals have actually special properties and can be classified by qualities of their angles and sides. Expertise the properties of different quadrilaterals can assist you in solving difficulties that show off this form of polygon.
You are watching: A parallelogram with four congruent sides and four right angles
Picking personally the name “quadrilateral” helps you understand what that refers to. The prefix “quad” way “four,” and “lateral” is obtained from the Latin word for “side.” for this reason a square is a foursided polygon.
Since the is a polygon, you know that that is a twodimensional figure made up of right sides. A quadrilateral additionally has 4 angles formed by its 4 sides. Below are some examples of quadrilaterals. Notification that each figure has four straight sides and four angles.
The amount of the inner angles of any kind of quadrilateral is 360°. Think about the two examples below.
You could draw many quadrilaterals such as these and carefully measure up the 4 angles. Friend would uncover that because that every quadrilateral, the amount of the internal angles will constantly be 360°.
You can likewise use your understanding of triangles as a method to understand why the sum of the inner angles of any quadrilateral is 360°. Any type of quadrilateral can be split into two triangles as presented in the images below.
In the first image, the quadrilaterals have actually each been split into two triangles. The angle measurements of one triangle are displayed for each.
These measurements include up come 180º. Currently look at the dimensions for the various other triangles—they also add up to 180º!
Since the sum of the inner angles of any kind of triangle is 180° and also there are two triangles in a quadrilateral, the sum of the angle for each quadrilateral is 360°.
Specific types of Quadrilaterals
Let’s begin by evaluating the team of quadrilaterals that have two pairs of parallel sides. This quadrilaterals are dubbed parallelograms They take a selection of shapes, yet one classic example is shown below.
Imagine extending the bag of the opposite sides. Lock would never intersect since they are parallel. Notice, also, the the opposite angles of a parallelogram space congruent, as are the the opposite sides. (Remember the “congruent” means “the same size.”) The geometric symbol because that congruent is , for this reason you deserve to write
and . The parallel sides are additionally the same length: and . These relationships room true for every parallelograms.There room two special situations of parallelograms that will certainly be acquainted to friend from her earliest experiences through geometric shapes. The an initial special situation is dubbed a rectangle. By definition, a rectangle is a parallelogram due to the fact that its pairs of the contrary sides are parallel. A rectangle likewise has the oneofakind characteristic that all of its angle are appropriate angles; all four of that is angles space congruent.
The other special instance of a parallelogram is a special form of rectangle, a square. A square is just one of the most simple geometric shapes. The is a special instance of a parallelogram that has four congruent sides and also four ideal angles.
A square is likewise a rectangle because it has two sets of parallel sides and four best angles. A square is also a parallelogram due to the fact that its the opposite sides are parallel. So, a square can be divide in any of these 3 ways, with “parallelogram” being the least details description and also “square,” the many descriptive.
Another quadrilateral that you can see is referred to as a rhombus. All 4 sides of a rhombus room congruent. Its properties encompass that each pair the opposite sides is parallel, likewise making that a parallelogram.
In summary, all squares space rectangles, yet not every rectangles space squares. Every rectangles space parallelograms, yet not every parallelograms are rectangles. And every one of these forms are quadrilaterals.
The diagram below illustrates the relationship between the different species of quadrilaterals.
You can use the properties of parallelograms to settle problems. Consider the instance that follows.
Example  
Problem  Determine the procedures of and .
 
 is the contrary is opposite  Identify the opposite angles. 
A building of parallelograms is that opposite angles are congruent.  
= 60°, for this reason = 60° = 120°, therefore = 120°  Use the provided angle measurements to recognize measures of the opposite angles.  
Answer  = 60° and = 120° 
There is an additional special type of quadrilateral. This quadrilateral has actually the building of having actually only one pair the opposite sides that are parallel. Here is one example of a trapezoid.
Notice the
, and that and are no parallel. Girlfriend can easily imagine that if you extended sides and , they would intersect over the figure.If the nonparallel sides of a trapezoid are congruent, the trapezoid is dubbed an isosceles trapezoid. Like the likewise named triangle that has two political parties of same length, the isosceles trapezoid has actually a pair that opposite sides of same length. The other pair of opposite political parties is parallel. Below is an example of one isosceles trapezoid.
In this trapezoid ABCD,
and . Which that the complying with statements is true? A) some trapezoids are parallelograms. B) every trapezoids are quadrilaterals. C) every rectangles room squares. D) A shape cannot be a parallelogram and a quadrilateral. A) some trapezoids room parallelograms. Incorrect. Trapezoids have actually only one pair of parallel sides; parallelograms have actually two pairs of parallel sides. A trapezoid deserve to never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals. B) all trapezoids room quadrilaterals. Correct. Trapezoids space foursided polygons, therefore they are all quadrilaterals. C) all rectangles are squares. Incorrect. Part rectangles might be squares, but not every rectangles have 4 congruent sides. All squares are rectangles however. The correct answer is the all trapezoids are quadrilaterals. D) A shape cannot it is in a parallelogram and a quadrilateral. Incorrect. Every parallelograms room quadrilaterals, for this reason if it is a parallelogram, it is likewise a quadrilateral. The correct answer is that all trapezoids are quadrilaterals. 
You can use the nature of quadrilaterals to solve problems involving trapezoids. Think about the example below.
Example  
Problem  Find the measure up of .

 
 = 360°  The amount of the measures of the inner angles the a square is 360°.  
 = 90° = 90°  The square symbol indicates a ideal angle.  
60° + + 90° + 90° = 360°  Since 3 of the four angle steps are given, friend can find the 4th angle measurement.  
+ 240° = 360° = 120°  Calculate the measurement of .From the image, you deserve to see that it is one obtuse angle, for this reason its measure have to be greater than 90°.  
Answer  = 120°  
Name that Quadrilateral  Quadrilateral  Description 
Parallelogram  2 pairs of parallel sides. Opposite sides and also opposite angles are congruent.  
Rectangle  2 bag of parallel sides. 4 best angles (90°). Opposite sides room parallel and also congruent. All angles room congruent.  
Square  4 congruent sides. 4 best angles (90°). Opposite sides are parallel. All angles are congruent.  
Trapezoid  Only one pair that opposite sides is parallel. 
A square is a mathematical surname for a foursided polygon. Parallelograms, squares, rectangles, and also trapezoids space all examples of quadrilaterals. This quadrilaterals earn their difference based on your properties, including the variety of pairs that parallel sides they have and their angle and side measurements.