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Types of Quadrilaterals Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to Types of Quadrilaterals

A four-sided geometric plane figure is called a quadrilateral . Because a quadrilateral has more sides than a triangle, there are more types. The allowable range of interior-angle measures is greater than is the case with triangles. With a triangle, an interior angle must always measure more than 0° (0 rad) but less than 180° ( π rad); with a quadrilateral, the measure of an interior angle can be anything up to, but not including, 360° (2 π rad).

The categories of quadrilateral are the square , the rhombus , the rectangle , the parallelogram , the trapezoid , and the general quadrilateral . Let’s define these and look at some examples.

It’s The Law!

There are two properties that a four-sided geometric figure absolutely must have—laws it is required to obey—if it is to qualify as a legitimate plane quadrilateral.

  • First, all four vertices must lie in the same plane.
  • Second, all four sides must be straight line segments of finite, positive length. Curves are not allowed, nor are points, infinitely long rays, or infinitely long lines.

For our purposes, we’ll add the constraint that a true plane quadrilateral cannot have sides whose lengths are negative.

The vertices of a triangle always lie in a single geometric plane, because any three points, no matter which ones you choose, define a unique geometric plane. But when you have four points, they don’t all necessarily lie in the same plane. Any three of them do, but the fourth one can get “out of alignment.” This is why a four-legged stool or table often wobbles, and why it is so difficult to trim the lengths of the legs so the wobbling stops. Once the ends of the legs lie in a single plane, and they define the vertices of a plane quadrilateral, the stool or table won’t wobble, as long as the floor is perfectly flat. (Later in this book, we’ll take a look at some of the things that can happen when a floor is not flat, or more particularly, what can take place when a geometric universe is warped or curved.)

Square

A square has four sides that are all of the same length. In addition, all the interior angles are the same, and measure 90° ( π /2 rad). Figure 3-1 shows the general situation. The length of each side in this illustration is s units. There is no limit to how large s can be, but it must be greater than zero.

 

Quadrilaterals Types of Quadrilaterals Square

Fig. 3-1 . Example of a square. Sides have length s , and the interior angles all measure 90° ( π /2 rad).

Rhombus

A rhombus is like a square in that all four sides are the same length. But the angles don’t all have to be right angles. A square is a special type of rhombus in which all four angles happen to have the same measure. But most rhombuses (rhombi?) look something like the example in Fig. 3-2. All four sides have length s . Opposite angles have equal measure, but adjacent angles need not. In this illustration, the two angles labeled x have equal measure, as do the two angles labeled y . Another property of the rhombus is the fact that both pairs of opposite sides are parallel.

 

Quadrilaterals Types of Quadrilaterals Rhombus

Fig. 3-2 . Example of a rhombus. Sides have length s , while x and y denote interior angle measures.

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