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.
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.
Fig. 3-2 . Example of a rhombus. Sides have length s , while x and y denote interior angle measures.
Rectangle
A rectangle is like a square in that all four angles have equal measure. But the sides don’t all have to be equally long. A square is a special type of rectangle in which all four sides happen to be the same length. But most rectangles look something like the example in Fig. 3-3. All four angles have the same measure, which must be 90° ( π /2 rad). Opposite sides have equal length, but adjacent sides need not. In this illustration, the two sides labeled s have equal measure, as do the two sides labeled t .
Fig. 3-3 . Example of a rectangle. Sides have lengths s and t , while the interior angles all measure 90° ( π /2 rad).
Parallelogram
The defining characteristic of a parallelogram is that both pairs of opposite sides are parallel. This alone is sufficient to make a plane quadrilateral qualify as a parallelogram. Whenever both pairs of opposite sides in a quadrilateral are parallel, those pairs also have the same length. In addition, pairs of opposite angles have equal measure. A rectangle is a special sort of parallelogram. So is a rhombus, and so is a square. Figure 3-4 shows an example of a parallelogram in which both angles labeled x have equal measure, both angles labeled y have equal measure, both sides labeled s are the same length, and both sides labeled t are the same length.
Fig. 3-4 . Example of a parallelogram. Sides have lengths s and t , while x and y denote interior angle measures.
Trapezoid
If we remove yet another restriction from the quadrilateral, we get a trapezoid. The only rule a trapezoid must obey is that one pair of opposite sides must be parallel. Otherwise, anything goes! Figure 3-5 shows an example of a trapezoid. The dashed lines represent parallel lines in which the two parallel sides of the quadrilateral happen to lie. (The dashed lines are not part of the quadrilateral itself.)
Fig. 3-5 . In a trapezoid, one pair of opposite sides is parallel.
General Quadrilaterals and Practice
In a general quadrilateral, there are no restrictions at all on the lengths of the sides, although the “nature of the beast” dictates that no angle can be outside the range 0° (0 rad) to 360° (2 π rad), non-inclusive. As long as all four vertices lie in the same geometric plane, and as long as all four sides of the figure are straight line segments of finite and positive length, it’s all right.
Of course, any quadrilateral can be considered “general.” A rectangle, for example, is just a specific type of general quadrilateral. But there are plenty of general quadrilaterals that don’t fall into any of the above categories. They don’t exhibit any sort of symmetry or apparent orderliness. These are known as irregular quadrilaterals . Three examples are shown in Fig. 3-6.
Fig. 3-6 . Three examples of irregular quadrilaterals. The sides all have different lengths, and the angles all have different measures.
Types of Quadrilaterals Practice Problems
PROBLEM 1
What type of quadrilateral is formed by the boundaries of a soccer field?
SOLUTION 1
Assuming the groundskeepers have done their job correctly, a soccer field is shaped like a rectangle. All four corners form right angles (90°). In addition, the lengths of opposite sides are equal. The two sidelines are the same length, as are the two end lines.
PROBLEM 2
Suppose a quadrilateral ABCD is defined with the vertices going counterclockwise in alphabetical order. Suppose further that ∠ABC = ∠CDA and ∠BCD = ∠DAB . What can be said about this quadrilateral?
SOLUTION 2
It helps to draw pictures here. Draw several examples of quadrilaterals that meet these two requirements. You’ll see that ∠ABC is opposite ∠CDA , and ∠BCD is opposite ∠DAB . The fact that opposite pairs of angles have equal measure means that the quadrilateral must be a parallelogram. It might be a special case of the parallelogram, such as a rhombus, rectangle, or square; but the only restriction we are given is the fact that ∠ABC = ∠CDA and ∠BCD = ∠DAB . Therefore, ABCD can be any sort of parallelogram.
Practice problems for these concepts can be found at: Quadrilaterals Practice Test.
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From Geometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.