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. 33. 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 .
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 34 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.
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 35 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.)
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), noninclusive. 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. 36.
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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