Now that we've seen several types of quadrilaterals that are parallelograms, let's learn about figures that do not have the properties of parallelograms. Recall that parallelograms were quadrilaterals whose opposite sides were parallel. In this section, we will look at quadrilaterals whose opposite sides may intersect at some point. The two types of quadrilaterals we will study are called trapezoids and kites. Let's begin our study by learning some properties of trapezoids.
Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Since a trapezoid must have exactly one pair of parallel sides, we will need to prove that one pair of opposite sides is parallel and that the other is not in our two-column geometric proofs. If we forget to prove that one pair of opposite sides is not parallel, we do not eliminate the possibility that the quadrilateral is a parallelogram. Therefore, that step will be absolutely necessary when we work on different exercises involving trapezoids.
Before we dive right into our study of trapezoids, it will be necessary to learn the names of different parts of these quadrilaterals in order to be specific about its sides and angles. All trapezoids have two main parts: bases and legs. The opposite sides of a trapezoid that are parallel to each other are called bases. The remaining sides of the trapezoid, which intersect at some point if extended, are called the legs of the trapezoid.
The top and bottom sides of the trapezoid run parallel to each other, so they are the trapezoid's bases. The other sides of the trapezoid will intersect if extended, so they are the trapezoid's legs.
The segment that connects the midpoints of the legs of a trapezoid is called the midsegment. This segment's length is always equal to one-half the sum of the trapezoid's bases, or
Consider trapezoid ABCD shown below.
The midsegment, EF, which is shown in red, has a length of
The measurement of the midsegment is only dependent on the length of the trapezoid's bases. However, there is an important characteristic that some trapezoids have that is solely reliant on its legs. Let's look at these trapezoids now.
Definition: An isosceles trapezoid is a trapezoid whose legs are congruent.
By definition, as long as a quadrilateral has exactly one pair of parallel lines, then the quadrilateral is a trapezoid. The definition of an isosceles trapezoid adds another specification: the legs of the trapezoid have to be congruent.
ABCD is not an isosceles trapezoid because AD and BC are not congruent. Because EH and FG are congruent, trapezoid EFGH is an isosceles trapezoid.
There are several theorems we can use to help us prove that a trapezoid is isosceles. These properties are listed below.
(1) A trapezoid is isosceles if and only if the base angles are congruent.
(2) A trapezoid is isosceles if and only if the diagonals are congruent.
(3) If a trapezoid is isosceles, then its opposite angles are supplementary.
Definition: A kite is a quadrilateral with two distinct pairs of adjacent sides that are congruent.
Recall that parallelograms also had pairs of congruent sides. However, their congruent sides were always opposite sides. Kites have two pairs of congruent sides that meet at two different points. Let's look at the illustration below to help us see what a kite looks like.
Segment AB is adjacent and congruent to segment BC. Segments AD and CD are also adjacent and congruent.
Kites have a couple of properties that will help us identify them from other quadrilaterals.
(1) The diagonals of a kite meet at a right angle.
(2) Kites have exactly one pair of opposite angles that are congruent.
These two properties are illustrated in the diagram below.
Notice that a right angle is formed at the intersection of the diagonals, which is at point N. Also, we see that ?K??M. This is our only pair of congruent angles because ?J and ?L have different measures.
Let's practice doing some problems that require the use of the properties of trapezoids and kites we've just learned about.
Find the value of x in the trapezoid below.
Because we have been given the lengths of the bases of the trapezoid, we can figure out what the length of the midsegment should be. Let's use the formula we have been given for the midsegment to figure it out. (Remember, it is one-half the sum of the bases.)
So, now that we know that the midsegment's length is 24, we can go ahead and set 24 equal to 5x-1. The variable is solvable now:
Find the value of y in the isosceles trapezoid below.
In the figure, we have only been given the measure of one angle, so we must be able to deduce more information based on this one item. Because the quadrilateral is an isosceles trapezoid, we know that the base angles are congruent. This means that ?A also has a measure of 64°.
Now, let's figure out what the sum of ?A and ?P is:
Together they have a total of 128°. Recall by the Polygon Interior Angle Sum Theorem that a quadrilateral's interior angles must be 360°. So, let's try to use this in a way that will help us determine the measure of ?R. First, let's sum up all the angles and set it equal to 360°.
Now, we see that the sum of ?T and ?R is 232°. Because segment TR is the other base of trapezoid TRAP, we know that the angles at points T and R must be congruent to each other. Thus, if we define the measures of ?T and ?R by variable x, we have
This value means that the measure of ?T and ?R is 116°. Finally, we can set 116 equal to the expression shown in ?R to determine the value of y. We have
So, we get x=9.
While the method above was an in-depth way to solve the exercise, we could have also just used the property that opposite angles of isosceles trapezoids are supplementary. Solving in this way is much quicker, as we only have to find what the supplement of a 64° angle is. We get
Once we get to this point in our problem, we just set 116 equal to 4(3y+2) and solve as we did before.
After reading the problem, we see that we have been given a limited amount of information and want to conclude that quadrilateral DEFG is a kite. Notice that EF and GF are congruent, so if we can find a way to prove that DE and DG are congruent, it would give us two distinct pairs of adjacent sides that are congruent, which is the definition of a kite.
We have also been given that ?EFD and ?GFD are congruent. We learned several triangle congruence theorems in the past that might be applicable in this situation if we can just find another side or angle that are congruent.
Since segment DF makes up a side of ?DEF and ?DGF, we can use the reflexive property to say that it is congruent to itself. Thus, we have two congruent triangles by the SAS Postulate.
Next, we can say that segments DE and DG are congruent because corresponding parts of congruent triangles are congruent. Our new illustration is shown below.
We conclude that DEFG is a kite because it has two distinct pairs of adjacent sides that are congruent. The two-column geometric proof for this exercise is shown below.