The broadest term we’ve used to describe any kind of shape is “polygon.” When we
discussed quadrilaterals
in the last section, we essentially just specified that they were polygons with four
vertices and four sides. Still, we will get more specific in this section and discuss
a special type of quadrilateral: the parallelogram. Before we do this, however,
let’s go over some definitions that will help us describe different parts of quadrilaterals.
Quadrilateral Terminology
Since this entire section is dedicated to the study of quadrilaterals, we will use
some terminology that will help us describe specific pairs of lines, angles, and
vertices of quadrilaterals. Let’s study these terms now.
Consecutive Angles
Two angles whose vertices are the endpoints of the same side are called consecutive
angles.
?Q and ?R are consecutive angles because Q and R are the endpoints of the same side.
Opposite Angles
Two angles that are not consecutive are called opposite angles.
?Q and ?S are opposite angles because they are not endpoints of a common side.
Consecutive Sides
Two sides of a quadrilateral that meet are called consecutive sides.
QR and RS are consecutive sides because they meet at point R.
Opposite Sides
Two sides that are not consecutive are called opposite sides.
QR and TS are opposite sides of the quadrilateral because they do not meet.
Now, that we understand what these terms refer to, we are ready to begin our lesson
on parallelograms.
Properties of Parallelograms: Sides and Angles
A parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.
Quadrilateral ABCD is a parallelogram because AB?DC and AD?BC.
Although the defining characteristics of parallelograms are their pairs of parallel
opposite sides, there are other ways we can determine whether a quadrilateral is
a parallelogram. We will use these properties in our two-column geometric proofs to help us deduce helpful information.
If a quadrilateral is a parallelogram, then.
(1) its opposite sides are congruent,
(2) its opposite angles are congruent, and
(2) its consecutive angles are supplementary.
Another important property worth noticing about parallelograms is that if one angle
of the parallelogram is a right angle, then they all are right angles. Why is this
property true? Let’s examine this situation closely. Consider the figure below.
Given that ?J is a right angle, we can also determine that ?L
is a right angle since the opposite sides of parallelograms are congruent. Together,
the sum of the measure of those angles is 180 because
We also know that the remaining angles must be congruent because they are also opposite
angles. By the Polygon Interior Angles Sum Theorem, we know that all quadrilaterals
have angle measures that add up to 360. Since ?J and
?L sum up to 180, we know that the sum of ?K
and ?M will also be 180:
Since ?K and ?M are congruent, we can define their measures
with the same variable, x. So we have
Therefore, we know that ?K and ?M are both right angles.
Our final illustration is shown below.
Let’s work on a couple of exercises to practice using the side and angle properties
of parallelograms.
Exercise 1
Given that QRST is a parallelogram, find the values of x and y
in the diagram below.
Solution:
After examining the diagram, we realize that it will be easier to solve for x
first because y is used in the same expression as x
(in ?R), but x is by itself at segment QR.
Since opposite sides of parallelograms are congruent, we have can set the quantities
equal to each other and solve for x:
Now that we’ve determined that the value of x is 7,
we can use this to plug into the expression given in ?R. We know that
?R and ?T are congruent, so we have
Substitute x for 7 and we get
So, we’ve determined that x=7 and y=8.
Exercise 2
Given that EDYF is a parallelogram, determine the values of x and y.
Solution:
In order to solve this problem, we will need to use the fact that consecutive angles
of parallelograms are supplementary. The only angle we can figure out initially
is the one at vertex Y because all it requires is the addition of
angles. We have
Knowing that ?Y has a measure of 115 will allow us to
solve for x and y since they are both found in angles
consecutive to ?Y. Let’s solve for y first. We have
All that is left for solve for is x now. We will use the same method
we used when solving for y:
So, we have x=10 and y=13.
The sides and angles of parallelograms aren’t their only unique characteristics.
Let’s learn some more defining properties of parallelograms.
Properties of Parallelograms: Diagonals
When we refer to the diagonals of a parallelogram, we are talking about lines
that can be drawn from vertices that are not connected by line segments. Every parallelogram
will have only two diagonals. An illustration of a parallelogram’s diagonals is
shown below.
We have two important properties that involve the diagonals of parallelograms.
If a quadrilateral is a parallelogram, then.
(1) its diagonals bisect each other, and
(2) each diagonal splits the parallelogram into two congruent triangles.
Segments AE and CE are congruent to each other because the diagonals meet at point
E, which bisects them. Segments BE and DE are also congruent.
The two diagonals split the parallelogram up into congruent triangles.
Let’s use these properties for solve the following exercises.
Exercise 3
Given that ABCD is a parallelogram, find the value of x.
Solution:
We know that the diagonals of parallelograms bisect each other. This means that
the point E splits up each bisector into two equivalent segments.
Thus, we know that DE and BE are congruent, so we have
So, the value of x is 3.
Exercise 4
Given that FGHI is a parallelogram, find the values of x and y.
Let’s try to solve for x first. We are given that ?FHI
is a right angle, so it has a measure of 90°. We can deduce that ?HFG is also a right angle by the Alternate Interior Angles Theorem.
If we look at ?HIJ, we notice that two of its angles are congruent,
so it is an isosceles triangle. This means that ?HIJ has a measure
of 9x since ?IJH has that measure.
We can use the fact that the triangle has a right angle and that there are two congruent
angles in it, in order to solve for x. We will use the Triangle Angle
Sum Theorem to show that the angles must add up to 180°.
Now, let’s solve for y. We know that segments IJ
and GJ are congruent because they are bisected by the opposite diagonal.
Therefore, we can set them equal to each other.
Because we can say that IJ and GJ
are congruent, we have
So, our answers are x=5 and y=4.
