Proving Quadrilaterals Are Parallelograms

In the previous section, we learned about several properties that distinguish
parallelograms from other
quadrilaterals. Most of the work we did was computation-based because
we were already given the fact that the figures were parallelograms. In this section,
we will use our reasoning skills to put together two-column geometric
proofs for parallelograms. We can apply much of what we learned in the previous
section to help us throughout this lesson, but we will be much more formalized and
organized in our arguments.

Using Definitions and Theorems in Proofs

The ways we start off our proofs are key steps toward arriving at a conclusion.
Therefore, comprehending the information that we are given by an exercise may be
the single most important part of proving a statement.

As we will see, there are different ways in which we can essentially say the same
statement. Recall, that many of our
angle theorems had converses. The converses of the theorems essentially
gave the same information, but in a reversed order. We will have to approach problems
involving parallelograms in the same way. That is, we must be conscious of the arguments
we make based on whether we are given that a certain quadrilateral is a parallelogram,
or if we want to prove that the quadrilateral is a parallelogram. Let’s take
a look at these statements so that we understand how to use them properly in our
proofs.

Given a Parallelogram

We can use the following statements in our proofs if we are given that a quadrilateral
is a parallelogram.

Definition: A parallelogram is a type of quadrilateral whose pairs of opposite
sides are parallel.

If a quadrilateral is a parallelogram, then…

Much of the information above was studied in the previous section. The purpose of
organizing it in the way that it has been laid out is to help us see the difference
in our statements depending on whether we are given a parallelogram, or if we are
trying to prove that a quadrilateral is a parallelogram.

Let’s look at the structure of our statements when we are trying to prove that a
quadrilateral is a parallelogram.

Proving a Parallelogram

Definition: A parallelogram is a type of quadrilateral whose pairs of opposite
sides are parallel.

If…

…the quadrilateral is a parallelogram.

Let’s use these statements to help us prove the following exercise. We will need
to use both forms of the statements above, because we will be given one parallelogram,
and we will have to prove that another one exists. This will give us practice using
regular theorems and definitions, as well as their converses.

Exercise

Solution:

As stated before this exercise, we need to be conscious of how to use theorems and
definitions, as well as their converses because we are given that NRSM
is a parallelogram, but we also want to prove that ERAM is
a parallelogram. We were also given that ?4??5, which will help us
prove our conclusion.

To begin, we know that ?R??M because they are the opposite angles
of parallelogram NRSM.

Knowing this allows us to claim that ?3??6 by the Angle Subtraction
Postulate. We see that ?R is composed of two smaller angles
(?3 and ?4). Likewise, we see that ?M
is composed of ?5 and ?6. Since the whole of the angles
are congruent, and two of the smaller angles in them are congruent, then their remainders
are also congruent.

Now, we have proven that one pair of opposite angles are congruent. If we can show
that ?2 and ?7 are also congruent, we can prove that
quadrilateral ERAM is a parallelogram.

Because NRSM is a parallelogram, we know that its opposite sides are
parallel. So, we have that segments NR and MS are parallel.
Considering these lines, we know that segments EM and RA
are transversals to the parallel lines, since they intersect both lines. Thus, we
can use the Alternate Interior Angles Theorem to prove that ?1??6
and ?3??8.

By transitivity, we can say that ?1 is congruent to ?8.
It is a bit difficult to imagine the chain of congruences that allows us to make
this claim, but it is as such:

Notice that ?3 and ?6 are congruent, opposite angles,
just as ?2 and ?7 are. Let’s look at our new illustration
to help us visualize what we’ve done.

We have proven that ERAM is a parallelogram because both pairs of
its opposite angles are congruent. The two-column proof for our argument is shown
below.