PARALLELOGRAMS and NEGATING SENTENCES

By definition, a parallelogram is a quadrilateral where both pairs of opposite sides are parallel.

If you're trying to show that a quadrilateral is a parallelogram, using the definition isn't always the easiest approach.
There are many different ways that a quadrilateral can be determined to be a parallelogram:

THEOREM equivalent characterizations of a parallelogram
Let [beautiful math coming... please be patient]$\,Q\,$ be a quadrilateral.
Then, the following are equivalent:
  • $Q\,$ is a parallelogram (i.e., both pairs of opposite sides are parallel)
  • the diagonals of $\,Q\,$ bisect each other
  • both pairs of opposite sides are equal
  • both pairs of opposite angles are equal
  • one pair of opposite sides are both parallel and equal

defn of parallelogram:
both pairs of
opposite sides are parallel
diagonals
bisect each other
both pairs of
opposite sides are equal
both pairs of
opposite angles are equal
one pair of
opposite sides are both
parallel and equal
a parallelogram; opposite sides are parallel a parallelogram; diagonals bisect each other a parallelogram; both pairs of opposite sides are equal a parallelogram; both pairs of opposite angles are equal a parallelogram; one pair of opposite sides are both parallel and equal
[beautiful math coming... please be patient]$\overline{AB}\parallel \overline{CD}$
and
$\overline{AC}\parallel \overline{BD}$
$AE = ED$
and
$CE = EB$
$AB = CD$
and
$AC = BD$
$m\angle CAB = m\angle BDC$
and
$m\angle ABD = m\angle DCA$
$\overline{AC}\parallel \overline{BD}$
and
$AC = BD$

If you've been progressing through these Topics in Geometry,
then you have all the tools you need to prove the previous theorem.

You could use a chain of implications together with a variety of prior results.
There are many routes you can take—here's one possibility:

both pairs of opposite sides parallel$\Rightarrow$both pairs of opposite angles are equal Hint:
You may need all these prior results:
 $\Rightarrow$one pair of opposite sides are both parallel and equal
  [beautiful math coming... please be patient]$\Rightarrow$ both pairs of opposites sides are equal
  $\Rightarrow$ the diagonals bisect each other
  $\Rightarrow$ both pairs of opposite sides are parallel

LOGICAL SKILLS FOR NEGATING SENTENCES

You should be able to negate each of the statements in the theorem above,
and phrase these negations in a variety of ways.

The skills for negating ‘and’ and ‘or’ sentences were explored in Logical Equivalences and Practice with Truth Tables.
They are reviewed here for your convenience:

How can a sentence ‘$\,A\text{ and }B\,$’ be false?
The only time an ‘and’ sentence is true is when both subsentences are true.
Therefore, an ‘and’ sentence is false when at least one of the subsentences is false.

Precisely, the truth table below shows that: [beautiful math coming... please be patient] $$ \text{not}(A\text{ and } B)\ \ \text{ is equivalent to }\ \ \bigl((\text{not }A)\text{ or }(\text{not }B)\bigr) $$

$A$ $B$ $A\text{ and }B$ $\text{not}(A \text{ and }B)$ $\text{not } A$ $\text{not } B$ $(\text{not }A) \text{ or } (\text{not }B)$
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T

How can a sentence ‘$\,A\text{ or }B\,$’ be false?
An ‘or’ sentence is false only when both of the subsentences are false.

Precisely, the truth table below shows that: [beautiful math coming... please be patient] $$ \text{not}(A\text{ or }B)\ \ \text{ is equivalent to }\ \ \bigl((\text{not }A)\text{ and }(\text{not }B)\bigr) $$

$A$ $B$ $A \text{ or }B$ $\text{not}(A \text{ or }B)$ $\text{not }A$ $\text{not }B$ $(\text{not }A) \text{ and }(\text{not }B)$
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T

NEGATING ‘For all’ and ‘There exist’ SENTENCES

Next, consider a sentence of the form:   ‘$\,\text{For all }x\text{, }P\ $’
To understand a (true) sentence of this form,
think of a population where each member has property $\,P\,$, as shown below:

What does it mean to say that the sentence ‘$\,\text{For all }x\text{, }P\ $’ is not true?
It means that at least one member of the population doesn't have property $\,P\,$, like this:

Of course, it could certainly be true that more than one member of the population doesn't have property $\,P\,$, like this:

Precisely:

[beautiful math coming... please be patient] $\text{not}(\text{For all }x\text{, }P\,)$ $\text{ is equivalent to }$ $\text{There exists }x\text{ such that }(\text{not }P\,)$

Similar reasoning shows that:

[beautiful math coming... please be patient] $\text{not}(\text{There exists }x\text{ such that }P\,)$ $\text{ is equivalent to }$ $\text{For all }x\text{, }(\text{not }P\,)$

NEGATING THE STATEMENTS IN THE
‘Equivalent Characterizations of a Parallelogram’ THEOREM

Here's an example of applying the negation tools:

These are all negations of:   $\,Q\,$ is a parallelogram (i.e., both pairs of opposite sides are parallel)

These are all negations of:   the diagonals of $\,Q\,$ bisect each other

These are all negations of:   both pairs of opposite sides are equal

These are all negations of:   both pairs of opposite angles are equal

Notice that ‘one pair of opposite sides are both parallel and equal’
can be re-phrased as
‘at least one pair of opposite sides are both parallel and equal’ or
‘there exists a pair of opposite sides that are both parallel and equal’.
These are negations of:   one pair of opposite sides are both parallel and equal

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Introduction to Area and Perimeter


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5
AVAILABLE MASTERED IN PROGRESS

(MAX is 5; there are 5 different problem types.)