audio read-through 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, however, then using the definition isn't always the easiest approach.

There are many different ways that a quadrilateral can be determined to be a parallelogram. Five of the most common ways are explored in this section.

THEOREM equivalent characterizations of a parallelogram
Let $\,Q\,$ 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
a parallelogram; opposite sides are parallel $\overline{AB}\parallel \overline{CD}$
and
$\overline{AC}\parallel \overline{BD}$
diagonals
bisect each other

a parallelogram; diagonals bisect each other $AE = ED$
and
$CE = EB$
both pairs of opposite sides are equal a parallelogram; both pairs of opposite sides are equal $AB = CD$
and
$AC = BD$
both pairs of opposite angles are equal
a parallelogram; both pairs of opposite angles are equal $m\angle A = m\angle D$
and
$m\angle B = m\angle C$
one pair of opposite sides are both parallel and equal
a parallelogram; one pair of opposite sides are both parallel and equal $\overline{AC}\parallel \overline{BD}$
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
  $\Rightarrow$ one pair of opposite sides are both parallel and equal
  $\Rightarrow$ both pairs of opposites sides are equal
  $\Rightarrow$ the diagonals bisect each other
  $\Rightarrow$ both pairs of opposite sides are parallel

Hint: You may need all these prior results:

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: $$ \begin{gather} \cssId{s53}{\text{not}(A\text{ and } B)}\cr \cssId{s54}{\text{ is equivalent to }}\cr \cssId{s55}{\bigl((\text{not }A)\text{ or }(\text{not }B)\bigr)} \end{gather} $$

$A$ $B$ $A\text{ and }B$ $\text{not}(A \text{ and }B)$
T T T F
T F F T
F T F T
F F F T
$\text{not } A$ $\text{not } B$ $(\text{not }A) \text{ or } (\text{not }B)$
F F F
F T T
T 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: $$ \begin{gather} \cssId{s59}{\text{not}(A\text{ or }B)}\cr \cssId{s60}{\text{ is equivalent to }}\cr \cssId{s61}{\bigl((\text{not }A)\text{ and }(\text{not }B)\bigr)} \end{gather} $$

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

a population where each member has property P

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:

a population where at least one member does NOT have property P

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

a population where at least one member does NOT have property P

Precisely: $$ \begin{gather} \cssId{s71}{\text{not}(\text{For all }x\text{, }P\,)}\cr \cssId{s72}{\text{ is equivalent to }}\cr \cssId{s73}{\text{There exists }\,x\,\text{ such that }(\text{not }P\,)} \end{gather} $$

Similar reasoning shows that: $$ \begin{gather} \cssId{s75}{\text{not}(\text{There exists }\ x\ \text{ such that }P\,)}\cr \cssId{s76}{\text{ is equivalent to }}\cr \cssId{s77}{\text{For all }x\text{, }(\text{not }P\,)} \end{gather} $$

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

Concept Practice