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:
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 |
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[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 |
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 |
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] $\text{not}(\text{There exists }x\text{ such that }P\,)$ | $\text{ is equivalent to }$ | $\text{For all }x\text{, }(\text{not }P\,)$ |
‘Equivalent Characterizations of a Parallelogram’ THEOREM
Here's an example of applying the negation tools:
- Negate: ‘Both pairs of opposite sides are parallel.’
- Rephrase as a ‘for all’ sentence: For all opposite sides $\,x\,$ and $\,y\,$, $\,x\parallel y\,$.
- Negate the ‘for all’ sentence: There exist opposite sides $\,x\,$ and $\,y\,$ such that $\,x\not\parallel y\,$.
- Rephrase the negation as an English sentence: There exists a pair of opposite sides that are not parallel.
These are all negations of: $\,Q\,$ is a parallelogram (i.e., both pairs of opposite sides are parallel)
- $Q\,$ is not a parallelogram
- it is not true that both pairs of opposite sides are parallel
- at least one pair of opposite sides are not parallel
- there exists a pair of opposite sides that are not parallel
These are all negations of: the diagonals of $\,Q\,$ bisect each other
- it is not true that the diagonals of $\,Q\,$ bisect each other
- at least one diagonal of $\,Q\,$ is not bisected at the point of intersection of the diagonals
- there exists a diagonal of $\,Q\,$ that is not bisected at the point of intersection of the diagonals
These are all negations of: both pairs of opposite sides are equal
- it is not true that both pairs of opposite sides are equal
- at least one pair of opposite sides are not equal
- there exists a pair of opposite sides that are not equal
These are all negations of: both pairs of opposite angles are equal
- it is not true that both pairs of opposite angles are equal
- at least one pair of opposite angles are not equal
- there exists a pair of opposite angles that are not 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
- it is not true that one pair of opposite sides are both parallel and equal
- each pair of opposite sides are either not parallel or not equal
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Introduction to Area and Perimeter