Why Parallelism Is So Important

Video Series: Incidence Geometry (Tutorials with Step-by-Step Proofs)

(D4M) — Here is the video transcript for this video.

00:00
hi everyone welcome back i’m dave in today’s episode we’re going to ask the
question why parallelism is so important so we’re going to go over what
parallelism is and start to ask this question here and yeah let’s do some
math let’s get started so i’m i’m first i’m going to point out that this episode
is part of the series incidence geometry
tutorials with step-by-step proofs check out the link below in the description
and we’re going to go over the incident axioms
and we went over them very quickly in the first episode of the series um very
slowly in the first episode of the series and so today right now i’m just
going to go over them pretty quickly so given any two points we have a unique
line through those two points and every line has at least two points on it
and there exists three non-collinear points
uh by collinear we mean three or more points are not arc are called collinear

00:01
if a line goes through all of them and they’re non-collinear if this does not
hold and three or more lines are called concurrent if they’re just a point
incident with all three of them and non-concurrent if this doesn’t happen
and then we have this definition of parallel right here and in the uh
theorems that we’ve proven so far i think we’ve only proven one theorem that
had uh talk about parallel in them so we’re
gonna start to do more with the parallel today so uh two lines are called
parallel if they’re not equal to each other
and no point is incident with both of them so another way of saying this is
two lines are parallel they have no points in common
so that’s just a briefer way of saying it this is more technical right but in
any case that’s what parallel means so parallel means distinct now keep in mind
that some people may define parallel lines uh slightly different they might
they might say that they’re either equal to each other um or they have no points

00:02
in common um you know for example if you’re say in a linear algebra class now
that might be something that you see oh yeah two lines are parallel you know
they’re the same line of course they’re parallel in any case um
uh we’re gonna say distinct lines and no point is incident with both of them so
uh if you’re you know studying some other text or doing some readings with
someone else you may want to pay attention because this isn’t a
necessarily a unanimous decision right here all right let’s get
on with what we’ve proven so far in the previous episodes so this is what we
proved in the second episode episode so non-parallel means just simply mean
not parallel and we have theorem two uh there exists three distinct lines
that are non-concurrent so we talked about non-parallel non-concurrent here
and then um in the next episode we talked about how every point um
has is it with at least one line and we strengthened that one up later

00:03
every line has at least one point no incident with it
every point has at least one line not incident with it
and every point is incident with two distinct lines so that makes
six makes three better and seven um i think in seven eight nine and 10 and
11 here i think we did all of these right here in the very in the previous
episode uh incident axioms and basic theorems
so again the link is below to the whole series playlist below
um check it out if you haven’t uh if you haven’t seen those yet
and so what we’re going to talk about today and so like i was saying earlier
only this one right here has something to do with parallel so we’re going to
look at parallel parallelism in this episode um and the way to probably approach
parallelism and to explain why it’s so important is
to look at these statements right here so we’re going to have three statements

00:04
right here the first one is for any line now and any point p
not incident with l right so i’m just going to draw a random line here line l
and any point p right so i have a point and so p is not on l
and this statement right here saying there’s no line that i could draw through p
there’s no line incident with p and it’s also parallel to l no line
right so is that a statement that you believe is true or false right
so uh that’s what an axiom right is something that we want to believe is true
or it’s just something that we want to have
give it some kind of meaning and then make a formal system up
another statement would be okay i have this line and this point not on it
um this statement is different than the first statement here the first statement
says there is exactly one line through p and we’ll call it m

00:05
so the second statement right here says there exists exactly one line incident
with p so no matter which line you draw l no matter which point you pick not on
l there’s exactly one line through p where these are parallel lines
and then the third statement says if l’s any line
uh it fails any line and p is not on that line then there has to exist more
than one and that would say like i have two lines both of them passing through p
so they’re distinct right so they both pass through p
and they’re both parallel to l so it may be the case that you’ve never
heard of something like this before because uh number two here is
going to be used for euclidean geometry which is what you use to study
free calculus and calculus and so maybe the case you’ve never heard
of these other two statements but in fact there are lots of types of

00:06
geometries out there some geometries uh follow this first statement um and the
first one is called the elliptic um parallel property so elliptic
parallel property and the second statement right here
which is the statement that we’re going to be the most interested in for this
video is going to be called the euclidean euclidean parallel property
and the third one is i didn’t make up these names here by
the way these these are well-known names uh appearing in lots of text and the
third statement is called the hyperbolic hyperbolic parallel property
and so which one should we add as an axia would you like to add the first

00:07
property right here number one the elliptic parallel property or number two
right and so in the next episode i’m going to give you some models i’m going
to explain what a model is and then we’ll start talking about
different models so models will be examples so we’ll see that some
geometries will have this property some geometries will have this property and
some geometries will have this property and so we’ll do that in the next episode
in this episode which is focusing on parallelism we’re going to focus on a
second statement right here and it’s called the euclidean parallel property
and basically what it says again is i think it’s important to see the words
but also to have your intuition right so
you want to say any line l and any point p
and the euclidean parallel property so they all have that in common right for
any line out for any line out for any line out right and so the euclidean
parallel property says there exists a unique line through p
that is parallel to l so m and m and l are parallel there exists a unique line

00:08
only one line passing through p and it’s uh parallel to l okay and so
there’s the euclidean parallel property and so now what i want to do is ask the
question here what is parallelism actually um
well actually first let me ask a couple questions here so the couple questions
are can any of these statements be proven from the incident axioms so so far we
have the instant axioms we have a1 we have a2 and we have a3
that might be your first question you might be scratching your head saying why
do we need to add these why don’t we just prove that they’re true or prove
that they’re false and so we’re going to answer that
question when we start looking at models so that’s a really good question uh to
ask right there if you ask that question you know that’s that’s a very
interesting question we’re going to get to that question here what’s another
question you might ask right so if not how would one show that one should take

00:09
um which one of these should we take as an
additional axiom and again we’re going to answer that with by looking at models
by looking at examples and so we’re going to come back to this question here
also um all right so before we answer those
questions we’re going to consider the three basic properties of parallelism
first i’m going to explain what parallelism is and then i’m going to
talk about these three properties right here reflexive symmetric and transitive
properties right here all right so let’s do that now
all right so what is parallelism so we’re going to take parallelism to be
this relation right here so parallelism um is this relation right here
is the relation which i’m going to denote by squiggle or
tilde um so i’m going to say two lines lines l and m and i’m going to say that

00:10
they’re related to each other um if line l is parallel to line m
or they’re equal so in other words this uh this this right here means
if and only if that l is parallel to m or they’re equal to each other
and it doesn’t matter if you write it equals first if they’re equal to each
other or they’re parallel to each other and remember some
people took this as the definition of parallel some people do take this as the
definition of parallel that they’re either equal or they have no points in
common so we’re using this symbol right here simply mean they have no points in
common and so i’m using this symbol to mean
parallel or they’re equal to each other so uh and this symbol right here this
relation right here is what i’m going to be calling parallelism is called

00:11
parallelism and yeah so parallelism all right so what about parallelism
right so we’re going to say first off that is reflexive
and so what does that mean right so this relation right here is reflexive
and so what that means is that if l is a line then it’s going to be related to
itself so this right here means that l is equal to l or l is parallel to l
now we know this case doesn’t hold because to be parallel with our
definition means they have to be distinct lines but this one obviously does hold
so this right here means either this one holds or this one holds and since we
know this one right here holds we know that this one right here holds and it
doesn’t matter what line we’re talking about so what we’re going to say is this

00:12
relation right here is reflexive and that’s because basically because the
equal sign is reflexive right so this is going to be extending the equals
the equality relation for lines and remember the equality of relation for
lines just means every point on this line is also on this line and every
point on this line is also on this line so it just means the lines have the same
points on them all right so symmetric symmetric so we’re going to say
two lines are symmetric if any time l is related to m here then we know that m
is related to l and let’s see why that’s true so so if i write this
how do i know that i can immediately also write this how do i know i can
switch them so let’s think about this for a minute
let’s think about what this right here means again so this right here means if

00:13
and only if l is equal to m or l is parallel to m
right now we know for equal sign that we can switch because
if these two lines have the exact same points well what does this right here
mean it just means the same thing these two lines have the exact same points so
if this is true we can also write this as true and this is relying upon the
fact that the equality relation is symmetric also right so so this part is clear
uh what about this part here can i switch this part right here can i
can i say that if l is um a parallel to m then m is parallel to l
just do these follow and they do actually because remember what this
right here means that l and m have no point in common
and so if l and m have no point in common then m and l have no point in
common either so we can just say um that you know this is um

00:14
going to be the same right here as if and only if m is related to l right here
so let’s just definition right there and this is true by uh symmetric
um because equality is symmetric and if we just look at this definition
right here and and look at the wording involved it’s you know l m have no
points in common all right so they’re symmetric we can see that this
relation which we’re calling parallelism
is we can see that it’s reflexive and we can see that it’s transitive now
when we talk about an equivalence relation which we are going to talk about
equivalence relation what we mean is something uh is a
relation in other words a set of ordered pairs think of it let me give it like
that is number one it’s reflexive and number two is symmetric
and then what’s the third condition to be an equivalent relation it’s transitive

00:15
so we’re going to see if this relation right here
parallelism is transitive if we have all
three and so far we have these two if we have all three then we can call it an
equivalence relation and we would like parallelism to be an equivalent relation
and that’s going to give the importance to the equivalence relation
uh parallelism and we’ll see why that is important in another episode so
this episode is you know what is parallelism and start to ask the
question why is it important and one of the steps in understanding why it’s
important is to first understand when it’s an equivalence relation because you
see parallelism isn’t always going to be an equivalence relation it’s going to
depend upon which of those three axioms that we’re going to add to the incident
axioms if we add one of those it’s actually going to become
all right so is parallelism transitive so we’re going to get this theorem here

00:16
uh and we’re going to work through this theorem right now if the euclidean
parallel property remember that was property number two
let me just refresh your memory on what that is if you draw any line l
and any point p not on l there exists a unique line through p
that’s parallel to l okay and so if if that holds
then this parallelism relation is in fact an equivalence relation
so let’s see how to let’s see how to prove this here um
so we need to show that this right here is transitive
so it remains to show that this right here is transitive because we already
talked about uh reflexive and symmetric um on the last screen right so we got to
show this right here is transitive and let’s write down what that means so

00:17
l is related to m and m is related to l uh or so if
right if if if l and m are related and m and l are related uh sorry m and n then
l and n are related so this is what we need to show
right so we need to show it’s transitive and this is what transitive means
okay so what i’m going to first do is outline the proof and kind of give you
the overall strategy that i’m going to take for the proof and then we’ll
write up a a proof in and see it kind of summarized
all right so i’m going to start off with this right here is my assumption right
here these two right here um and so let me put it like right here
l is related to m and m is related to n and we need to show that that implies it
must then follow that l is related to n and what i’m going to do is i’m going to

00:18
also suppose that that that’s not true so what does
it mean to say that this is not true so i’m going to assume for a
contradiction that it’s not true so what i want to do is write this out
and what and write out write out this right here and then we’ll piece together
the proof right so what does this right here mean right so this part right here
means that l is equal to m or l is parallel to m
and what does this right here mean m is equal to n or m is parallel to n
right so we got some cases here to think about now what does it mean to be not
you know to l to not be related to n right so

00:19
that means so that’s if and only if l is not equal to uh n and and um
it’s not true and so let’s see if i can put this down here a little bit more i’m
trying to save myself some space here so if and only if
so so let’s just work this out slowly right here in fact i’ll put it right here
so nut negation right here l is related to n what does that mean it means that l
is equal to n or l is parallel to n right just writing
out the definition just like i wrote this definition right here and i wrote
this definition right here so it’s just the definition inside the negation right
here now how do we negate an or so we’re going to say l is not equal to n and
l is not parallel to n all right so that’s if and only if now um
so what we have here is if and only if l is not equal to n and

00:20
and now now i’m going to say not if and only if then so i’m going to say implies
l is not equal to n and now let’s remember back to theorem one
um if two lines were not parallel then they have a unique point in common so
i’m going to use that theorem one right here so l is not in and by theorem one
so i’ll just say theorem 1 there exists a point let’s call it p
there exists a point p on l and in because these are not parallel and on l and n
um and so what i’m thinking about here is uh and so that was just scratch work
there all right and so i got these two written out and we’re
going to work through some cases here in a moment and i’m going to assume that

00:21
this is not happening right here if i assume that this is not happening
then when i work through all my cases every case should give me a contradiction
every case right because we have cases this may hold this may hold or this may
hold this may hold so i’m going to work through those four cases in every one of
those cases i’m going to get a contradiction with this right here
meaning this cannot hold so the negation of the negation
which is l is related to n which is what we want so that’s the idea behind this
proof so i got this point p by theorem one so l and n go through p
let’s call that line l and this line n and line m is going to be some line down
here and but we don’t know if they’re equal
to each other or whatever right here so that’s just kind of a guiding diagram
right here so remember we’re just trying to get some intuition do some scratch
work we’re not doing our proof yet we’re just kind of showing you the idea i’m
just kind of showing you the idea of how my proof is going to work

00:22
all right so um like i said i’m going to have cases right here so the first case
is going to be l is equal to m and m is equal to n
so why is this case a contradiction well if they’re both if all three lines
are equal to each other then that says that uh well that contradicts that right
there right away doesn’t it so l is equal to n l is not equal to n by that
right there so if if remember if this holds then this holds right here so
l is not equal to n but these are contradiction right here so this case is
a contradiction it cannot be happening that these are both holding and this
right here is holding all right so here we go so here’s
another case l is equal to m and um and let’s look at this one right
here now m is parallel to n all right and so what i’m going to do is
i’m going to look at this point p remember p is on l p is on n
um but what if l is equal to m so we have p is on m [Music]

00:23
p is on n right p is on l already and p is on m [Music] and
that’s a contradiction to this m and n have no points in common and so
let me just say that right here also uh p is on n right p is on n
p is the point because these two lines are are not parallel right negation
um and so p is on m and p is on l and that’s going to contradict this right here
that they have no points in common right there so these two cannot happen right
here and then now the case is very similar to if we have these two right
here holding right here so now we’re going to have l is parallel to m
and m is equal to n so what are we going to have here p is on
these two lines are equal so p is on m right here again because n and m are

00:24
equal p is on m and p is on why would p be on uh l well p is on l
by what by theorem one right so p is on l and so these two are parallel that’s a
contradiction right here l and l and m are supposed to be parallel no points in
common so this is the contradiction right here
and so this case cannot hold either and so then the last case is what if
they’re both parallel to each other so l is parallel to m and m is parallel to n
and so what do we have going on in this last case here and the last case is
you know we have two lines going through the p that are both parallel to m
and this is exactly what the euclidean parallel property says cannot happen
remember the euclidean parallel property
says that if you take a point off a line
there’s only one line going through that point that is parallel to m

00:25
but actually have two lines so l l and n um are different lines
and they both pass through p and they’re both parallel to m so this one right
here is a contradiction by the euclidean parallel prop property
i’ll just abbreviate that euclidean parallel property
so all these cases are a contradiction which means that cannot hold
so if this holds then so if if these two hold then
uh the negation has to hold of this because this right here leads to
contradictions so the negation of the negation right is just l is related to n
so this is the exactly the transitive property there that we’re trying to show
and on the previous screen we did reflexive and symmetric and this is the
argument for the transitive property and so then we’re going to have everything
we need to get an equivalence relation right there

00:26
so um let me just uh write or walk through the proof now the proof should
be easy much easier to follow when you can kind of see
how i broke this up into four cases like that and how i got a contradiction in
each case all right so here we go with the proof now
so first i’m going to assume that the euclidean parallel property holds
and i’m going to suppose that l is related to m and m is related to n
and i’m going to suppose that this right here does not hold and i want to get a
contradiction in every single case and then we’ll see that
i mean you know if you get a contradiction in every single case
and one of the cases has to hold right so then you know
the negation right which is it does hold all right and so
since this right here does not hold so we have l is not equal to n and

00:27
l and n are parallel uh are sorry are not parallel and so by
theorem one if if two lines are not parallel then there exists a unique
point and and i’m not going to necessarily use i’m just using there
exist a point uh incident with lines l and m
all right so now i got some cases to happen here the cases are coming from
this right here and this right here right because this says two things may
happen either uh their parallel lines or their equal lines and i have two cases
right here also so all together we have four cases right
and we we went through all four cases so if these two right here happen remember
this was case one if this these two lines are equal
well that can’t happen because we’re assuming negation happens right here so
we’re assuming that there’s or not equal right here so these cannot happen
all right so now suppose that l is equal to m and m is parallel to n
so then p lies on in because um because it already um yeah so p

00:28
already lies on n and l but if l is equal to m then p is on m now
and but we can’t have p on both of these lines because they’re parallel
and similarly we cannot have this case right so so that’s uh you know
enough there said because everything is symmetric here uh so the remaining case
uh both lines are parallel to the m cannot occur either since the euclidean
parallel property holds there has to be a unique line and we know
because we have distinct lines and they’re both parallel to m so that’s
going to violate the euclidean parallel property which we assume to exist
all right so all cases considered it follows that if these are related and
these are related then l and n must be related also
and so that gives us reflexive symmetric and transitive as needed
so there we go uh let’s do some more math together

00:29
check out this episode right here right here

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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