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## Worked examples

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# CA Geometry: More proofs

## Video transcript

All right, we're on problem
number seven. And when I copied and
pasted it I made it a little bit smaller. So I'm going to read it for you
just in case this is too small for you to read. It says, use the proof to answer
the question below. So they gave us that angle 2
is congruent to angle 3. So the measure of angle
2 is equal to the measure of angle 3. I'm trying to get the knack of
the language that they use in geometry class. Which, I will admit, that
language kind of tends to disappear as you leave
your geometry class. But since we're in
geometry class, we'll use that language. So angle 2 is congruent
to angle 3. Which means that their
measure is the same. Or that they kind of did the
same angle, essentially. Fair enough. So let's see. Statement one, angle 2 is
congruent to angle 3. That's given, I drew that
already up here. Statement two, angle 1 is
congruent to angle 2, angle 3 is congruent to angle 4. So they're saying that angle
2 is congruent to angle 1. Or angle 1 is congruent
to angle 2. That's right there. And that angle 4 is congruent
to angle 3. Fair enough. And they say, what's the reason
that you could give. And I forgot the actual
terminology. But in my head, I was thinking
opposite angles are equal or the measures are equal,
or they are congruent. And you could just imagine two
sticks and changing the angles of the intersection. You'll see that opposite
angles are always going to be congruent. Let's see, that is the
reason I would give. Opposite angles are congruent. Let's see which statement of the
choices is most like what I just said. Complements of congruent
angles are congruent. Complements. Vertical angles are congruent. I think that's what they
call opposite angles. Vertical angles. I think that's what they mean
by opposite angles. Supplements of congruent
angles are congruent. That's not true. Corresponding angles
are congruent. These aren't corresponding. I think this is what they
mean by vertical angles. Complements of congruent
angles are congruent. You know what, I'm going
to look this up with you on Wikipedia. Let me see. Vertical angles. As you can see, at the age of
32 some of the terminology starts to escape you. What matters is that you
understand the intuition and then you can do these Wikipedia
searches to just make sure that you remember
the right terminology. Let's see what Wikipedia
has to say about it. Vertical angles. A pair of angles is said to be
vertical or opposite, I guess I used the British English,
opposite angles if the angles share the same vertex and are
bounded by the same pair of lines but are opposite
to each other. Right. So somehow, growing up in
Louisiana, I somehow picked up the British English
version of it. Maybe because the word opposite
made a lot more sense to me than the word vertical. With that said, they're
the same thing. Wikipedia has shown
us the light. And so my logic of opposite
angles is the same as their logic of vertical angles
are congruent. Next problem. I'll start using the
U.S. terminology. Although I think there are a
good number of people outside of the U.S. who watch these. So maybe it's good that I
somehow picked up the British English version of it. Once again, it might be
hard for you to read. I'll read it out for you. Two lines in a plane
always intersect in exactly one point. Fair enough. Which of the following best
describes a counter example to the assertion above. I like to think of
the answer even before seeing the choices. So can I think of two lines in
a plane that always intersect at exactly one point. Well, what if they
are parallel? What if I have that line
and that line. They're never going to intersect
with each other. They're parallel. That's the definition
of parallel lines. The other example I
can think of is if they're the same line. I guess you might not want to
call them two the lines then. But I would. This line and then
I had this line. Well that's parallel, but
imagine they were right on top of each other, they would
intersect everywhere. So either of those would be
counter examples to the idea that two lines in a plane
always intersect at exactly one point. And if we look at their choices,
well OK, they have the first thing I just
wrote there. Parallel lines, obviously they
are two lines in a plane. But they don't intersect
in one point. Problem eight. I'm going to make it a little
bigger from now on so you can read it. OK, this is problem nine. Which figure can serve as the
counter example to the conjecture below? If one pair of opposite sides
of a quadrilateral is parallel, then the quadrilateral
is a parallelogram. So once again, a lot
of terminology. And I do remember these
from my geometry days. Quadrilateral means
four sides. A four sided figure. And a parallelogram
means that all the opposite sides are parallel. For example, this is
a parallelogram. Let me see how well
I can do this. If you ignore this little part
is hanging off there, that's a parallelogram. And if all the sides were
the same, it's a rhombus and all of that. But that's a parallelogram. And that's a parallelogram
because this side is parallel to that side. And this side is parallel
to that side. All the sides are parallel. Now they say, if one pair
of opposite sides of a quadrilateral is parallel, then
the quadrilateral is a parallelogram. So I want to give a
counter example. Let me draw a figure that has
two sides that are parallel. Let's say that side and that
side are parallel. And I don't want the other
two to be parallel. Then it wouldn't be
a parallelogram. Let's say the other sides
are not parallel. Let's say they look like that. And like that. And once again, just digging in
my head of definitions of shapes, that looks like
a trapezoid to me. So let's see. Yeah, good, you have a trapezoid
as a choice. Trapezoid. All the rest are parallelograms.
A rectangle, all the sides are parellel. And we have all 90
degree angles. Rectangles are actually a subset
of parallelograms. Rhombus, we have a parallelogram
where all of the sides are the same length. All the angles aren't
necessarily equal. Square is all the sides are
parallel, equal, and all the angles are 90 degrees. So all of these are subsets
of parallelograms. This is not a parallelogram. Although it does have two
sides that are parallel. So this is the counter example
to the conjecture. I think you're already
seeing a pattern. In a lot of geometry,
the terminology is often the hard part. The ideas aren't as deep as the
terminology might suggest. Given, TRAP, that already
makes me worried. Given TRAP is an isosceles
trapezoid with diagonals RP and TA, which of the following
must be true? OK, let's see what
we can do here. So an isosceles trapezoid means
that the two sides that lead up from the base to
the top side are equal. Kind of like an isosceles
triangle. So let me draw that. Actually, I'm kind
of guessing that. I haven't seen the definition
of an isosceles triangle anytime in the recent past.
An isosceles trapezoid. But it sounds right. So I'll go with it. And that's a good
skill in life. So I think what they say when
they say an isosceles trapezoid, they are essentially
saying that this side, it's a trapezoid,
so that's going to be equal to that. They're saying that this side
is equal to that side. An isosceles trapezoid. And they say RP and TA
are diagonals of it. Let me draw that. So let me actually write
the whole TRAP. So this is T R A P
is a trapezoid. Let me draw the diagonals. RP is that diagonal. And TA is this diagonal
right here. OK. All right, let's see
what we can do. Which of the following
must be true? RP is perpendicular to TA. Well, I can already tell
you that that's not going to be true. And you don't even
have to prove it. Because you can even
visualize it. If you squeezed the
top part down. Imagine some device where this
is kind of a cross-section. If you were to squeeze the top
down, they didn't tell us how high it is. Then these angles, let me
see if I can draw it. That angle and that angle, which
are opposite or vertical angles, which we know is
the U.S. word for it. Those are going to get smaller
and smaller if we squeeze it down. And in order for both of these
to be perpendicular those would have to be 90
degree angles. And we already can
see that that's definitely not the case. All right. RP is parallel to TA. Well that's clearly not the
case, they intersect. All right, they're
the diagonals. RP is congruent to TA. Well, that looks pretty
good to me. Because it's an isosceles
trapezoid. If we drew a line of symmetry
here, everything you see on this side is going to be kind
of congruent to its mirror image on that side. So both of these lines, this is
going to be equal to this. And I can make the argument, but
basically we know that RP, since this is an isosceles
trapezoid, you could imagine kind of continuing a triangle
and making an isosceles triangle here. Then we would know that that
angle is equal to that angle. Well, actually I'm not going
to go down that path. But you can almost look
at it from inspection. Although, maybe I should do
a little more rigorous definition of it. But RP is definitely going
to be congruent to TA. Because both sides of
these trapezoids are going to be symmetric. And so there's no way you
could have RP being a different length than TA. Since this trapezoid is
perfectly symmetric, since it's isoceles. And then D, RP bisects TA. Let's say if I were to draw
this trapezoid slightly differently. If it looks something
like this. If this was the trapezoid. This is also an isosceles
trapezoid. And then the diagonals
would look like this. So here, it's pretty clear that
they're not bisecting each other. In order for them to bisect each
other, this length would have to be equal
to that length. And that's clear just by
looking at it that that's not the case. That is not equal to that. So they're definitely not
bisecting each other. So you can really, in this
problem, knock out choices A, B and D. And say oh well choice
C looks pretty good. But you can actually deduce that
by using an argument of all of the angles. Anyway, that's going
to waste your time. But that's a good exercise
for you. Is to make the formal proof
argument of why this is true. Although, you can make a pretty
good intuitive argument just based on the symmetry
of the triangle itself. Anyway, see you in
the next video.