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# Angles of similar quadrilaterals

Video transcript

In the diagram below,
quadrilateral EFGH-- so that's this one
here-- was obtained by performing a sequence
of transformations on quadrilateral ABCD. And then they tell
us the information that's already written here. Measure of angle
A is 69 degrees. Measure of angle
B is 102 degrees. Measure of angle
G is 145 degrees. And measure of angle
H is 44 degrees. Which of the following
additional facts is sufficient to conclude that
the only transformations used were translations, rotations,
reflections, and dilations, and not any other kind
of transformations? So this is another
way of saying, if we only used translations,
rotations, reflections, and dilations, then that means
that these two figures are similar. So if this is the
case, then we're dealing with similar
quadrilaterals. So another way of saying this
is, which of these facts-- so either this first
fact, or this second fact, or both of them, or neither
of them-- which of these facts are sufficient in order to
determine that these two quadrilaterals are similar,
which is another way of saying that you could go from
one to another only through translations, rotations,
reflections, and dilations? So let's look at this first
set of facts right over here. So they give us the measure
of angle C is 145 degrees. So they tell us that right
over there is 145 degrees. And they tell us that
the measure of angle E is 69 degrees. So they tell us that
this is 69 degrees. So with that piece of
information right over here, in each of these quadrilaterals,
we know three of the angles. And because we know three,
we can figure out the fourth. We can figure it
out because the sum of the angles in a
quadrilateral are going to add up to 360 degrees. So if we wanted to figure
out the measure of angle F right over here--
let's say that is x-- we could say that x
plus 69 degrees plus 145 degrees plus 44 degrees
is equal to 360 degrees. So let's get the calculator
out right over here. Our x would essentially
be 360 minus these things. So 360 minus 69 minus 145
minus 44 is 102 degrees. So this angle right over
here is 102 degrees. Now notice, these two already
have three angles in common. They have a 69 degree--
let me color code it-- so this angle
right over here has a measure of 69 degrees. So does this one. Then you move one over. This angle right over here is
102 degrees, so is this one. Then you move to the next
angle is 145 degrees, well, that would correspond
to this one. And if you know
three of the angles, you know that this
last one right over here, this fourth one, was
going to have to be 44 degrees. You could take 360 and from
it subtract 145, 102, and 69, and you will get 44 degrees. So you have two quadrilaterals
where all of these angles are congruent to each other. So if you have these
two shapes and you have all the
corresponding angles are congruent to
each other, then you know that they are similar. And if you know that
they are similar, you know that you
can go from one to another purely through some
combination of translations, which is really just shifting
it in two dimensions, rotating it around some
points, reflecting it, which you can kind of use flipping
it over around some line, and dilations, which is
scaling it up or down. So the first one
is definitely good. Let's see if the second one
could also independently help us determine that these
two things are similar. So what I'm going to
do is clear this out so I have some
space to work with. Let me clear it out. Let me clear this
right over here. And now let's think about
what the second statement is telling us. The second statement tells
us the measure of angle D is 44 degrees. So they're saying
that is 44 degrees. And that the measure of
angle F is 102 degrees. So once again, it's given us
one more angle in each of them. And from that, we can
figure out the last angle. And so if you know all of the
angles in two quadrilaterals, you can determine
if they are similar. They're going to be
similar only if you have four corresponding
angles that are congruent to each other. Let me just show
it to you just so I don't say that it's enough. Let's calculate this last
angle right over here. We can get our
calculator back out. It's going to be 360
minus 102 minus 145 minus 44, which is
equal to 69 degrees. So this right over
here is 69 degrees. So notice, 102 degrees, then
69 degrees, then 44 degrees. And so this last one has
to be 145 degrees, just like this one. If three of the
angles are the same, that fourth one
has to be the same. So this angle corresponds
to this angle. This angle corresponds
to that angle. This angle corresponds
to that angle. And then finally, this angle
corresponds to this angle. You have two quadrilaterals. The interior four angles
are all congruent. They correspond to each other. So they must be similar. And so you can get
from one to another through translations, rotations,
reflections, and dilations. So this one is also good. So actually, either of
these are sufficient in order to conclude that the
only transformations used were these. So I would click, both
answers are correct.