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## 8th grade foundations (Eureka Math/EngageNY)

### Unit 3: Lesson 2

Topic B: Foundations- Writing proportions example
- Worked example: Solving proportions
- Solving proportions
- Complementary & supplementary angles
- Complementary and supplementary angles (visual)
- Vertical angles
- Vertical angles
- Identifying supplementary, complementary, and vertical angles
- Finding missing angles
- Unknown angle problems (with algebra)

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# Vertical angles

By using our knowledge of supplementary, adjacent, and vertical angles, we can solve problems involving the intersection of two lines. Including this one! Created by Sal Khan.

## Want to join the conversation?

- What is the difference between intercept and intersecting? anyone?(96 votes)
- Intercept and intersect are similar terms used in different math subjects. An intercept, as in x or y intercept, is a term commonly used in graphing, where a line crosses your y or x axis at zero. An intersection is used in geometry, where a line segment or a ray crosses another. Hope this helped!(153 votes)

- How is CEF adjacent (1:30) ?(14 votes)
- Adjacent refers to angles that add up to 90 degrees not straight lines. They are not adjacent, They are supplementary.(4 votes)

- What are adjacent, supplementary and complimentary angles? anyone?(7 votes)
- Supplementary angles are two angles whose angle measures sum to 180 degrees. For example, if one angle is 120 degrees and another angle is 60 degrees, the two angles are supplementary. Complementary angles are two angles whose angle measures sum to 90 degrees. For example, if one angle is 30 degrees and another angle is 60 degrees, the two angles are complementary. Adjacent angles share a side, share a vertex, and don't overlap. This link might help you picture adjacent angles. https://www.mathsisfun.com/geometry/adjacent-angles.html(23 votes)

- Can angles be complementary and/or supplementary without being adjacent? Thx in advance(9 votes)
- Yes if you have two parallel lines and a transversal, there are all sorts of supplementary angles (same side interior, same side exterior) that are not adjacent. In a right triangle, the two acute angles will always be complementary. They do not have to even be related to each other in any way, they can be drawn independently.(8 votes)

- So what is exactly a vertical angle?(4 votes)
- It is vertical angles (plural) - a pair of non-adjacent angles formed when two lines intersect. There are two pair of vertical angles with intersecting lines, they are across from each other.(10 votes)

- How are angles:CEA and CEB Adjacent? They aren't connecting to one line..(6 votes)
- The angles CEA and CEB are indeed adjacent. As you can see by looking at the letters they have both C and E. Both angles share a common vertex of E and they also share the line segment of C, therefore, they are touching, or adjacent.(4 votes)

- Are there any tricks to remember complimentary, vertical, and supplementary angles?(5 votes)
- complimentary and supplementary are not quite related to vertical, so you should not put all three together. Vertical angles are across from each other on any two intersecting lines and are always congruent.

If you draw a line across the C, it sort of looks like a 9, so it is two angles adding to be 90, If you draw a line across the S, it sort of looks like an 8 to remind us that it is two angles adding up to 180.(2 votes)

- I still don't get the difference between the verticle angle v.s supplementary & complementary angles.(3 votes)
- vertical angles are very specific - you have two have two intersecting lines to form two sets of vertical angles which are across from each other and congruent.

Both supplementary angles and complementary angles are much broader - they do not even have to be touching or near each other, but they could be.

Any two angles that add to be 90 degrees are complementary. This includes the two angles of a right triangle, any time you put a ray between the two sides of a right angle, or just two angles in two different places.

Similarly, any two angles that add to be 180 degrees is complementary. This includes linear pairs (any two angles that form a line), same side interior, same side exterior, or just any two angles in two different places.(6 votes)

- What if an angle is 0 degrees. Will the angle be 180 degrees because it will pretty much be a line like
*______*(4 votes)- 0º angles are called "degenerate angles" and 180º are called "straight angles". They look very similar when drawn, but can have different properties depending on where they show up.(2 votes)

- Do supplementary and complementary angles have to be next to each other?(2 votes)
- No they don't need to be adjacent. 2 angles just have to add up to 90 or 180.

http://www.mathwarehouse.com/geometry/angle/images/complementary/picture-of-complementary-angles2.png

http://www.mathwarehouse.com/geometry/angle/images/supplementary/supplementary-not-adjacent.png(5 votes)

## Video transcript

Let's say I have two
intersecting line segments. So let's call that segment AB. And then I have segment CD. So that is C and that is D. And
they intersect right over here at point E. And let's
say we know, we're given, that this angle right over here,
that the measure of angle-- That B is kind of, I don't know
why I wrote it so far away. So let me make that
a little bit closer. Let me make that B
a little bit closer. So let's say-- I'll
do that in yellow. Let's say that we know that
the measure of this angle right over here,
angle BED, let's say that we know that
measure is 70 degrees. Given that information,
what I want to do, based only on what
we know so far and not using a protractor,
what I want to do is figure out what the other
angles in this picture are. So what's the measure of angle
CEB, the measure of angle AEC, and the measure of angle AED? So the first thing
that you might notice when you
look at this, I've already told you that
this is a line segment and that this is a line segment. You see that angle BED and
angle CEB are adjacent. And we also see that if
you take the outer sides of those angles, it
forms a straight angle. And we also see that angle
CED is a straight angle. So we know that these two angles
must also be supplementary. They're next to each other
and they form a straight angle when you take their outer sides. So we know that angle BED and
angle CEB are supplementary, which means they add up to,
or that their measures add up to 180 degrees. Supplementary angles. Which tells us that
the measure of angle BED plus the measure
of angle CEB-- and I keep writing measure here. Sometimes you'll just
see people write, angle BED plus angle CEB
is equal to 180 degrees. Now we already know the measure
of angle BED is 70 degrees. So we already know that
this thing right over here is 70 degrees. And so 70 degrees plus
the measure of angle CEB is 180 degrees. You subtract 70 from
both sides, and we get the measure of angle
CEB is equal to 110 degrees. I just subtracted 70
from both sides of that. So we figured out that this
right over here is 110 degrees. Well, that's interesting. And I went through
more steps than you would if you were doing
this problem quickly. If you did this problem quickly
in your head, you'd say, look this is 70 degrees,
this angle plus this angle would be 180 degrees, so
this has to be 110 degrees. So now let's use the
same logic to figure out what angle CEA is. So now we care about the
measure of angle CEA. And we can use the exact same
logic that we used over here. Angle CEA and angle
CEB, they are adjacent. They form a straight angle,
if you look at their outsides, so they must be supplementary. They form a straight
angle right over here. So they're supplementary. So they must add
up to 180 degrees. So the measure of angle CEA
plus the measure of angle CEB, which is 110 degrees, must
be equal to 180 degrees. So once again, subtract
110 from both sides. You get the measure of angle
CEA is equal to 70 degrees. So this one right over
here is also 70 degrees. And what we'll learn
in the next video is that this is no coincidence. These two angles, angle
CEA and angle BED, sometimes they're called
opposite angles-- well, I have often called
them opposite angles, but the more correct term
for them is vertical angles. And we haven't proved it. We've just seen a
special case here where these vertical
angles are equal. But it actually turns out that
vertical angles are always equal. But we haven't proved it to
ourselves for the general case. But let me just
write down this word since it's a nice new word. So angle CEA and angle
BED are vertical. And you might say, wait, they
look like they're horizontal, they're next to each other. And the vertical
really just means that they're across
from each other, across an intersection
from each other. Angle CEB and angle
AED are also vertical. So let me write that down. Angle CEB and angle
AED are also vertical. And that might even make
a little bit more sense, because it literally is, one
is on top and one is on bottom. They're kind of vertically
opposite from each other. But these horizontally
opposite angles are also called vertical angles. So now we have one angle left
to figure out, angle AED. And based on what
I already told you, vertical angles tend to be,
or they are always, equal. But we haven't proven that to
ourselves yet so we can't just use that property to say
that this is 110 degrees. So what we're going to do
is use the exact same logic. CEA and AED are
clearly supplementary. Their outsides form
a straight angle. They're clearly
supplementary, so CEA and AED must add up to 180 degrees. Or we could say the
measure of angle AED plus the measure of angle CEA
must be equal to 180 degrees. We know the measure
of CEA is 70 degrees. We know it is 70 degrees. So you subtract 70
from both sides. You get the measure of angle
AED is equal to 110 degrees. So we got the exact
result that we expected. So this angle right over
here is 110 degrees. And so if you take any
of the adjacent angles that their outer sides
form a straight angle, you see they add up to 180. This one and that
one add up to 180. This one and that
one add up to 180. This one and that
one add up to 180. And this one and that
one add up to 180. If you go all the way
around the circle, you'll see that they
add up to 360 degrees. Because you literally are
going all the way around. So 70 plus 110 is
180, plus 70 is 250, plus 110 is 360 degrees. I'll leave you there. This is the first
time that we've kind of found some interesting
results using the tool kit that we've built up so far. In the next video,
we'll actually prove to ourselves using pretty
much the exact same logic here, but we'll just do it with
generalized numbers-- we won't use 70
degrees-- to prove that the measure of
vertical angles are equal.