GEO.1.6
Summary
TLDRThe video explains various angle relationships, including vertical, adjacent, complementary, and supplementary angles. It covers how to identify these angles using examples and visual aids, emphasizing key concepts like congruence and linear pairs. The instructor also provides tips on distinguishing between complementary and supplementary angles, using memorable tricks. The video concludes with exercises on solving for unknown angles using algebra, applying the angle relationships discussed. Viewers are encouraged to practice on their own and check their answers.
Takeaways
- ๐ Vertical angles are congruent angles located across from each other on intersecting lines, like angle 1 and angle 2.
- ๐ฏ Adjacent angles are two angles that are next to each other and share a common side, such as angle 1 and angle 2.
- ๐ Complementary angles are two angles whose sum equals 90 degrees, for example, angle 1 and angle 2 summing up to 90 degrees.
- ๐ Supplementary angles are two angles whose sum is 180 degrees, like one angle being 135 degrees and another 45 degrees.
- ๐ Linear pairs are adjacent and supplementary angles that form a straight line, with their sum always being 180 degrees.
- ๐ Vertical angles are congruent, meaning they have equal measures, like two angles each measuring 112 degrees.
- ๐งฎ Complementary angles add up to 90 degrees; for example, if one angle is 68 degrees, the other must be 22 degrees.
- ๐ A linear pair is different from supplementary angles because they must be adjacent and supplementary, not just add up to 180 degrees.
- ๐ Angle relationships can be used to find unknown angle measures by applying the properties of vertical, complementary, and supplementary angles.
- ๐ For linear pairs or complementary angles, setting up equations using their properties helps in solving for unknown angles.
Q & A
What are vertical angles?
-Vertical angles are two angles that are directly across from each other on intersecting lines. They are always congruent, meaning they have the same measure.
How can you remember what vertical angles are?
-You can remember vertical angles by imagining an 'X' shape. The angles directly across from each other through the vertex are the vertical angles.
What are adjacent angles?
-Adjacent angles are two angles that are next to each other and share a common side.
What are complementary angles?
-Complementary angles are two angles whose sum is 90 degrees.
How can you distinguish between complementary and supplementary angles?
-Complementary angles sum to 90 degrees, while supplementary angles sum to 180 degrees. A helpful way to remember is that complementary angles are the 'right' thing to do (90 degrees forms a right angle).
What is a linear pair of angles?
-A linear pair consists of two adjacent angles that are supplementary. They form a straight line when combined, adding up to 180 degrees.
Do supplementary angles have to be adjacent to each other?
-No, supplementary angles do not have to be adjacent. They can be separate, but their angle measures must add up to 180 degrees.
How do you find the measure of an unknown angle using complementary angles?
-To find the measure of an unknown angle using complementary angles, subtract the measure of the known angle from 90 degrees. For example, if one angle is 68 degrees, the other would be 90 - 68 = 22 degrees.
What does it mean for angles to be bisected?
-When an angle is bisected, it is divided into two equal angles. If a line bisects an angle, the two resulting angles are congruent.
How do you solve for an unknown angle in a linear pair if given one angle's measure?
-If you know one angle in a linear pair, you can find the other by subtracting the known angle's measure from 180 degrees since the angles are supplementary. For example, if one angle is 135 degrees, the other is 180 - 135 = 45 degrees.
Outlines
๐ Introduction to Angle Relationships
This paragraph introduces the concept of angle relationships, starting with vertical angles. It explains that vertical angles are formed when two lines intersect and are always congruent. An example is provided using four angles, where pairs of opposite angles are congruent. The paragraph then transitions to adjacent angles, which share a common side, with an example illustrating two adjacent angles.
๐ฏ Exploring Supplementary and Linear Pair Angles
This paragraph delves into supplementary angles, which add up to 180 degrees, and how they differ from linear pairs. It explains that supplementary angles donโt need to be adjacent, whereas linear pairs are both adjacent and supplementary, forming a straight line. The paragraph ends with an exercise encouraging the reader to identify various types of angles based on the information provided.
๐ Solving for Angle Measures Using Relationships
This paragraph focuses on using angle relationships to determine specific angle measures. It walks through examples involving vertical, complementary, and supplementary angles, demonstrating how to calculate unknown angles. The reader is guided through the steps to find angle measures, with explanations on how to set up and solve equations based on the given relationships.
๐ Understanding and Solving Angle Equations
In this final paragraph, the script continues with exercises on solving angle equations, specifically using the concept of angle bisectors. It explains how to approach problems where angles are bisected, leading to congruent angles that can be solved using algebra. The reader is encouraged to practice on their own and check their work with a provided key.
Mindmap
Keywords
๐กVertical Angles
๐กAdjacent Angles
๐กComplementary Angles
๐กSupplementary Angles
๐กLinear Pair
๐กCongruent
๐กAngle Bisector
๐กPerpendicular
๐กEquation Setup
๐กAngle Relationships
Highlights
Introduction to angle relationships using a graphic organizer.
Definition and identification of vertical angles as angles across from each other on intersecting lines.
Explanation of why vertical angles are always congruent.
Mnemonic for remembering vertical angles as 'directly across from each other through the vertex'.
Introduction to adjacent angles and their shared common side.
Definition of complementary angles as two angles whose sum is 90 degrees.
Practical example of identifying complementary angles in a diagram.
Definition of supplementary angles as two angles whose sum is 180 degrees.
Memory aid for differentiating between complementary and supplementary angles using the concept of compliments.
Explanation of a linear pair as two adjacent and supplementary angles forming a straight line.
Difference between linear pairs and supplementary angles in terms of adjacency.
Guidance on identifying angle relationships in a given set of angles.
Method for finding angle measures using vertical angle congruency.
Process for solving for angles when given complementary angles.
Approach to calculating angles in a linear pair using the sum of 180 degrees.
Strategy for solving angle measures using algebra and angle relationships.
Example of setting up equations for angles when given perpendicular lines.
Technique for finding angles when given a linear pair and the measures of some angles.
Challenge problem involving finding angles when given a relationship between their measures.
Introduction to using angle bisectors to solve for unknown angles.
Method for solving angles when given they are bisected and their relationship.
Transcripts
so now we're going to look at actual
angle relationships that exist amongst
angles so first we're going to look at
this graphic organizer right here
because we're going to break apart what
they all are okay so we're gonna start
at the top and then we're going to work
our way counterclockwise okay so
vertical angles are two angles across
from each other on intersecting lines so
for example
let me actually draw
me some vertical angles it's right there
okay so I'm going to use this color so
let's say we have one two three and four
that's what I might call them so
vertical angles would be like angle
angle one being congruent to angle two
there across from each other right and
angle three being congruent to angle
four okay they're always across from
each other on intersecting lines they're
always congruent now the reason they're
called vertical if you can imagine is
this is the firt X and they are directly
across from each other through the
vertex okay so that helps you remember
vertical a little bit better so going
counter counterclockwise we're going to
come over here to adjacent angles
okay so adjacent angles the two are two
angles that are next to each other and
share a common side so for example all
right so I got some drawn up there so
make my life easier
so Jason angles like right here
they share a common side so this would
be like angle 1 angle 2 those are
adjacent and this is the side that they
share
all right so now let's move down over
here to complementary angles those are
two angles whose sum is 90 degrees so
let's draw one of them here's an example
of a complementary angle the two angles
whose sum is 90 degrees so like this
would be one
this would be angle wine this would be
equal to 90 degrees so you can say that
angle 1 plus a 2 is equal to 90 degrees
okay next we have
supplementary angles
and those are two angles whose sum is
180 degrees so let me get that drawn so
here we have an example of supplementary
angles so this for example let's say
that this was 135 degrees and that was
45 degrees and then you would end up
with a hundred and eighty degrees okay
now somehow I remember the difference
between complementary and supplementary
supplementary is a little bit harder to
necessary remember so to me the biggest
way or the easiest way to remember the
difference is to remember what
complementary angles are because then
you know that one is 90 you know that
one is 180 so if I know which one is 90
I'm able to figure out the other one is
180 so I think of complementary and I
think of this might be a silly way to
remember but it helps me
I think of complements and they're the
right thing to do to give a person a
compliment so complementary angles are
the right thing to do so they're 90
degrees
it might be silly way but it works now a
linear pair there are two angles that
are adjacent and supplementary and they
form a straight line so basically what I
have down here so I'm an actually
bring that up there as well okay so this
is a linear pair so I would say one two
you could say I have 1 plus angle 2 is
equal to 180 degrees now the difference
between a linear pair and supplementary
is that supplementary angles do not
necessarily
have to be attached so for example
remember what I drew out here
so this is our hundred and thirty five
degree angle
then
this would be
let me make this a little smaller so
that you can see it so here we have them
separated so this is still a hundred and
thirty-five degrees and this is still 45
degrees so in a supplementary angle they
don't actually have to be attached to
each other
they don't even have to be near to each
other but if they equal 180 degrees when
you add their angle measures together
they're supplementary but a linear pair
what's up here they have to be adjacent
to each other and supplementary in order
to be a linear pair
it's now coming on down here identifying
the types of angles pause this and then
see if you can identify the angle
relationships that are taking place
between these angles and then we'll come
back and see how you did
all right so now looking at one
those are vertical angles two are
adjacent and complementary three
adjacent supplementary and a linear pair
remember a linear pair is a linear pair
because it's adjacent in supplementary
so you can't leave those out number four
those are just jacent number five
vertical and complementary look at that
again if you didn't get the
complimentary and six is vertical and
supplementary look that again if you
didn't get supplementary now let's look
at using these angle relationships to
actually find angle measures right so
for a number well I'm just gonna scroll
through them we're gonna look at these
women write down what we have okay for
number one these are vertical angles
vertical angles are congruent so that
must also be a hundred and twelve
degrees number two these are
complementary angles remember complement
so the right thing to do ninety degrees
so just doing 90-68 is going to give you
X being twenty two degrees three those
are supplementary angles it's a linear
pair as well so because they're
supplementary they're 180 degrees so 180
minus 120 four
gives you 56 degrees now when we start
further on it's asking you to think a
little bit more on what you're looking
at right just let's look at number five
we're going to skip number four and go
to five
right so first and foremost you will
notice this 90-degree angle there and
this straight line if this is 90 degrees
then this is 90 degrees so Z is just
going to be 90 minus 72 which is going
to give you 18 degrees right
and now if you look at why
those are vertical so Y is going to be
72 degrees and now finally the next one
I notice is that X is vertical to Z plus
the 90 degrees so 90 plus 18
is 108 degrees okay so hopefully you can
see how we broke that down so try six
through nine on your own and then check
the key
so let's come down to using algebra
let's do a couple these together set up
a few and then you can try the rest on
your own and see how you do it's all
pretty much the same process recognizing
the angle relationships and using those
angle relationships in order to write
your equations from the expressions so
in ten let's see if we're gonna do this
one or set this one I'm looking at P Q T
versus s Q R those are vertical angles
so because of that I'm not going to do
this one with you but you can see that
they're vertical so you just have to set
them equal to each other in order to
solve because vertical angles are
congruent let's look at eleven it's
telling me that a B is perpendicular to
C D that's the first thing it's saying
so when I go over here can I find a B
and C D I'm getting a note
that it's 90 degrees okay and it's
telling me that DCE
gives me that value and then it gives me
ECB so this one
this angle Plus this angle are going to
be equal to 90 degrees solve for that
and then see what you get
twelve let's look at 12 help you break
these down
12 gives me angle K and let's see what
that K n m and M and J so we have M and
J and we want to find the measure of K M
on this well this angle Plus this angle
is equal to how many degrees that's a
linear pair so they're equal to 180
degrees because it's supplementary so I
just have to add those together and set
them equal to 180
now try 13 on your round it's a little
bit more of a challenge and then see how
you do let's see well 14 it's telling me
that they're complementary so if they're
complementary they're equal to 90
degrees
add them together set it equal to 90 15
are supplementary add together set equal
to 180 now these ones are the ones again
that people have the most trouble with
because it doesn't give you actual
numbers it's telling you something so
again I ask myself one end and a one or
two form a linear pair the measure of
angle 2 is 6 more than twice the measure
of angle 1 find the measure of angle 2
looking at number 16 I can't write first
thing I ask myself is what angle does it
tell me nothing about okay
well since the measure of angle 2 is 6
more than twice the measure of angle 1 5
measuring 2 it tells me nothing about
the measure of angle 1 the measure of
angle 1
that's your x-value okay so that's where
your starting and the measure of angle 2
it says it is 6 more than twice the
measure so I have 6 more than twice the
measure so twice the measure would be 2x
right and then if I have 6 more I'm
gonna add 6 to that and it's a linear
pair so I'm gonna add those together and
set them equal to 180
so in 17 let's look at 17 J and K are
complementary
which one does it tell me nothing about
it tells me absolutely nothing about the
measure of angle K so the measure of
angle K is your X and then the measure
of angle J is 18 months than the measure
of angle K and when something is 18 less
if I say I have 18 less than you you
don't do 18 minus whatever you have no
you take whatever you have and you
subtract 18 from it so X minus 18
they're complementary complements are
the right thing to do right angle so add
those together set them equal to 90 and
then solve you can check the key when
you're done so now these ones are using
angle bisectors so let's look at 18 UW
is bisecting so if this is bisecting
that it means that this angle is
congruent to this angle and let me make
that
right and if they're if they're
congruent then I just have to set them
equal to each other let's see if that's
what it's asking me it's giving me tea
eww and it's giving me w u V just set
those equal to each other and solve for
x
and let's see not too much different
over here right for 19 and 20 sews try
those on your own and then check the key
to see how you did
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