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Angles & Line Relationships | Vertical, Adjacent, Complementary & More!
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May 29, 2025
Section 11.1 ( Pre Algebra) In today’s lesson, we’re breaking down one of the most important geometry foundations you'll need for Algebra 1—Angle and Line Relationships! Whether you're reviewing or learning this for the first time, this is the perfect place to start. What You’ll Learn: What angles are and how they relate to one another Vertical Angles: What they are and why they're always congruent Adjacent Angles: What it means for angles to be “next to” each other Complementary Angles: Pairs that add up to 90° Supplementary Angles: Pairs that total 180° Angle relationships formed by intersecting lines and parallel lines Special terms like alternate interior angles, alternate exterior angles, and corresponding angles.
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so we need to know about angle and line
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relationships There's a lot of things
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that we need to know about angles right
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and what I want you to know first is
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this Uh angles can be classified by
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their relationship with one another
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right to each other They can be
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classified like that So the first thing
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that I want you to learn is this
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Whenever you see this sign right this
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little sign like this that
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means
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measure This mean measure right the sign
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the symbol stands for the word measure
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So if I put this here that means that
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the measure of one is equal to the
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measure of what two This is the measure
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of an angle Measure of the angle Okay
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When you see this is the measure
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of
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the angle that is being spoken of Right
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so now you see these two lines here
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Right these are called line that are
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intersecting Right so if you have two
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intersecting lines they form how many
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angles how many angles do you count
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there four right you have two
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intersecting lines and they form four
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angles We have 1 2 3 and four Okay So
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those two intersecting lines form four
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angles And actually we're going to be
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doing homework on this because I do want
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to still give some Oh my god So we have
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one two three and four right it's
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actually Wednesday so you can't
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Wednesday I know but I'll give it to you
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tomorrow So that's no one two three four
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guys So we have four angles Now there's
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something interesting about these angles
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right if you look at this one and two
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these are called vertical angles They're
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called what vertical angles Okay They
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meet at the side You see one and two are
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vertical angles And then three and four
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will also be called what
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vertical angles They're also called
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vertical angles Yes Why are they
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horizontal and angles now that's the
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term that we use vertical angles They're
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called vertical angles right angles
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These are called vertical angles You
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have two seeant line that meet they form
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at the summit vertical angles So these
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two are called vertical angles These are
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also called vertical angles So three and
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four are called vertical angles And the
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relationship between them is this They
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are
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congruent What does that mean when I say
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they are congruent they are the same So
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measure of one is equal to measure of
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two and measure of three is also equal
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to measure of four So those two
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excuse me it said measure of three is
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equal to four Yeah measure of four Thank
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you for realizing that Right Measure of
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three is equal to measure of four So
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those are called variable angles So in
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any given case variable angles are
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always going to be what they always
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going to be congruent They're always
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going to be the same So anytime you see
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two lines like this automatically know
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that this angle and this angle are going
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to be the same and this angle and this
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angle are going to be the same cuz they
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are called vertical angles right those
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are called vertical angles now the next
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uh type of angles we're going to talk
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about are obviously what adjacent angles
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right you know the word adjacent what
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does that mean no I don't you don't know
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who knows the word adjacent what does
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that mean
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English people Right you don't know why
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something is adjacent to something else
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I don't know is like
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connected next to right So these two
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angles how many angles do I have here
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one No Three I have three angles Right I
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have five six and I also have ABC Right
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I have five angle five I have angle six
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And I also have angle ABC which is the
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whole angle Five Right Which is the
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whole angle Is that four that's three
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angles This angle by itself is by itself
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Right five is by itself six is by itself
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And then the whole thing is by it's also
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part of it So I have three angles here
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Right these are called adjacent angles
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because they share the same vertex This
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is called a vertex A line a vertex here
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So these two angles five and six share
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the same vert vertex So they are called
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adjacent angles For example
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if I'm here right next to this table so
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I'm adjacent to the table Okay so we
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share the same line here this line share
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the same So I'm adjacent to this table
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So the same thing with these two angles
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five and six share the same vertex So
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they adjacent angle They are next to
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each other Now the the relationship here
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is this The measure of ABC which is the
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the the bigger angle is equal to the
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measure of five plus the measure of six
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That's self explanatory You can see that
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for yourself If I take this whole angle
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to find it I'm just going to add that
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and this five and six is going to give
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me the whole angle Are we good on that
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okay Now the next thing we're going to
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learn is complimentary angles
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Complimentary angles right so if you
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look here what do you see what kind of
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angle do we have here a right angle
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right a right angle So a right angle So
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these are called complimentary angle
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because if you add seven and eight what
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do you get 90° 90° If you add two angles
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and they add up to 90 those two angles
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are called complimentary angles because
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they add up to 90 Okay and then so
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that's pretty easy stuff It's just a
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language So you need to know vertical
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angles Vertical angles are these angles
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where you have top vertical angles These
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are called um adjacent angles because
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the sum here does not give you 90 Okay
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so these are adjacent angles And then
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here the sum gives you 90 These are
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called complimentary angles
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complimentary angles because they add up
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to 90 Now the next type of angle we have
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is this one here I have nine and 10 So
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if I join this together what do you
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think the whole thing is going to give
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me if I was to put this together right
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if I was to put this together here I put
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these two lines together what would you
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get between 9 and 10 180 180 Because
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this angle and this angle are going to
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give you 180 So if you have that what do
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you call this angles supplementary
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angles So the supplementary angles add
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up to 180 and the complimentary angles
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add up to 90 and the adjacent angles are
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angles that share the same vertex and
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then the vertical angles are the one
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that are congruent by way of the seeant
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lines that form them Yes So just to be
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clear adject angles are equal to what
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complimentary angles the sum of the two
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angles is 90 Okay Supplementary angle
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the sum of the two angles is 180 And
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what adjacent angles there is nothing
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You just know that this whole angle is
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this plus that they are less than 90
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right here They are less than 90
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They can also be less than 180 As long
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as it's not 180 or 90 they are called
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angles Um it's like it's a one's right
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angle and then that one's straight line
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Yeah Well that's not acute I can still
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have an adjacent angle like this right
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this is still adjacent It's five and six
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It's still more than 90 but it's just
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not 80 180 or 90 Okay Adjacent angles
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can be either less than 90 or more than
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90 As long as they are not 90 or 180
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they're not less than 90 and more than
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180 More than 90 Okay More than 90 is
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adjacent Less than 90 is still adjacent
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90 is complimentary And then 180 is
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supplementary Right okay Now I'll give
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you a quick example here So let's say
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you have this paper right and you want
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to make snowflakes And to make the
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snowflakes you want to cut out like
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triangular shapes right if I want to cut
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out this triangular shape out of it So
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if I cut this out right so if I cut this
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out I get that I get this here right i
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get this So now I cut out this this
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piece for me I get that So now the
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question is uh what is the relationship
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between y angle and x angle they're
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divorced What what's the relationship
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between them what would it be
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supplementary Supplementary Why is it
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supplementary
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because if you look here if they put
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together what is it going to give you
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180 180 Because measure of y right
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measure of y plus measure of x will give
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you 180 right right this is so that's
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going to be a supplementary because if
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you look here if I take this angle and I
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add it to this angle it's going to give
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me 180 So these are all supplementary
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angles These are going to be
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supplementary 180 right they're going to
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give you 180 because it's going to be a
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flat or a straight angle Okay Now the
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question is this If y if this one is 135
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how would you find x
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how would you do it subtract Subtract
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Right if this is 135 and we are trying
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to find x we just have to do what
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subtract from 180 So I'm going to take
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it out right i'm going to take it out of
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here And I'm going to find the measure
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of x to be equal to
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45° So x will be 45° Okay because this
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is 135 We don't know what this is but we
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know that the whole thing is 180 So if
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you take this out of that you get 135
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and that gives you 45° Yep Would the
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paper be 180 where are we getting the I
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know I understand the supplementary
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angle but I do not see as a picture
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where um you get the 180 the angles
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would be because now look so I cut this
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one out right so suppose I put it back
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together So I won't have this If I was
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put together I would just have that
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right and now how much is this oh okay I
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see 180 Right Well if I cut this out now
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I just have that If I cut this one out I
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just have this space here So now this is
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Okay I see this is this is uh x and this
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is y and y is 135 So the whole thing is
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180 So if you put it together it's 180
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If you take it apart this one is
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different What is 135 you just have to
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take it out of the 180 and that gives
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you 45 Okay All right So now the next
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set of angles we going to run uh we're
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going to learn that and that should be
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the end of the lesson It's not that hard
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is we're going to talk about
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uh this type Oh let me
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uh all right let me draw a transver line
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here right one two three four five six
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seven and eight Now tell me what you see
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here What do you observe what kind of
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line are these two line these two lines
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here Excuse me No these two lines here
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No just this two lines Parallel lines
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right these are parallel lines So that's
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important right because when you have a
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par line and you have this is called a
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transversal Transversal right he's going
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this way He slashes it So now how many
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angles do you count there eight Eight We
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have eight angles right we have eight
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angles So now we're going to learn some
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a new a new language here So we we're
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going to learn about
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alternate interior angle
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That's the first part that we're going
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to learn Alternate interior angles So
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we're going to first identify them Right
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It just looks like it's making two of
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the vertical lines Yeah we do have two
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vertical lines Well we are three and two
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are vertical angles One and four also
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vertical angles But I want to introduce
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some new angles These are called the
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alternate interior angles Right so
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alternate interior angles So if you look
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at it carefully right so those angles
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they're basically opposite of one
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another but they're the inside of the
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parallel lines Okay so if you look at
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this right which angles are going to be
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alternate but inside of the parallel
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lines tell me if you look inside of the
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par line four and what four and three
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four and no four and six alternate right
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four and six So measure of four is going
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to be equal to measure of six Right
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measure of four is going to be equal to
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measure of six These are called
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alternate interior angles because they
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are alternate because one is here one
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the other one is on the other side and
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they are inside that's why the word
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interior is used So it would also be
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three and five Three and five is also
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going to be alternate interior angle So
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three and five two and three also be two
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and three Two and three No why two is
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where is it inside or outside outside
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Outside So we call that one alternate
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what interior
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Exteriors right one and eight That's
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still uh exterior right so now we have
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the second type is alternate exterior
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angle Can I get hold on a second let me
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get to the drop I don't have any more
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left but I I'll see if I get you some
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Right Alternate exterior angle That
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would be what one and what one and eight
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One and eight Right One and eight
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because they are alternate angle and
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they are on the outside of the parallel
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lines So 1 and 8 what else two and seven
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Two and
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seven Two and seven So two and seven are
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also going to be what alternate exterior
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angles Does that make sense now the next
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one which is the last one is called
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corresponding angles Corresponding
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angles right they look the same
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Corresponding
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angles Corresponding angles Right So
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where do you think angles are
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corresponding to each other one and what
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one and two One and six right they're on
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the same side right so one and six are
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going to be corresponding angle This and
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this one are going to be corresponding
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angles What else keep thinking Why are
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they corresponding excuse me Why are
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they corresponding corresponding because
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they are um they share the same
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transversal line and they are on they
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form the same angle here See this angle
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and this angle are the same because
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these two are paral Okay So shouldn't
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couldn't it also be one and seven no one
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and seven is not going to be It's going
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to be what two and five
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Two and five Anything else three and
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seven Three and seven Seven And then
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what else four and eight Four and eight
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Right Four and eight will also be
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corresponding angles So really it's
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really easy stuff It's not that hard All
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right So tomorrow we're going to work on
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we're going to do a class work and I'm
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going to let you all work on that So
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that's pretty much this So today we
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learned about vertical angles adjacent
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angles complimentary angles
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supplementary angles and then we did a
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problem on this and then we also worked
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on alternate interior alternate exterior
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and then corresponding angles Language
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is really easy You just have to figure
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out what they are Right so tomorrow
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we're gonna do a class work on that and
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then that's we're gonna finish that and
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then maybe