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CCSS.Math:

let's say I have an angle ABC and it looks something like this so its vertex is going to be at B the vertex is that be maybe a sits right over here and C sits right over there and then also let's say that we have another angle called da B angle actually let me call it D be a DBA I want to have the vertex once again at B so let's say let's say it looks like this so this right over here is our point D that is our point D and let's say that we know that the measure of angle DBA let's say that we know that that is equal to 40 degrees so this angle right over here it's measure is equal to 40 degrees and let's say that we know that the measure of angle ABC is equal to 50 degrees it is equal to 50 degrees so there's a bunch of interesting things happening here the first interesting thing that you might realize is that both of these angles share a side if you view these as Rays they could be lines line segments or rays but if you view of MS rays they both share this Ray ba and when you have two angles like this that share the same side these are called adjacent angles because the word adjacent literally means next two adjacent these are adjacent the they are adjacent angles now there's something else that you might notice it's interesting here we know that the measure of angle DBA is 40 degrees and the measure of angle ABC is 50 degrees and you might be able to guess what the measure of angle DBC is measure of angle DBC if we drew a protractor over here I'm not gonna draw it'll make my drawing all messy but if well let me I'll draw it really fast if you had a protractor right over here clearly this is opening up to 50 degrees then this is going another 40 degrees so if you wanted to say what the measure of angle DBC is it would be the it would essentially be the sum of 40 degrees and 50 degrees and let me delete all of this stuff right here to keep things clean so the measure of angle DBC would be equal to 90 degrees and we already know that 90 degrees is a special angle this is a right angle this is a this is a right angle there's also a word for two angles whose sum add up to 90 degrees and that is complementary so we can also say we can also say that angles DBA and and angles ABC our complementary are complementary and that is because complementary and that is because their measures add up to 90 degrees so the measure of angle DBA plus the measure of angle ABC is equal to 90 degrees they form a right angle when you add them up and this is another point of terminology that kind of related to right angles when you fort when you have a right angle when a right angle is formed the two rays that form the right angle or the two lines that form that right angle or the two line segments that form that right angle are called perpendicular so because we know that measure of angle DBC is 90 degrees or that angle DBC is a right angle this tells us we know we know so this tells us that dB if I call them maybe I could say the line segment DB is perpendicular is perpendicular perpendicular to two line segment BC BC or we could even say we could say Ray BD ray BD is instead of using the word perpendicular user sometimes this symbol right here which really just shows two perpendicular lines BD is perpendicular to BC so all of these all of these are true statements here and these come out of the fact that the angle formed between DB and BC that is a 90 degree angle now we have other words when our two angles add up to other things so let's say for example I have one angle over here let's say have angle over here that is I'll just make up let's just call this angle so let me just let me put some letters here so we can specify so let's say this is X Y & Z and let's say that the measure of angle X Y Z is equal to is equal to 60 degrees and let's say that you have another you have another angle that looks like this you have another angle that looks like this and I'll call this I'll call this let's say maybe M m n o MN o and let's say that the measure of angle MN o is 120 degrees so if you were to add the two measures of these so let me write this down the measure of angle m and o plus the measure of angle plus the measure of angle of angle X Y Z is equal to this is going to be equal to 120 degrees plus 60 degrees which is equal to 180 degrees so if you add these two things up you essentially are able to go all halfway around the circle or you can go all you know throughout the entire half circle or a semicircle for a protractor and when you have two angles that add up to 180 degrees we call them supplementary and I know it's a little hard to remember sometimes 90 degrees is complementary they're just complementing each other and then if you go if you add up to 180 degrees you have supplementary you have supplementary angles supplementary angles and if you have two supplementary angles that are adjacent so that they share a common side so let me draw that over there over here so let's say you have one angle that looks like this and that you have another angle so let me put some letters here again and I'll start reusing letters so let's say that this is a b c and you have another angle that looks like this you have another angle that looks like this that looks like that and i already used to see that looks like this notice and and let's say once again let's say that this is 50 degrees and let's say that this right over here is 130 degrees clearly angle DBA plus angle a BC if you add them together you get 130 degrees plus 50 degrees which is 180 degrees so they are supplementary so let me write that down angle DBA DBA and angle ABC and angle ABC are supplementary they add up to 180 degrees but they are also they are also adjacent angles they are also they are also adjacent and because they're supplementary and they're adjacent if you if you look at the broader angle the angle used from the sides that they don't have in common if you look at angle DBC angle D BC this is going to be essentially a straight line which we can call a a straight which we can call a straight angle so I introduced you to a bunch of words here and now I think we have all of the tools we need all of the tools we need to start doing some interesting proofs and just to review here we talked about adjacent angles or I guess any angles that add up to 90 degrees are are considered to be complementary this is adding up to 90 degrees if they happen to be adjacent then the two outside sides will form a right angle when an it when you have a right angle the two sides of a right angle are considered to be perpendicular and then if you have two angles that add up to 180 degrees they're considered supplementary and then if they happen to be adjacent they will form a straight angle or another way if you said if you have a straight angle and you have you have one of the angles the other angle is going to be supplementary to it they're going to add up to 180 degrees so I'll leave you there