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CCSS.Math:

let's say I have two intersecting line segments so let's call that line that's segment a B and then I have a segment and then I have segment C D 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 I mean 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 be a little bit closer so let's say then yellow let's say that we know let's say that we know that the measure of this angle right over here angle B edie let's say that we know that that measure is 70 degrees given that information what I want to do based only on what we know so far 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 a D 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 you see that angle B II D and angle CEB and angle CEB are adjacent they 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 CED is a straight angle straight angle so we know 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 you know we know that angle ve D angle B e D and angle CEB are supplementary which means they add up to or that their measures add up to 180 degrees supplement supplementary supplementary angles which means which means which tells us that the measure of angle be e d+ the measure of angle CEB and i keep writing measure here sometimes you'll just see people right angled B II D plus angle CEB is equal to 180 degrees now we already know the measure of angle B II D 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 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 we care about the measure of angle C e a and we can use the exact same logic that we used over here angle CEA and angle CEB they for 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 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 C e a is equal to 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 B II D sometimes they're called opposite angle well I've often called them opposite angles but the the more correct term for them is vertical angles and what we'll 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 c e a and angle B II D and angle B edy angle bed our vertical our vertical and you might say way they look like they're they're horizontal they're next to each other and the vertical really just means that they're cross 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 angle AED are also are also vertical and that might may 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 so are these horse these horizontally opposite angles are also called vertical angles so now we have one angle left to figure out angle AED angle AED and based on what I already told you vertical angles tend to be or they are always equal you this was probably we haven't proven that to ourselves yet so we can't just use that property to say that this is 110 degrees so 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 plus the measure of angle C e a must be equal to 180 degrees we know the measure of CEA is is 70 degrees we know it is 70 degrees so you subtract 270 from both sides you get the measure of angle a IDI 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 they're out saw outer sides form a straight angle you see they add up to 180 this one and that one add up to 180 this one in that one add up to 180 this one in 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're 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 and the next video will actually prove to ourselves using pretty much the exact same logic here but we'll just do it with generalised numbers we won't use 70 degrees to prove that vertical length that vertical angles that the measure of vertical angles are equal