11 5 Parallel Lines and Triangles

hello again guys we're back for 115 discussing parallel lines and triangles we start here with postulate that we've seen before it's our parallel postulate it tells us that through point not on line so our line here is line through point our Point there's only one line parallel to the given line if we skew this at all it will become not parallel so this has to be perfectly the same direction as our other line and we're actually going to use this postulate to work forward then from there dealing with triangles and it might confuse you how parallel lines relate to triangles but you will see here in just one moment so this theorem right here may seem little simple the triangle angle Sum Theorem states that the three angles of triangle will always sum to 180° but some people still don't quite understand why and there's neat little trick with piece of paper where you would make triangle and piece of paper and then cut it across one way cut it across the other way and then match up and now there's another way that we can actually do that and we're going to prove why this theorem is true all right so you guys have this proof solved out in your notes but would love if you would to take sheet of paper or something cover up the statements and the reasons and try to think through it on your own before uncovering it so the first thing that always happens is we look at our given and we look at what we're trying to prove so our given is we have triangle BC that's it then have angles 1 2 and 3 Now 1 and three are exterior angles to the triangle but I'm trying to show that they all equal 180° when added up together you should be able to start to think about well we have straight line here PR and these three angles are the only three angles on that straight line so we drew PR through parallel to AC now parallel to AC is very important here for how one and three are going to work in my triangle coming up later so then what we can see because of that straight line is if combine angles one and two and just look at the angle PBC so I've kind of highlighted that angle there and angle three we can make that supplementary angle pair because angles that form linear pair are supplementary now of course we have three but you can always combine angles to look at them as there were fewer now because of the definition of supplementary angles what it means to be supplementary is that they add up to 180° so we can go ahead and write it out like this because because we had originally combined angles one and two we can now use the angle addition postulate to restate angle PBC as being the addition of angle one and angle two which then because up here used PBC can use that same statement but instead replace angle one and angle two in for angle PBC and add that with angle three to equal 180 and all we did was substitute moving on angle one is going to be equal to to angle and angle 3 is going to be equal to angle now if we come back up here and look at this for second this might be confusing to begin with but if we extend couple of these lines so I'm going to extend this that way and we have parallel line up top and what we have here is our transversal so angle is actually the angle that sits on the inside right here and then we're comparing it to angle one now why would these be the same because they are alternate in interior angles we know that alternate interior angles of parallel lines are always congruent so we can say that one is equal to and three likewise would be equal to angle which would be right down in here would be your angle on the inside and again that is because if lines are parallel then alternate interior angles are congruent and the last no second to last step if angle one is equal to and angle 3 is equal to actually now we're just stating the measure of the angles so remember that little and the angle symbol means the measure of the angle so congruent angles have congruent measures and then just by using the substitution property we can go ahead and plug in that angle plus angle 2 plus angle will equal 180 we've got here there and the angle two is really like our angle doesn't matter we can just call it two so there we have it we've proven that in any triangle our three angles will add add up to 180 and that'll work no matter what as long as you have base line here and you draw another line parallel to that and work through the proof just like this now be ready to do that on the on test if ask you about that but because we know this theorem to be true we can now use this theorem to always find the third angle of triangle now hopefully you guys have already had some practice doing that but if not that is why we're allowed to do that so now go ahead and apply that theorem to find your missing angles here in example one and example two come back and get the answer so then by applying that theorem we can set up that my two angles that are am given and my third angle will then equal 180° so then add up my like terms and still have angle one can subtract from both sides and then find that the measure of angle one is 30° likewise over here we can do the same thing but hopefully for you it's easy to see that have an angle of 70 and an angle of 30 so if were to sum those together that would give me 100° which leaves my third angle here having to equal measure of 80° and because and share vertical relationship would also be 80° so now we pause for just moment to talk about the exterior angles in polygons and more specifically triangles because that's what we're dealing with here so we know that all three angles in triangle add up to 180 so this purple the missing angle and angle two and angle three would add up to 180 but what we also know is that because this purple angle and angle one form linear pair they would add up to 180 so what we see is that and let's talk about some how we name things really quick so remote interior angles pause for second to read through this make sure this makes sense so now I've got some color coding here our exterior Ang Le has then two remote interior angles that correspond with it and the relationship here because they share the purple angle the purple angle plus one would be 180 and or the purple angle plus 2 and three would be 180 that this angle the extra angle is going to equal the two remote interior angles the sum of those remote interior angles we are going to look at that theorem here next so here's our theorem that nicely States what was just trying to get at we we have angle one is our exterior angle angles two and three are remote interior angles and angle one will equal the sum of our remote interior angles so we have example five here and now we're not quite there in your notes so let's go ahead I'm going to skip ahead to examples three and four and then we'll come back and look at example five so in three and four you want to figure out what the measure of the exterior angle here is and then the measure of this remote interior so go ahead apply theorem 17 write out the math that fits this and solve for angle one and two so pause the video now so here all we are applying is our triangle exterior angle theorem so we know that the sum of the two remote interior angles will equal the third so we can set it up just like that and what we see is when add up those two angle one is going to equal 98° then we come over and apply the triangle exterior angle theorem again but here this is going to set up little differently because have my exterior angle and then only one of my Interiors so can set it up with my exterior angle and one of my Interiors added with my unknown interior then would just subtract 59 from both sides getting that measure of angle 2 would equal 65° so now we're going to wrap up this lesson by looking at our example five here we're going to say which of the numbered angles are exterior angles now careful some people will really quickly say okay four 5 6 7even and eight because they're all outside the triangle that's not true remember that an exterior angle must have corresponding set of remote interior angles so when you're looking for an exterior angle you need to be able to match it up with its two remote interior angles so pause the video try to answer all three of these questions I'm going to come back and answer them one at time all right my exterior angles then would be angles five six and 8 if look at angle five its remote Interiors would be 1 and three if look at angle six the remote Interiors would be 1 and two and if look at angle 8 the remote Interiors are angle one and angle two also remember the remote Interiors are always the angles that are not right next to it so you would go ahead and have those listed out here in that five matches one and three 6 one and two and eight matches one and two also and again to restate 6 and 8 have the same ones because they share the interior angle not remote interior but they share the interior angle right next to them so that makes their remote interior angles both one and two which means that angle six and angle 8 would have the same measure which makes sense because they are vertical angles so in case was moving too quick for you there there it is all typed out so you can see the exterior and the remote interior angles that matches it and that is all have for today's lesson for 115 so please make sure you've worked through all the Pearson work you've done your practice work and you come to class and let me know if you have anything at all that is bugging you about this material we are actually going to skip right through 116 we're going to cover that in class so your next lesson would actually be to move on to chapter 12 so again please let me know if there's anything can do for you other than that will see you guys in class have great day