good day everyone and once again we are back together uh just looking at uh euclidean i've been waiting i have to just start with this section so we're starting with euclidean geometry this time around uh so if you're new to the channel please just click that subscribe button and just make sure that you're part of the family all right and um uh obviously for some of you who need assistance with mathematics or physical science uh please just consider just sending us an email and our email address is info at lumisengosi.co.za [Music] right so i'm just going to simply start here but just please remember when it comes to euclidean geometry you know many people just think uh you know it's all about just uh um you know circles and you know lines and all of that remember that it it starts all the way back from great i think grade 8 if i remember in fact even before that when you're talking about acute angles and so on it might even be great grade 7 you know it builds up so the the problem is that many people some tend to segment it you know into this and compartmentalize it in a sense but what i want you to do is and this is going to be our approach we're just going to start and build up our lesson okay just from the grade you know eight stuff and then just build it up so that it becomes one full body of knowledge okay so i want you to stay with me and another thing is that for you to be any good at euclidean geometry means that you have to practice the only way that you can defeat this monster is if you try to do at least one geometric rider a day and uh that usually just helps with you you know just perfecting the skill um you know of knowing how to answer questions and so on so without any further ado okay i'm just gonna start today and um what i want us to uh quickly go back to is to look at um you know just a couple of things just as a form of revision all right now let me start with a straight line so what we know from geometry is that the sum of angles on a straight line so if i take any particular straight line all right and i try to find any angle or rather the angles around that particular straight line so say for argument's sake we tell you that we've got angle x you've got angle 60 over there and um yeah and you've got angle whatever that that else is right so in this case uh let's say this is going to be 40 okay and so they say to you find angle x all right so we would know in this case we know that the sum of angles on a straight line are equal to 180 so in this case i would simply say x plus 40 plus 60 is equal to 180 and what is the reason around that okay so i would simply say these are angles on a straight line okay the sum of angles on a straight line a straight line are equal to 180 and so what would i do to get the value of x all i simply do uh 60 plus uh 40 that would give me a hundred if i take that to the other side it becomes negative so that's 180 minus a hundred and so x would be 80 degrees okay so this would be the sum of angles around a straight line so just remember whenever we've got angles on a straight line they're equal to 180 of course we can apply this in so many other different ways okay right now what else do we know we know also that the sum now let's go to triangles this time around okay so when you're looking at triangles just to remind ourselves we know that the sum of angles on a triangle are equal to 180 so if i've got triangle abc all right and i know that uh you know i've got uh angles there let's say just for argument's sake this is 65 degrees uh this is going to be uh let's just say just for argument's sake that's going to be 45 degrees and so what would be that angle over there right so all you just simply do uh if we wanted to know what the angle or angle c is we know once again we'll say angle x plus 65 plus 45 this is equal to 180 okay but again what why is that what's the reason behind that these would be angles on a triangle okay you can say sum of angles on a triangle okay any examiner would know when you are talking about angles on a triangle of course their sum is equal to 180 of course we're going to do exactly the same thing you know just find the angle x in this case we're going to say well 65 plus 45 i think that would give us a 110 okay right so that would be one one zero um in fact uh you see now i'm making a mistake there so that would be x plus a hundred and ten which would be a hundred and eighty and of course what would be our value for x if i take this to the other side that would simply be 70 degrees okay right so in this case just keep in mind that every time that we're looking at angles on a straight line they're equal to 180 but what else do we know about a straight line we know that the exterior angle on a straight line now please please please you know we talk about this in this case remember that all of these things find application in what we're going to do when it comes to you know euclidean geometry it is part of euclidean geometry you're supposed to know it okay so suppose i've got uh again let's say angle pqr and in this case uh let's just say we extend line pr to s so that would mean that's a straight line okay now what you need to keep in mind is that the sum of your interior angles of a triangle now for instance if i know that that is angle uh now literally let's keep it to angle p and angle q right so in this case angle q r s angle q r s which means we are looking at this angle over here i'm just going to make it in yellow so this angle over here which is an exterior angle so that angle qrs okay is equal to the sum of the angles of the interior opposite interior angles of the triangle now this means qrs is equal to both angle p and angle q the sum of them so this will be angle p plus angle q if you don't mind i'm just calling them okay by their points in this case so uh a qrs that's qrs not ors qrs is equals to the sum of the interior opposite angles so in this case that's what we know about triangles okay and remember we can say it's the exterior angle of a triangle all right any examiner would know that right whenever you've got an angle outside the triangle it would be equal to the sum of the interior opposite angles in that case all right now what i want us to quickly move on to so we know some angles on a straight line we know the exterior angles of a straight line i mean of a um triangle sum of angles on a triangle we know obviously uh as far as the exterior angles of a triangle as well as the interior angles of a triangle now what i want us to quickly move on to our you know parallel lines and obviously i'm going to come back to triangles okay but i want us to quickly look at parallel lines all right now having a look at parallel lines suppose i draw two parallel lines there okay and of course we'll have a transversal over there let's say our lines are a b and c d okay let's say our transversal is e ef now just keep in mind in this case there are a couple of things that i would know about parallel lines okay so we always kind of indicate that the lines are parallel there okay so that's how we indicate that the lines are parallel now what do we know about um parallel lines so first of all we've got what we call um co interior angles in fact you know what let's let's talk about a couple of things especially also uh with a straight line uh as well as parallel lines so first of all if i take um yeah so we call that ef in fact let's call that g and h i think so that it becomes easier for us to mention the angles so what are the things that we know we know that first of all we've got what we call vertically opposite sides now if i look at side f h b okay or angle rather fhb all right which is the one the ones that have a of sort of marked in the yellow so that fhb there it would be equal to angle a ahg so a h g all right so which is that angle over there but why would those angles be equal we say these are vertically opposite angles so keep that in mind so they are vertically opposite okay so they're vertically opposite angles so in this case we just simply say vertically opposite angles okay right of course we would have uh had to express that a b is parallel to to ct which is given okay right what else do we know of course that would automatically mean that the other two angles which is this one here and this angle so when i want you to look take a look at that those two green angles are also equal why in this case so that's angle a f a h f and b h g okay so those would be n uh equal again y again uh because they are vertically opposite angles all right so in that case uh that really helps us now which other angles are equal i want you to note so i'm gonna take the one that's in the mustard color this one g right but we say it is also equal to this angle over here so they kind of make a z um sort of uh shape okay so it's equal to this angle over here so this angle is equal to that angle so what do we call those angles now we say well in this case we know that a f a h g angle hg would be equal to angle now i think it's confusing if i'm writing it in green because it seems like i'm talking about those ones let me not you know confuse you in this case let me just keep it to that mustard color right so i'm going to say a h g right so i've got my angle ahg okay just gonna keep it to the white that's fine ahg that would be equal to angle um hgd so that's h g d okay and why is that because we've got two lines that are parallel in this case so then we call them alternating angles so we say these are alternating angles why because we've got line a b parallel to line cd now remember every time that you are doing an uh euclidean geometry you must provide a reason why you are saying angles are equal or whatever the case may be right so and i want you to keep in mind we are building up okay so we're making sure that you've got the full body of knowledge that you need because remember eukaryan geometry is just one body okay of knowledge it's just that obviously we learn the different parts of it you know in the different grades but it's all one thing okay right so in this case we know that we've got alternating angles okay so we've indicated them and then maybe i should have taken a different color okay so this angle here is kind of equal to that angle there so we call them alternating angles right and then we've got what we call corresponding angles now please i want you to note i'm going to indicate that um yeah let's take a a much different color in this case let's indicate that with the purple okay so in this case i'm going to say well look i have got that angle there okay equal to this angle over here now look at this what's happening okay i've got my parallel lines over there okay so this side is equal to that side okay that angle and in this case we say that those are corresponding angles of course similarly we can say this angle over here should be equal to this angle over there okay right i don't want to include a lot of them okay so in this case we know angle f h b once again so f h b should be equal to angle um h g d h g d all right and in this case what should be the reason behind that why are they equal we call them corresponding angles so again so these are corresponding angles and why is that because a b is parallel to c d okay so we've got uh when it comes to lines that are parallel we've got alternating angles we've got corresponding angles and finally what we have is what we call co interior angles i think i wanted to start with that one um but nonetheless it's it's it's okay right so let's take another color all right so i'm going to take the ultra yellow color so in this case if i have for instance within that sort of you know inside those parallel lines that's why we call them co-interior because you've got angles that are inside or between within the parallel lines so i'm going to take this angle here so let me not draw them in the same color because then i might see i might actually create an impression that they're equal so if i take that yellow angle and this green one over here so those two angles they are within the uh you know the parallel lines so in this case what does that tell me it says if i take this angle here maybe it would be better to label them differently so it means that a h g so if i take angle a h g uh sorry yeah i think it's a h e e g yeah you can call it g so a hg right should be a rather plus angle as c g h so meaning that green angle there from c to g to h so it's that angle over there so if i take that angle plus this angle over here and both of them give me 180 so plus cgh they're both equal to 180 and in this case we simply call them co interior angles all right just keep that in mind of course we know a b is parallel to cd in that case because we are given the fact that they are parallel so in this case we've got co-interior angles so keep that in mind so we've talked about vertically opposite angles we've talked about alternating angles you know corresponding angles as well as what you call co interior angles all right so whenever you've got straight line uh i think they usually use this uh you know sort of f-u-n uh sort of principle okay i remember we say these ones okay so let me just indicate those ones in the uh yellow so this angle and that angle if those are parallel lines remember we said these are corresponding angles okay corresponding angles and then we said oh but we remember this guy here and that guy are inside within the parallel lines okay so we're having fun so in this case these are called co interior angles okay they're called operating with each other to make 180 and then obviously we've got this angle here which is sort of our set okay all right this angle here equal to that angle in that case we call these uh alternating angles so these are alternating angles all right so this is as far as um what you're going to have in parallel lines all right um you know just uh maybe just something to to also add there always just keep in mind of course there's a relationship between straight lines see if that line is a straight line there remember that the angle between this angle here plus that angle also are equal to 180 and it talks to what we've already spoken about that's angles on a straight line okay right so i think what i want to do right now is just to look at just a couple of more things when it comes to just revision okay we're going to go back to triangles because i want us to be able to prove um you know what when when angles are similar and when they're congruent okay right we'll just look at that in just a few all right so we've got three types of triangles so we're talking about triangles now and we're just going to lead up to congruency and similarity but first of all before i start with that remember that we've got three types of triangles okay so we've got what we call a scalene triangle okay so scaling triangle uh essentially none of the sides are equal okay but we do know for any type of triangle uh the sum of angles on a triangle would be equal to 180 so this would be a scalene triangle but we also have what we call a an isosceles triangle okay so let me try that one we my triangle is el terrible so in this case if i were to consider now at um and i saw silly's triangle an isosceles triangle is a triangle which uh two sides are actually equal so if i tell you that those two sides are equal all right so that makes it isosceles so only two sides are equal so meaning the third one would not be equal to any of the two sides so in that case this we call an isosceles triangle now please i want you to note in an isosceles triangle we always say that the base angles of an isosceles triangle are equal so if i've got those two lines now it makes though the the angles at the bottom [Music] of the two lines to be equal so we say the base angles of an isosceles triangle so just remember that so base angles of an isosceles triangle triangle are equal okay so we need to keep that in mind but the converse is true meaning if the base angles would be equal then automatically it makes the lines the sides also equal so if this is a b c right so if angle a is equals to angle c then in that case it would simply mean that side a b is equal to side uh bc now just note that it's the sides that are opposite those angles that are equal right uh each of those and that is opposite each of those angles right and then we've got another triangle that we call an equilateral triangle so if i've got a triangle of course i'm not going to make my try my diagrams uh quite accurate okay but just for you to get the you know just the gist of what i'm trying to explain now this we call an equilateral triangle uh right equilateral triangle so in an equilateral triangle all it simply means is that all the signs are equal automatically when all the sides are equal then it makes the angles equal as well now if you think about it if the angles are equal and the sum of them is 180 then it means how do you get the size of each angle it's 180 divided by three because we've got three angles so as a result that would be 60 degrees so in this case you know this would be 60 60 and 60. so we call that an equilateral triangle we know that all sides are equal and as a result all angles are equal okay right so that is in as far as triangles are consent now let's talk about congruency okay now first of all what do we mean when we say that triangles are congruent okay what we simply mean uh by congruent is that um if i take two triangles and i put them exactly on top of each other you know they would be exactly equal it means one each of the sides would be equal to the side of the other and each of the angles of this of the one would be equal to another so it means that that lie exactly on top of each other as though it's just one thing in a sense now there are three ways of proving congruency okay and the first way of proving congruency is if suppose i've got two triangles and i've got triangle abc and i've got another triangle uh d e f okay suppose i'm given these two triangles and they tell me that the sides are equal let's say i've got side a b equal to signed d e and let's say side b c is equal to e f okay let's make that a double line and i've got the dead side ac equal to the f now i've got three sides equal here so what can i therefore say right we say well therefore because i've got side a b equal to d e all right which is given so that's given to me and i've got side bc equal to ef that's also given um put that in brackets whenever you give a reason and then i've got side ac equal to side df which is also given now in this case what can i conclude because those three sides are equal it means therefore now you need to always be careful um you know when you know just writing down the vertices okay meaning the different sides of your triangle the vertices of your triangle so in this case it means that triangle therefore triangle a b c now notice i've started with the uh i should have said triangle abc so i've started with this vertex over here okay so it's vertex when it's one but it's vertices when it's uh plural okay so triangle abc should therefore be congruent now the sign that we use for congruent are the three lines you think of it as equal okay just with an extra line okay so in this case um a triangle abc is congruent to triangle now note when i name the other one it should always be in the same order of lines so i should say d e f now please be very careful about this one because we're going to use it when we are talking about proportionality okay and it it always makes a huge difference how you order that around okay right so the um abc's try is congruent to triangle def but why is that we simply say because side side side okay so side side side so that means that three sides are equal and so that makes those triangles um also congruent all right now let's take another triangle let's say you've got another set of triangles okay so that's the reason for one reason for congruency okay so let's say i've got triangle p q r okay and another triangle s t u okay so there are my two triangles again now another reason for congruency is if i have got two angles and one side so suppose they gave me two angles all right so uh let me just make two angles in one side so say they've got that angle equal to that and i've got that yellow one equal to that yellow one and suppose they would tell me one of the sides is equal okay so let's say that side is equal to that side so again um in this case this is another reason for congruency when i've got two angles and one side equal so in this case we know that triangle okay let's start with the reasons so i'll say angle p is equal to angle s okay and that's given and then i've got angle q equal to angle t and by the way they may not necessarily be they they may not correspond like that okay so you can have the sides um you know on different parts okay so for instance uh this angle here could have been equal to that one there but the main thing is that you've got angles equal okay so angle q is equals to angle t and again it's given and you've got side p r equal to s u and that's also given okay um in this particular case so now what can we conclude therefore it means that now again if i decide to start at p it means i'm gonna start at s for this one okay so the different vertices which are the vertices that are equal of course it's p is equal to s if i started at u for this uh for if i if i started r for this one means i must start at u for this one okay so uh in this case it's going to be triangle q p r notice i started at the top went to this one and went to this one so and triangle i keep forgetting to write a triangle there is congruent to triangle in this case it's going to be tsu so that's going to be tsu okay and what is the reason of course this now is going to be angle angle side okay so that tells us that uh you know the triangle is congruent um i want to just simply say you know uh later on i'm going to be talking about similarity so just remember that all congruent triangles are similar so by virtue of being congruent they are automatically similar okay right now let's go to the third one okay and the third reason for triangles to be congruent in this case suppose i've got my triangles again all right sure i don't want this video to be too long or to this lesson to be too long but it is necessary for us to go through this right so suppose i've got again uh let's say we've got d e f okay and we've got another triangle pqr there's our triangle that the two triangles there okay so i've got just say for argument's sake now we've got another reason for congruency we say well this is side angle side now this is very important because it has to be the two sides and the angle in between them okay remember when we talked about two sides and an included angle so in this case if i give you this side d e and say well it's equal to that side in fact you know what let's let's bring a bit of a twist to this okay so if i say uh i've got this side here equal to that side and i say this side is equal to that okay yeah i'm just bringing it you know so that it's a little bit different so now which angle should be equal it must be the angle in between okay so the angle in between now say we were told that d e is equal to uh so we said the e is equals to p q okay uh so in this case remember that was given to us so that's given and then we've got side d f equal to q r this time okay again it was given all right and the angle in between them and the two sides angle d in this particular case is equal to angle q and again that's given all right so what can we conclude in this case right now please i want you to note so i'm going to start so triangle therefore triangle let's say d e f d e f okay d e f would be congruent now note i started at the vertex with the equal angle all right and then i went for the side with the sort of mustard color okay so in this case i'm i have to start side with the equal angle so that would be q and then i went to the mustard side so this would be q p r okay whereas here it was d uh what was it d e f right d f so d f would be equal to i mean uh congruent to q p r and please note we always put it like this we say side angle side and nothing wrong with writing angle as an a right just to show that it is the angle that is equal so that's our third reason for congruency and then the last one uh just to quickly talk about that one this is when we've got a 90 degree triangle okay so suppose we've got uh here are 90 degree triangles okay sorry that my triangles don't look triangular okay so suppose we've got those 90 degree sides okay so let's call this a b c let's call this d e f okay right so now what would be the third reason okay so the third reason would be if i have one side so let's say this side equal to that side but also my hypotenuse so let me make that a different color all right my hypotenuse equal but in this case i also have a 90 degree now i've made it very deliberate that i didn't put it i didn't put the 90 degree in between any of the sides so it could have been any of the two sides okay if it was for argument's sake this side over here equal to that one it doesn't necessarily need uh for the 90 degree to be in between uh here is an exception to the one where we've got uh um side angle side it could be any of the two sides but if we have a 90 degree in this case then we know that it is congruent so in this case what did we have we had side a ac equal to d f that's given okay and i have the hypotenuse bc equal to ef and again that's given and now i also know that angle a is equal to angle d okay this is a right angle try i mean this is a a 90 degree triangle right so i can say that's a right angle okay so then what can i say again you make sure that uh you know how you label that so what it means is that therefore triangle let's say b a c okay so triangle bac would be congruent to triangle e d f e d f and what would be our reason again in this case it would be right angle hypotenuse side okay right so that is as far as the 90 degree is concerned now ladies and gents remember i said all congruent triangles are similar right but we do have similar triangles that are not necessarily congruent so what do we mean when we say triangles are similar you can get a triangle let's say i take those two triangles there right definitely you can see that in terms of size they are not equal but they may be sort of similar in terms of scale uh i mean in in terms of you know um if if you take in proportion to each side okay right so in this case suppose i've got those triangles a b c and i've got d e f over this side okay right now if i i am given that three angles are equal so suppose we've got angle a equal to that angle there angle b equal to that angle there and of course this one automatically and by the way just please note if two angles are equal it automatically makes the third angles to also be equal okay right so if we've got three angles that are equal it does not mean congruency this time it means similarity and the sign that we use is going to be different so in this case well they told us angle a is equal to nld angle b should be equal to angle e and angle c is equal to angle f all of those are given okay okay so all of those would be given now in that case what do i say i say well triangle so that's what it means triangle a bc now note so that would be similar now for similarity what you use are lines like that okay remember we said uh if they are congruent we use sort of the horizontal lines and in this case it's going to be the vertical lines so this abc is similar to triangle d e f yeah that's d e f okay right i'm going to come back to this because we're going to use it especially when it comes to grade 12 trick i mean geometry because you know obviously that one uses proportionality quite a lot and of course it it's it is quite important okay so the last one when we've got we did say that you know when lines are similar or rather when triangles are similar they are also not when they're congruent they're similar but not the other way around so this triangle the last one that we did um it might be similar okay but in this case it's not congruent so when i take triangle df and place it on abc they might not not necessarily sit on top of each other okay right i want to kind of reach towards the conclusion uh but just the last concluding remarks right so the last thing that i just want us to remind ourselves of uh is if given a 90 degree triangle okay just remember that we can always apply the theorem of pythagoras so if we say that's triangle abc please remember that we can apply a theorem of pythagoras and of course the square of the hypotenuse so in this case it would be bc squared would simply be the square of the hypotenuse would be the sum of the squares of the other two sides so this would be a b squared plus ac squared so just keep that in mind but another thing that i just wanted to sort of mention if we already know that this angle here is um 90 now suppose this angle was x okay so what would this angle be all right so those things are quite uh useful right so what would be the size of this angle just keep in mind so angle c in that case would simply be 90 minus x y because between angle b and c you now have 90. so if you wanna you know you really want to be pedantic about it so a plus b plus c uh all those angles are equal to 180 okay so in this case you know our a value is 90 our b value is x and so we're looking for angle c right so if i take this to the other side of course angle c would simply be equal to 180 minus 90 minus x so what you end up with it's 90 minus x just keep it in mind it's something that really usually comes in very handy all right when it comes to uh you know triangles and when we are doing dealing with triangles and so on okay ladies and gents i want to leave this lesson here i know and some of you are going to be complaining and thinking ah but you know we're looking for geometric riders and here's this guy you know starting with all of these other mushy mushy stuff listen at the end of the day remember as i did say to you that um euclidean geometry is one body of knowledge it's unfortunate that you know you have to sort of teach it in stages but remember i'm giving you this so that it becomes the basis of what we build on okay so next time i will be looking at uh quadrilaterals okay so four-sided figures and obviously proving different sides of uh quadrilaterals and in that case we'll then move on gradually to cyclic quads and then obviously will get into circle geometry and all its other nitty gritties okay right otherwise uh from me for now i'll see you guys next time please don't forget to hit that subscribe button hit that notification bell of course tell as many people as you can you know about this channel and i'll i promise you by the time that we are done okay you will be a champion when it comes to youtube euclidean geometry i'll see you guys next time