 Hello everyone welcome again to another session on gems of geometry in the previous few series of lectures you saw different types of properties of different geometrical entities like triangles and circles so if you remember we started with the sign law in triangles followed by steward theorem Siva's theorem and multiple other theorems related to triangles we studied also we saw things like Simpson's line and you know we which talked about collinearity of points and other things right we also discussed various properties of oddity woods medians and angle bisectors perpendicular side bisectors of triangles carrying on with that you know we are going to start new section now where we are going to discuss more on collinearity and concurrence of lines and visa v let's say quadrangles and other geometrical polygons right so that's what the agenda of this video as well as the subsequent videos are is that we are going to discuss as I told you collinearity conference and we are also going to discuss properties of quadrangles now and also various properties related to the diagonals and multiple other things we'll start with a small introduction on some you know some definitions and some nomenclature has to be understood before we actually jump on to studying all those theorems because all those terms will require some kind of a proof and those proofs will require some kind of prerequisite understanding of some definitions and some notations so that's what the objective of this session is so today we are just going to discuss and this session particularly only about the various types of quadrangles how do we define them what are different types and you know how do we classify them and then what is meant by area of a polygon so as if if you remember your any polygon n-sided polygon in a in geometry we have what inverteces and with inverteces there are n sides as well right so any n-sided polygon which is called n-gon so we have learned that n-gon is nothing but a polygon with n sides and whenever there are n sides there are n vertices and so hence n n-gon is a polygon closed figure with n vertices and n sides and we have studied about triangles we have studied about quadrilaterals we have been calling these as quadrilaterals but in case of triangles we call them triangle we don't call them trilateral in common day-to-day this things why are quadrilaterals called quadrilaterals so we actually should use the word quadrangles in quadrilaterals for quadrilaterals as well because later on when you take up a course on projective geometry you'll come to know that there we don't talk about laterals we don't just talk about lines and hence wherever the lines are intersecting they form an angle and hence wherever we get four angles in a closed figure we call it quadrangle is it so that's what is the you know understanding is so we'll be calling them quadrangle and not quadrilateral for this session at times because of my own let's say you know the way we have been taught during our school days and all that so we have been calling it quadrilateral that's there's no wrong or nothing is wrong about it but yes now calling it quadrangle would be more appropriate so that's what we are going to do and we'll start with understanding of basic quadrangles and their varieties so let me you know introduce you to different types of quadrangles which you have already seen so let me just turn these axis of you don't require them so let me switch it off okay now let's talk about a polygon and specially we are going to talk about quadrangles right so here is a quadrangle so if you see I have drawn up I'm sorry let me just redo it okay so here is I'm going to do a raw polygon this is the polygon and yes so ABCD is a polygon and you have you notice that all the angles here if you see are less than what less than 90 degrees or less than 180 degrees all of them if you see now of them are reflex at max they can be obtuse but none of them are reflex and this is called a convex polygon right so what is this called this is called a convex polygon you already know this call X in this case quadrangle quadrangle fair enough right this is what we have learned and one more thing if you notice if I just turn these off and let me just you know join the diagonals so if I join the diagonal with segment KC so AC is a diagonal BD is also a diagonal and what do we observe both the diagonals are within the polygon is it it so hence if you see or the quadrangle both both diagonals writing both diagonals are within are within the quadrilateral is it so let me write only quadrilateral or quadrangle right now this is a one variety another variety could be if I just again switch it off and show you another polygon so let me draw another polygon so here is the polygon and let me draw it here so point this and let's say this one EF and let's say G and H and again back to E so what difference do you see guys so if you see yeah so if you notice what is what is the difference between these two so one angle here clearly is sorry these angles have been misplaced from their positions never mind so this is the another variety of quadrangle and here this angle H angle H it is reflex angle right it's greater than 180 degrees within the polygon or within the quadrangle the angle is greater than 180 degrees this type of quadrangle is called a reentrant reentrant quadrangle okay quadrangle and what is the observation here if I turn this off and join the let's say join your diagonal so if you see out of the two diagonals one is within the quadrilateral or quadrangle right so one one diagonal one diagonal is out is outside is outside or not contained in the in the quadrangle outside the outside the quadrangle right this is another observation now let me draw one more type okay and how about this that I am taking one point here I let's say and this is J and let's say K and L and back can you see this is another type of kind of twisted you know you have just taken a quadrangle and given it to twist around one of the diagonals like that and it is appearing to be like that isn't it so this is called if you see this one is called crossed quadrangle crossed as the name suggests name is pretty much appropriate crossed quadrangle okay crossed quadrangle and again the specialty about this is let me draw the diagonals here so if I draw the diagonals J to L and I to K if you see nothing you know very surprising both are both diagonals are outside both diagonal are outside outside the quadrangle correct so this is the third type of quadrangle we are going to study yep and lots of properties related to this crossed polygons are going to be there later on we are going to see all of that now another point I would like to make here is related to the concept of area of a polymer okay so guys for that matter I have not discussed what is the vertex what is the side and all that you already know all of this and you have learned previously what's side vertex and all that adjacent sides opposite sides I have not covered those things here you know it already I'm assuming now we are going to talk about something about area right so here there are two things we need to understand one is there is concept of positive and negative in case of areas as well as far as geometry is concerned just like vectors directed line segments we have positive line segment and negative line segment in Milano's theorem we'll see that we do talk about negative line segments right so in so we generally try to maintain the sense of it right so in the order in which the points are encountered by you start traveling in particular line so hence we call it directed line segment so if you go in one direction and come back these two segments are negative or opposite in sign okay so similarly in areas also we say that if we are tracing the points in anti-clockwise so as I told you there are two points one now we have positive and negative areas as well okay and second is the way it is done or how do we decide which is positive and what is negative so clockwise direction of points if you trace in a polygon let's say area is what area is a space planar space covered by a polygon now if you take the vertices of the polygon and start tracing them in anti-properwise direction so in this case if I take and this quadrilangle ABCD so ABCD so if you see we are tracing with these points in anticlockwise direction isn't it so from A to B to C to D anticlockwise direction then we say we are we are talking about positive area of the quadrangle ABCD and we write like this so within brackets you write just a minute within brackets you have to write the vertices so A B C D this is how I would mean or in literature mathematical literature this will mean positive area of this quadrangle right so if you change the order in in let's say you go clockwise direction ADCB so the area will simply be minus A B C B ADCB right I hope you understood this similarly if you take this triangle ABC right so if you simply write ABC it is positive area correct but the area of the triangle is written as minus CBA if you go in the opposite that is clockwise direction right so ABC is equal to negative of this CBA I hope this is understood right similarly here if this is point O right so hence in this triangle LOK so LOK is actually anticlockwise direction so area will be simply LOK but if you go the other way around LKO so it will be negative LKO I hope you understood these you know notations and definitions about and types of different quadrilaterals and how do we mention area okay positive negative clockwise direction negative anticlockwise direction positive do remember this because we are going to use these concepts multiple number of times in the subsequent sessions so in that case let's say if you you know whenever we are discussing any theorem and you are confused you can always come to this video and understand what are the meanings of different notations now what we are going to do is we are now going to take up the very first theorem related to quadriangles and that's called Weregnans parallelogram you would have studied this in your previous grades or something you know some geometry course but we will be taking up the Weregnans it was a French mathematician basically contemporary of Lipnitz and Isaac Newton and you will be studying about the theorem which he gave for the areas of quadrilaterals and quadrangles and associate quadrangles and after that we will be going to discuss something called Ramabukta's theorem which many of you would have already studied about it so we are going to see all these theorems one by one followed by multiple other varieties of different geometrical features okay so if you like this video please do share and subscribe to our channel and that would like to reach out to many many more people so that they can also enjoy or let's say you know get this knowledge of geometry which is there for all of us okay so see you in the next video and we will be going to discuss Weregnans theorem in the next video okay see you bye bye and thanks for watching this