 Hello friends I welcome you all to a new session on geometry and in this series of video sessions we are going to understand a very interesting topic that is line and angles. Lines and angles is a part of elementary high school geometry and it forms a basis for all understanding in higher geometry later on. So we must be very very thorough with the concepts and definitions around lines and angles and different properties of both these geometric elements. Now the history goes history suggests that you know the lines and angles have been part of human civilization and all its pursuits for a very long long time. So we get a lot of references and evidences of you know these elements being aggressively used in different applications in early history and we know that Greeks were pioneers in collating all this information and there was a mathematician called Euclid which who collated all this information all this information in around third century BC in his book called The Elements. So since then we are you know using the concepts of these basic geometrical elements for furthering our knowledge in mathematics especially geometry. So we'll begin with what is aligned the definition and then we'll go to definition of an angle and we'll try and understand in this session what are different parts and other important information around angles like interior and exterior of an angle. So what is the line guys if you see look around yourself let's say if you look at the ceiling and the point the the region where the ceiling meets the wall it represents a line you already are familiar with let's say you know various examples of line around you so the notebook on which you are writing right now probably you know the edges represent a line so the computer on which you are watching this session the edges again would represent a line the pen which you are holding probably you know the edge would again you know represent a line and you know if you take a sheet of paper and try to fold it across you know across any crease so that crease represents a line so there are various such examples of line around you so how do we basically define a line so if you see line is nothing but a collection of points if you see what is aligned a collection of points but not just any random collection of points isn't it there is a there's a trend or there's a pattern a collection of points and what is this collection of points you know so special why is it so special because if you see all these points all these points when you know put in order all these points are oriented in one particular direction or oriented slash let's say aligned aligned in a in a particular direction in a particular direction okay so this is what we see so all those examples which we just discussed are actually like that so if you take you know points arrange them in and move them move these points or let's say you know just plot these points in one particular direction one particular direction and and if you join all of them like that this becomes an example of a line other important feature about line is it is it it extends it extends it extends infinitely infinitely in both the directions in both the directions isn't it so if you see this line can extend in this direction as well and this line can extend in this direction as well infinitely there's no end to it right so hence these are the characteristic properties of line so line basically is a collection of points oriented in a particular direction and this line extends in both direction extensively so we how do we you know representation of line so obviously we can't draw an infinitely long line on a piece of paper so how do we represent it so representation representation of a line how do we how do we express a line while doing mathematics so we simply draw a symbolic part of the line and then put two arrows around it so if you see there are two arrows these arrows these arrows signify the direction okay signifies the direction signifies the direction of the line okay later on when you will be dealing with concepts on vectors and other things you will then realize that the concept of this direction is very important so hence this line is represented like this and we also have to you know put a name around it so we put them as a and b so we take two points on the line and we say this is line a b line a b okay this is how a line is represented many a times small roman letters are also used for example they can express this as l so line l is another way of expressing or representing line on piece of paper okay now if you also note is there is another characteristic of this line the characteristic is it is one dimensional so line is only one dimensional one dimensional isn't it dimensional what does it mean it means that there is no line has line has has only length only length right there is no breadth no breadth so thickness is almost negligible and no height as well so neither it is wide nor it is high so hence only length so hence it extends infinitely in both direction and it is given by this length so there is a length of this line okay so it's a one dimensional geometrical element okay so these are the characteristic features of line now once we understand line and we saw the examples of line you can just you know turn around look around yourself and you'll see different varieties of examples of line and once you know you'll be also noticing that there are you know if you are comparing let's say two or more lines let's say we take two lines now so two lines two lines right now how and you know what kind of these lines are now these lines can be in the same plane so we call them coplanar coplanar coplanar lines what are coplanar lines let's say this is in our notebook so I draw two lines over this notebook and this these two lines a b a b and c d lie on the same notebook same same surface so they are coplanar lines but if you observe there are non-coplanar lines as well right for example if you look at the ceiling of your room so one line which is you can imagine any of these creases there any of these you know uh edges around which the ceiling meets the wall let's say this is uh this is this is the wall and this wall meets the ceiling along this line okay now this line and then imagine the room the room is like that the room's ceiling is like that this is another edge of the ceiling and this these are the other walls okay so if you see this line here is and this line these two lines are not in the same plane correct they are not in the same plane so hence lines could be on the same plane and hence they will be called coplanar and they are non-coplanar lines as well in in these uh sessions we will be considering only coplanar lines now there are two coplanar lines what are the possibilities and how would they appear so their possibility is case one one line is like that let's say a b a b and i have another line coincident on the same line the same line in this case we call they call them as coincident lines coincident lines there are just one top of the other case two would be case two would be a line like that and a line like that right so they are such that any any time you find out the distance between them you have the same distance right let's say d these type of lines you already know are called parallel parallel lines never meet they never meet or intersect and the third variety you by now would have guessed is nothing but when two lines are not parallel on the same plane they intersect right so what are these these are intersecting lines intersecting lines now intersecting lines creates you know new features new geometric features and the feature which we are not which we are now going to talk about is angles okay so two intersecting lines always create an angle okay so let us take another example so these are two intersecting lines let's say a b and cd so if you notice they intersect at a point at a point and this point is called point of intersection or the vertex of the angle okay vertex of the angle right this is nothing but point of intersection point of intersection intersection now right now at the same point of intersection there could be you know many such many such you know lines intersecting for example there could be another line like that now how do we differentiate these you know geometrical elements now if you see let me call this point of intersection as o right so we have now angles formed out of there and let's say this is the other line is pq okay pq so this shape if you see this is let's say d i'm redrawing it and this is b and this point was o now this particular geometric element is called an angle so angle is formed by two intersecting lines one vertex so what are the parts of the angles now let let us you know so this is a vertex of the angle okay this is one arm or the side and this is another arm or side so there are two arms and side or sides of a triangle right and then we put a curve like this to represent an angle okay so this is what is an angle okay so we we saw that two angles uh sorry one angle is formed when two lines intersect okay so on and that too on a plane so please remember what all did we discuss two lines two intersecting coplanar lines coplanar lines form an angle form an angle this is what we right understood now how do we naming now name it so naming an angle or how do we identify naming an angle so angles are named and you will ask why do we require to name it because every time you want to express something about the angle you don't need to draw the angle shape again and again so you can just refer to it by some name so for example the different types of the way or different ways how angles are named are like this so we draw this angle sign and we say b o d which is also same as saying angle d o b so you see the vertex is in the middle and the two edges are on the other other two sides of the vertex okay many a times this angle can also be represented simply by its vertex angle o if if the vertex vertex uh is on only one is on only one angle only one angle what do i mean i mean for example in this case you can't call angle o because then it will be confusing which angle you mean do you mean this angle you mean this angle because all these angles have same vertex oh so only when there is only one angle on a vertex you can also represent them as angle o or by the name of its vertex another way of representing angles is by let's say Greek letters alpha beta theta like that okay so hence what happens is you take two arms two sides of the angle and you represent it as like let's say alpha right so hence we say it as an angle alpha we can simply write angle alpha okay many a times people write numbers as well so let's say this is and they will write one and we represent it as represent it as angle one okay this is how angles are represented another way in this case also instead of having a side angle people do prefer to write like that right put a hat over it so that also represents an angle or dob and like that right so hence there are one two three four and five ways of representing angles we saw now another concept which we have to discuss in this is what is meant by interior and exterior of an angle so let's say now this is an angle let's say this is angle a b c vertex is b right now any point within the folds of the two arms will be interior or in other way let's say from this point p you want to reach both the sides so hence if you you start moving like that so you reach point c isn't it or the side b c and for reaching side b a you have to go in the opposite direction isn't it so when this happens that means you have to go in opposite direction to meet the two sides then the point is within the angle so let's say this this point here here also if i want to meet bc bc is nothing but you have to extend it like that then i have to move like that okay and if you want to move move towards let's say a b then you have to go in the opposite direction isn't it something like that opposite direction so then that point is considered to be interior this is interior of this region this entire region will be called this region is nothing but interior region this is interior region of of the of the angle correct this is integer interior region is that okay so hence you're moving into opposite direction but let's say if there is a point here there's a point here now to meet both the sides you will have to move in the same direction isn't it same direction so hence this region outside outside the outside the region of interior angle of interior of an angle is now exterior of an angle right so if you see this is this is nothing but this is the exterior of the angle so this part is exterior exterior of an angle exterior of an angle so interior of an angle and exterior of an angle so if you see an angle an angle angle divides the plane divides the plane that means the points on the plane are now distributed okay in three parts into three parts what three parts one is first is angle itself there are lots of points on the angle itself angle itself for example any point here q r all are on the s or on the angle itself second is interior of an angle interior of an angle and third is exterior of an angle exterior of an angle okay so please keep it in mind that an angle divides entire plane into three regions now all this discussion was about angles on a plane so a natural question comes do we also have angles which are not on a plane yes we do have and we then we call them as solid angles solid angles now these are 3d angles example when you go to an ice cream parlor you order of ice cream in cone so the angle made by the cone at the vertex here this is the solid angle okay so this is not a 2d it's a 3d angle we'll discuss about all this again in later uh sessions and as well as in our coordinate geometry series so i hope you understood the concept of a line and an angle and elements attached to line and angle different parts of it and other such definitions we will move on to the next part of uh uh concepts related to angles in the subsequent sessions