 Hello friends welcome to the next session on lines and angles in the last session. We discussed definitions of Line angles and we've also discussed about interior and exterior of an angle in this session We are going to understand what is meant by Measure of an angle. How do we measure an angle and why do we need to measure it in the first place? So, you know that in mathematics, there will be Cases where you need to compare and then do some operations on angles For example in arithmetic also when you need to compare which is greater two or three So the quantum the magnitude of two and three makes, you know Them comparable that is we can very easily say that two is less than three, right? Because there is a magnitude there's a measure attached to the number now similarly if we have these angles for example This one is the first one and then we have this another one. So there are two angles here a b c and D e f let's say now if the question is which one is a bigger angle, right bigger angle means which is You know, which is covering more Let's say interior of an angle. So let's say the angle which has Major area inside the angle or the interior is more we call it a bigger angle so if you see this angle is more than this but then how much more and How do we quantify this? So for that we have something called a measure of an angle and we measure the angle in three different ways right so we will be discussing them here and You have also seen during your childhood you or today also you use a tool called a protractor Protractor to measure the angles, isn't it? So what you do is you simply, you know drawing for example This is an exact an angle of a bc over here and what you basically do is you take the protractor and Put the zero zero and the zero line around the put the zero line around the Oh Around the zero Like axis is it so here is what I'm trying to do the same thing. So I am putting this protractor on 200 okay now once done you measure where is the other side pointing? So here we check this one. So if you see we check this side and this side is Coinciding with 40 degrees if you see it starts from zero zero ten twenty Thirty forty so it's around forty right so what is the unit of this? So the first one as I told you there are three systems We are measuring the angle here in the first system and this is called sex a decimal System sex a decimal system Where we measure angles in terms of degrees Okay in degrees degrees are the so hence you you say 20 degrees or in this case we measured 40 and we put a small dot onto the right-hand side right 40 degrees This is how you have you measure angles in sex a decimal Systems, okay, what are other features around sex a decimal system? So we say that around the point around a Point the angle is angle measure angle measure is Angle measure is 360 degrees You must be knowing it already that within a circle. Let's say This is the point. This is the first ray and let's say there are two coincident ray over You know here and then one of the rays starts moving like that like that Okay, and it is it is going anticlockwise direction Anticlockwise direction and after some time it comes back to its original Original position so hence if you see this full angle swept is nothing but 360 degrees Okay, so the entire plane is divided into 360 degrees with respect to a given First line. Okay, so this is what is hexadecimal system Now there are other two systems which are very much in use the second one is very much in use the third one is Not that much in use the second one is Is called angle a circular measure? circular measure where what we say is We measure the angle in terms of something called radians okay radians and How do we define one radian one radian is nothing but the ratio of? Ratio of Arc length arc length of unit length or rather arc of unit length arc of unit length divide by radius of Unit length what does it mean? It means that you know what is an arc. So if you take a circle Let's say we have a circle Okay, this is a circle and this is a center and let's say this is This length let's say is one unit one unit and This radius also is of one unit Then the angle described here is One radian okay, so let's say the angle is theta. Okay, so theta. I'm saying is one upon one This is the ratio isn't it ratio of arc length divided by the radius is one radian you will later on understand that in a sector for example in this sector this is this is called a sector of the circle Sector of the circle. So in a sector central angle subscribed inside the Sector is nothing but arc length arc length divided by Radius Okay, this is another way of measuring Angles this we will be Studying a lot in your higher grades. So hence radians, right? So please remember. What is the relation between radian and degrees? so we say that two pi radians is Equal to 360 degrees Okay, two pi radians is equal to 360 degrees. This is the conversion Relation, so what will be let's say pi radian? Pi radian and we also denote it by small only RAD rad Pi radians will be equal to nothing but 180 degrees half of it Pi by two radians It's equal to 90 degrees and so on and so forth Okay, one more interesting thing from here if you see What is if let's say arc length becomes full circumference? You'll write two pi R Is a circumference that's the arc length is the circumference You you are you are taking a arc which is a full circle Let's say we are instead of considering only this we consider the full circle So when you consider full circle, what is the arc length? Clearly the circumference is just two pi R and when divided by R you see what is the angle you get you get two pi Which matches with the point that inside the circle the the center at the center the angle is 360 degrees, isn't it? So if you see this is what we get So when you take the full circumference you get two pi radians at the center, which is equivalent to 360 degrees Okay, so please remember This conversion just for an example, let's say if you have 45 degrees You can check 45 degrees is equal to pi by four radian Okay There is one more method of measuring Angles and that is called Centesimal system Centesimal Centesimal system What is what is centesimal system guys? So in this case instead of we we have a conversion factor like that at 90 degrees 90 degrees is considered to be equal to 100 grades Okay, 100 grades So basically we measure Centesimal system in grades Right, so we talk about angles and so hence for example, a right angle right angle is 90 degrees is nothing but 100 grades Okay, similarly, 360 degrees or the angle inside a circle will be equal to 400 grades Okay. Now this system is not very much in Vogue or uses You know in the current era we use mostly The circular measure which will be you'll be seeing a lot of uses in your academic pursuits and The degrees which you have been anyways using so far So these are the two system mostly in you know Uses these days so so in this session We learned how to measure an angle and what was the use of what is the use of measuring an angle? So basically we need to quantify Two angles and then compare and then do lots of operations which we'll see In subsequent sessions, so hence measurement of angle was necessary and we have three ways of measuring angles Okay, in the in the in the subsequent sessions we'll deal with how measurement of an angle and how other operations on angles are done