 Hello friends So in this question, it's given that angles of a triangle are in the ratio two is to three is to four We have to determine the three angles, right? So Here is a triangle right, so let's say a b see ABC is a triangle and the ratio of the three angles, so let's say This angle is or let's say this looks the smallest one. So let's say this is 2x and This is 3x and This is 4x So now they are in the ratio two is to three is to four so you can always start like that, right now by angle some property We know So by angle some property by angle some Property of A triangle what we had discussed in the previous sessions My angle some property of a triangle. What do we know? That some of the three angles must be 180 degrees, right? So angle a plus angle b plus angle c is 180 degrees So angle a is 2x angle v is 3x Or whatever whichever name a is actually 3x never mind, but Total sum is this only so this is let's 180 degrees. Okay, so 2 plus 3 plus 4. This is 9x 180 degrees Isn't it so what is x then x is 20 degrees? So that is so then now I know angle a is equal to how much 3x So hence 3 into 20 degrees 60 degrees All right angle B was 4x How much so 4 into 20? 80 degrees right and angle C is equal to 2x which is 2 into 20 degrees Hence 40 degrees right so the three angles are 40 60 and 80 and you can verify by what the sum should be 180 so indeed 40 degrees plus 60 degrees plus 80 degrees is indeed 180 degrees Isn't it so this is verified and the ratio is also 2 is to 3 is to 4 correct So whenever there is a ratio related problem learning is you can write them as this so three angles could be That whichever x is unknown. Okay, so the ratio of 2x 3x 4x is 2 is to 3 is to 4 Right, so that's how you can solve this problem So here is another problem guys in this problem. It's been given that a b is parallel to DC so a b is parallel to DC and X is equal to 4 y upon 3 right so x and y are given so you can see this is x this is y and Y is 3 z by 8 so this is z Right find BCD So BCD that means we have to find out z a BC a b c is nothing but x plus y if you look from the figure and BAD angle bad BAD this angle Is that it so we will be using some concepts of Parallel lines and transversals and alternate interior angles and all that And if you see angle ADB has been given as 36 degrees So let's solve Okay, so what is given angle ADB is 36 degrees Is it that means Y is equal to angle ADB Why? No, we can't say that because this these are not parallel so we can't say that so no Okay, but a b is parallel to DC. That means what let's try to Yeah, so So we know that X angle X Will be equal to this angle. Is it it? Right. So this angle is also XY Because AB is parallel to CD. So you can write since AB is parallel to CD therefore angle ABD ABD is equal to angle BDC Is equal to X and what is the reason the reason is alternate Interior Angles I Hope this is clear to you now Intrangle Intrangle BDC Let's take BDC Y because there is all XY and Z. Can we not say that X plus Y plus Z is 180 degrees and why is this this is because of angle some Property of our triangle We have been learning this and using this for a while now So hence and now let's also reduce it to one variable if you see Okay, so let me write it here so that it becomes clear to you. What do I mean is that X is 4 Y by 3 isn't it? It's given and Y is 3 Z upon it Correct. So can I not write X is 4 by 3 into this Y which is here. You can replace it by this way So this will become 3 by 8 Z Isn't it? So this 3 and this 3 will go this will cancel 2. So this is Z upon 2 So X I got a Z upon 2 Y 3 Z upon 8 So I can remove or replace all these 3 X Y and Z in this equation. I can replace X Y and Z in this equation by only Expressions on Z. So X can be replaced by Z by 2 Plus Y can be replaced as 3 Z by 8 and then we have got Z and this is 180 degrees correct So dear friends now you can take the common denominator LCM as 8 so this will become 4 Z This will stay 3 Z and this will be 8 Z is equal to 180 degrees Now let me solve this one here. So you will get 4 plus 3 7 plus 8 15 by 8 Z is 180 Right. So what is Z dear friends 180 degrees into 8 upon 15 Right. So 15 times 15 is 12 times this right so 15 times 12 is 180. Okay, so This is 96 degrees. So Z comes out to be Z comes out to be 96 degrees. So Z comes out to be 96 degrees Okay, so if I know Z now I can know X X is how much Z upon 2 which is 96 degrees by 2 which is 48 degrees Okay, and what is Y? Y was if you see this is 3 Z upon 8 so 3 into Z that is 96 upon 8 Right. Now. This is nothing but 12. So 36 degrees So we got all X Y and Z and you can verify X plus Y plus Z should have been 180 degrees. Is it so? So let's check 96 plus 48 plus 36 degrees is indeed So our X is correct Y is correct Z is correct Now what is to be found out the question wanted to us to find out Z first angle BCD is Z So hence these are the three demands of the question BCD ABC BAD, right? So BCD was what Z. So hence angle BCD is clearly Z Is equal to 96 degrees got the first one angle ABC was X plus Y Right, which is nothing but X was 48 Y was 36 So this comes out to be How much 74 is it? One of 84 degrees Yeah, 84 So ABC is 84 an angle BAD BAD what was BAD? Let's check the figure once again. So BAD is this angle Right. So don't you think that BAD is nothing but 180 minus X minus 36 So in this triangle this one Right What is BAD so let me write here itself so that becomes easier for you to connect Okay, so I'm writing here right in triangle a BD and X plus 36 degrees plus angle BAD is how much 180 degrees ASP again angle some property. So this is the reason Right. So hence what will be angle BAD BAD is clearly 180 degrees minus 36 degrees minus X Right and X we already found out X was 48 degrees Is it so hence it is 180 degrees minus 36 degrees minus 48 degrees Which happens to be how much? 96 Right 36 plus 48. Anyways, we had seen that is 84 So 180 Right, so we found out all the three angles. I hope this solution is clear to all of you