 Hello and welcome to the session in this session. We discussed the following question. It says in the given figure Proof that x is equal to alpha plus beta plus gamma Before we move on to the solution. Let's recall the fact which says that if a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior Opposite angles. This is the key idea for this question Now we move on to the solution We are given this figure in which we have this angle a is alpha degrees angle b is beta degrees angle c is gamma degrees and This angle adc is x degrees Now to start with the solution first we will join b and d and we produce this bd to e Now consider this angle as p degrees this as q degrees and this whole was given to us as beta degrees now Let this be s degrees this be t degrees and This whole was given to us as x degrees so here we have p plus q is equal to beta and s plus t is equal to x Now if you look at this triangle a bd side bd is produced to e and This s is the exterior angle of triangle a bd So using the key idea, we would say this s is equal to sum of the two interior opposite angles which are alpha and p so it would be s is equal to alpha plus p Let this be equation one Now next if you consider the triangle cbd in This triangle also the side bd is produced to e and here the exterior angle formed is T So using the key idea, we would say that t is equal to the sum of the interior opposite angles And this would be q plus gamma Let this be equation two Then we have adding one and two We get s plus t is equal to alpha plus p plus q plus gamma That is we have f plus t is equal to p plus q plus alpha plus gamma Now we know that s plus t is equal to x this gives us x is equal to p plus q which is beta plus alpha plus gamma That is we get x is equal to alpha plus beta plus gamma So Hence proved So this completes the session. Hope you understood the solution for this question