 welcome friends to another problem solving session on triangles now we have a problem here which says we have to prove that the angle between internal bisector of one base angle and the external bisector of the other base angle of a triangle is equal to one half of the vertical angle so let's try and understand this first so internal bisector of one base angle so here is a triangle abd abc triangle abc and be be is the bisector is the bisector of angle abc correct and ce so abc happens to be the internal angle right so this is the one base angle so one base angle is angle abc whose internal bisector is be or whose bisector is be now acd is the external base angle and ce is the bisector bisector of angle external angle that is acd acd and within brackets I'm writing external angle isn't it so acd is external angle no doubt about it abc is the internal angle now be bisects angle abc so I have shown here as x this is x and this is x because be is the bisector of angle abc see similarly I have shown this as y is also why and why is why because ce bisects acd I hope that's clear now we have to prove what is to prove let's write that statement first so to prove what we need to prove we need to prove that angle between internal bisector of one base angle so we are angle between internal bisector so these are the angle between the internal bisectors internalized angle bisector and external angle bisector the angle between the two is this given let's say this is z okay so they are saying we have to prove that what do we need to prove that z is equal to one half half of angle a this is what we need to establish okay small z is equal to so let me write it as small z small z okay z is half angle a this is what we need to prove so let's see how we can approach this again there's a triangle and angles are involved and clearly we can see there is a case of exterior angle as well external angle that is so can we not say that two angle y two times angle y that is y plus y which is acd right is equal to interior opposite angle sum of interior opposite angle so angle b ac plus two times x right because abc is this angle abc plus b ac will it be equal to acd exterior angle theorem let me write this as exterior angle theorem very good right so this is one another one is we can also say if you take in triangle so let me write it here in triangle which triangle be ac again angle ecd angle ecd will be equal to so ecd happens to be the so this is ecd this is the exterior angle guys so in terms interior angle will be for this this one and this one so can I not write that as x plus z or ecd was what why so I can write y is equal to x plus z y is equal to x plus z and 2 y is equal to angle a plus 2x so let's say this is equation 1 and this is equation 2 okay now let's look at them together okay so can I not substitute y from 2 in 1 so can I say 2 times angle x plus z so this was why check 2 times x plus z is equal to angle b ac so I'm just writing that as a plus 2x is it so this implies 2x plus twice z is equal to angle a plus 2x correct and clearly you can see 2x will disappear can be cancelled so hence 2z will be angle a and hence this will mean z is equal to half angle a right and this is what was the demand of this particular question so we could prove that the angle formed by the internal bisector of one base angle and the external angle of the other base angle is equal to half the vertical angle in this case vertical angle was a so in one snapshot is this right so in the entire proof is this I hope you got the solution