 Hi and welcome to the session. I am Arsha and I am going to help you with the following question that says in figure 6.44 the site QR of triangle PQR is produced. Two point is if the bisector of angle PQR and angle PRS meet at the point T then prove that angle QTR is equal to half of angle QPR. So first let us learn that if we have a triangle, suppose we have a triangle ABC then if a site of a triangle is produced to a point D, suppose BC is produced to a point D then the exterior angle so-called is equal to the sum of two interior opposite angles that is angle ACD is equal to the sum of angles A and B which is theorem 0.8 of your book. So this is the key idea we are going to use in this problem to prove that angle QTR is equal to half of angle QPR. Let us now begin with the solution in triangle PQR site QR is produced to a point S therefore we can write angle PRS is equal to the sum of angles PQR and angle QPR. Since this is by 0 number 6.8 of your book. Similarly in triangle TQR site QR is produced S therefore we can write angle TPRS is equal to the sum of angles QTR and angle TQR. So this is the equation number one and this is the equation number two. Now on dividing both sides of equation one by two PRS upon two is equal to angle PQR upon two plus angle QPR upon two. Now angle PQR upon two is equal to angle TQR since QT is the bisector of angle PQR. So this implies these two angles are equal and thus this can be written as half of angle PQR. This as it is QPR upon two and left hand side also as it is that is PRS upon two. The PRS upon two can be written as angle TRS bisector of angle RS. Now TRR QR as it is plus angle QPR upon two. This the equation number three. Comparing equation two and equation three left hand sides are equal so right hand sides are also equal. So we have QTR plus angle TQR is equal to angle TQR plus angle QPR upon two. TQR is common on both the sides so on cancelling we have angle QTR is equal to angle QPR upon two. Thus we have proved that angle QTR is equal to angle QPR upon two. So this completes the solution hope you enjoyed it take care and bye for now.