 Hi and welcome to the session I am Arsha and I am going to help you with the following question which says in figure 6.31 if pq is parallel to st angle pqr is equal to 110 and angle rst is equal to 130 degree find angle qrs also we are given a hint draw a line parallel to st through point R. Let us now begin with the solution and we are given pq is parallel to st angle pqr is equal to 110 degree and angle rst is equal to 130 degree and we have to find angle qrs which is this angle now according to the hint let us draw a line through the point R which is parallel to the point line st this line we have drawn through the point R such that it is parallel to the line st. Let us name this line as xy let us write the construction draw xy parallel to st through point R. Now since st is parallel to xy SR is any transversal so angle tSr will be equal to angle SRx this angle since they alternate interior angles let us write down xy is parallel to st and SR is a transversal this implies angle tSr is equal to angle SRx now tSr is equal to 130 degree so this implies angle SRx is equal to 130 degree so this whole angle is 130 degree. Now since pq is parallel to st and this is given to us and st is parallel to xy this is by construction this implies that pq is parallel to xy since two lines which are parallel to the same given line parallel to each other thus we have pq parallel to xy. Now pq is parallel to xy this implies that pqr plus angle SRq is equal to 180 degree since sum of consecutive interior angles is supplementary let us write down pq is parallel to xy and qr is a transversal this implies angle pqr plus angle qrx is equal to 180 degree since sum of consecutive interior angles is 180 degree. Now angle pqr is equal to 110 degree plus angle qrx we will find out is equal to 180 degree this implies angle qrx is equal to 180 degree minus 110 degree which is equal to 70 degree thus this angle is 100 and sorry this 70 degree. Now we will find angle qr is which we will get on subtracting angle qrx from angle SRx so we have qrx is equal to SRx minus angle qrx so this is equal to 130 degree minus 70 degree which is equal to 60 degree thus angle qrx is equal to 60 degree this completes the solution hope you enjoyed it take care and have a good day