4. In Fig. 6.31, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS. [Hint : Draw a line parallel to ST through point R.]
Solution: First, construct a line XY parallel to PQ. We know that the angles on the same side of transversal is equal to 180°. So, ∠PQR+∠QRX = 180° Or, ∠QRX = 180°-110° ∴ ∠QRX = 70° Similarly, ∠RST +∠SRY = 180° Or, ∠SRY = 180°- 130° ∴ ∠SRY = 50° Now, for the linear pairs on the line XY- ∠QRX+∠QRS+∠SRY = 180° Putting their respective values, we get, ∠QRS = 180° – 70° – 50° Hence, ∠QRS = 60°
