Two parallel lines ℓ and m run across the page with ℓ above m, and the point P lies in the strip between them. Q is a point on ℓ and R is a point on m, both of them to the left of P. At Q the ray QP makes an angle of 35° with the part of ℓ that runs to the right of Q, and at R the ray RP makes an angle of 48° with the part of m that runs to the right of R. What is the measure of ∠QPR?

Draw through P the line n parallel to both ℓ and m. With QP as a transversal of ℓ and n, alternate angles make the angle between PQ and the left-hand part of n equal to 35°. With RP as a transversal of m and n, the same argument gives 48° between PR and the left-hand part of n. Q lies above n and R below it, so the two angles sit on opposite sides and add: ∠QPR = 35° + 48° = 83°. Subtracting gives 13°, and taking the supplement gives 97° — the angle on the far side of P, not ∠QPR.