The angles that occupy the same relative position at each intersection where a straight line crosses two others is called corresponding angles. If the two lines are parallel, the corresponding angles are equal.
From the prove above, it was given that line KL is parallel to line MN.
Also, angles KRQ and MSQ occupy the same relative position at the intersections R and S.
Therefore, the missing justification for the second line of the proof is "corresponding angles theorem".