Final answer:
The information in the question does not pertain directly to the proof of angle 1 and angle 2 being supplementary and the resulting parallel lines p and q. A correct proof would involve geometric theorems related to supplementary angles and transversals, such as the converse of the corresponding angle postulate or the alternate interior angles theorem.
Step-by-step explanation:
The question appears to be requesting a proof that if two angles, angle 1 and angle 2, are supplementary, then lines p and q are parallel (p ll q). However, the given information in the question seems to be unrelated to the proof required. The given expressions describe a rotation transformation in a coordinate system, and there are also statements concerning momenta, distances between objects at a certain angle, and trigonometric identities associated with triangles, such as the law of sines and cosines.
To prove the original statement that if angle 1 and angle 2 are supplementary, lines p and q are parallel, one would typically rely on the fact that if the sum of the measures of two angles is 180 degrees (supplementary), and they are consecutive angles formed by a transversal cutting two lines, then the lines are parallel. This is because according to the converse of the corresponding angle postulate or the alternate interior angles theorem in geometry, if a pair of corresponding or alternate interior angles are congruent, the lines cut by the transversal must be parallel.