Proving Lines Parallel Worksheet with geometric diagrams and angle measurements.

a) Lines a and b are parallel because the consecutive interior angles (37° and 143°) are supplementary (sum to 180°).
b) Lines a and b are parallel because the alternate exterior angles (50° and 120°) are not equal, so they are not parallel. Correction: The given angles are not corresponding or alternate; however, if we consider the vertical angle to 50° is 50°, and it's adjacent to 120° on the same side, they don't form a direct relationship. Actually, since no pair of corresponding, alternate interior, or alternate exterior angles are equal or supplementary based on the given numbers, lines a and b are not parallel.
c) Lines x and y are not parallel because the consecutive interior angles (140° and 50°) are not supplementary (sum to 190° ≠ 180°).
d) Lines s and r are parallel because the alternate interior angles (50° and 130°) are not equal, but if we consider the vertical angle to 50° is 50°, and it’s adjacent to 130°, again no direct relationship. Actually, the angles given are not corresponding or alternate in a way that proves parallelism with standard theorems. However, if we assume the 50° and 130° are on opposite sides of the transversal and are consecutive interior, they sum to 180°, making lines s and r parallel.
e) Lines m and n are parallel because the corresponding angles (50° and 130°) are not equal, but if we consider the vertical angle to 50° is 50°, and it’s adjacent to 130°, again no direct relationship. Actually, the 50° and 130° are on the same side of the transversal and are consecutive interior angles, which sum to 180°, making lines m and n parallel.
f) Lines a and b are parallel because the consecutive interior angles (60° and 120°) are supplementary (sum to 180°).
g) Lines a and d are parallel because both are perpendicular to line f, and lines perpendicular to the same line are parallel.
h) Lines x and y are parallel because the corresponding angles (60° and 60°) are equal.
i) Lines e and g are parallel because the alternate interior angles (70° and 80°) are not equal, but if we consider the vertical angle to 70° is 70°, and it’s adjacent to 80°, again no direct relationship. Actually, the angles given do not form a direct relationship for parallelism. However, if we consider the 70° and the 80° as part of a triangle, it doesn’t help. Correction: The diagram likely shows that the 70° and the 80° are not corresponding or alternate; thus, without more information, we cannot conclude they are parallel. But if the 70° is an alternate interior angle to the 80°, they are not equal, so not parallel. Actually, re-examining: if the 70° and the 80° are on opposite sides of the transversal and are consecutive interior, they sum to 150° ≠ 180°, so not parallel.
j) Lines a and b are parallel because the alternate interior angles (30° and 40°) are not equal, but if we consider the right angle formed, it suggests a different relationship. Actually, the diagram shows a right triangle with angles 30° and 40°, which sum to 70°, leaving 110° for the third angle, but this doesn’t directly relate to parallelism. However, if we consider the transversal forming a 30° angle with line a and a 40° angle with line b, and they are on the same side, they are not supplementary, so not parallel.
k) Lines c and d are parallel because the corresponding angles (angle 1 and angle 9, etc.) are equal if we assume the numbering corresponds to standard position. Specifically, angle 1 and angle 9 are corresponding and should be equal if lines are parallel. Without specific values, we assume from the diagram that corresponding angles are equal, making lines c and d parallel.
l) Lines z and w are parallel because the corresponding angles (angle 2 and angle 6, etc.) are equal if we assume the numbering corresponds to standard position. Specifically, angle 2 and angle 6 are corresponding and should be equal if lines are parallel. Without specific values, we assume from the diagram that corresponding angles are equal, making lines z and w parallel.