Geometry - Parallel Lines

The relationship of slopes between parallel lines is that they are equal. This means that if two lines are parallel, their slopes will have the same value.

Vertical angles are always congruent, regardless of whether the lines are parallel or not. Vertical angles are formed by two intersecting lines and are opposite to each other. They have equal measures and are congruent because they are formed by the same pair of intersecting lines. This property holds true for any pair of vertical angles, regardless of the orientation of the lines.

The method to prove two lines are parallel when cut by a transversal is by showing that the alternate interior angles are congruent. This means that the angles on the inside of the two lines, but on opposite sides of the transversal, are equal in measure. If the alternate interior angles are congruent, it indicates that the lines are parallel.

Perpendicular lines have slopes that are opposite reciprocals of each other. This means that if one line has a slope of m, the perpendicular line will have a slope of -1/m.

When parallel lines are cut by a transversal, corresponding angles, alternate interior angles, and alternate exterior angles are congruent. Corresponding angles are formed when a transversal intersects two parallel lines and are located in the same relative position on each line. Alternate interior angles are located on opposite sides of the transversal and inside the two parallel lines. Alternate exterior angles are located on opposite sides of the transversal and outside the two parallel lines. Therefore, all three angle pairs mentioned in the answer (corresponding, alternate interior, and alternate exterior) are congruent when parallel lines are cut by a transversal.