Parallel and perpendicular lines have many different relationships. Several of them involve parallel lines being cut by a transversal, which forms congruent and complementary angles. With the correct knowledge of angles, we can use a known angle to find the unknown measurements.
Transversal: A line that intersects a set of lines (may or may not be parallel).
Interior: Area between two lines.
Exterior: Area outside of two lines.
Corresponding Angles: Two angles that have corresponding positions (the “same place” with respect to the transversal but on different lines).
Alternate Interior Angles: Two angles that are on the interior but on opposite sides of the transversal.
Alternate Exterior Angles: Two angles that are on the exterior but on opposite sides of the transversal
Consecutive Interior Angles: Two angles that are on the same side of the transveral and on the interior of the two lines. Also called same-side interior angles.
If parallel lines are cut by a transversal perpendicular to the lines, then all angles formed are congruent and measure 90°.
There are several postulates and theorems related to the angles formed by the transversal cutting two parallel lines:
The converses of the Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, and the Consecutive Interior Angles Theorem can be used to prove lines are parallel when given a diagram with lines, transversals, and congruent angles
If two lines are cut by a transversal, there are several conditions that will prove the lines are parallel.
Theorem: If two lines intersect to form a linear pair and the angles are congruent, then the lines are perpendicular.