Geometry

Angles and Transversals

Big Picture

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.

Key Terms

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.

Angles Formed by Transversals

Examples:

  • Interior: Area between / and m
  • Exterior: Area outside / and m
  • Adjacent angles:  3 and 4
  • Vertical angles:  1 and  6
  • Alternate interior angles:  3 and 6
  • Alternate exterior angles:  1 and  8
  • Consecutive interior angles:  2 and  3
  • Corresponding angles:  5 and  7

Perpendicular Transversal

If parallel lines are cut by a transversal perpendicular to the lines, then all angles formed are congruent and measure 90°.

Angles Formed by Transversals

Parallel Lines

There are several postulates and theorems related to the angles formed by the transversal cutting two parallel lines:

  • Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
         •  Example: \(\angle 2 \cong \angle 6\)
  • Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
         •  Example:  \(\angle 4 \cong \angle 5\)
  • Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
         •  Example:  \(\angle 1 \cong \angle 8\)
  • Consecutive (Same Side) Interior Angles Theorem: If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary.
         •  Example:  \(\angle 3 \cong \angle 5\)
Parallel Lines

Geometry

Angles and Transversals cont.

Parallel Lines (cont.)

Proving 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.

  • Converse of Corresponding Angles Postulate: If corresponding angles are congruent when two lines are cut by a transversal, then the lines are parallel.
  • Converse of Alternate Interior Angles Theorem: If alternate interior angles are congruent when two lines are cut by a transversal, then the lines are parallel.
  • Converse of Alternate Exterior Angles Theorem: If alternate exterior angles are congruent when two lines are cut by a transversal, then the lines are parallel.
  • Converse of Consecutive Interior Angles Theorem: If consecutive interior angles are supplementary when two lines are cut by a transversal, then the lines are parallel.

      Perpendicular Lines

      Theorems:

      • If a transveral is perpendicular to one of two parallel lines, it is perpendicular to the other parallel line.
             a. If \(l || m\) and \(l \perp n\), then \(n \perp m\).
      • If two lines in a plane are perpendicular to the same line, they are parallel to each other.
             a. If \(l \perp n\) and \(n \perp m\), then \(l || m\).
      Perpendicular Lines

      Proving Perpendicular Lines

      Theorem: If two lines intersect to form a linear pair and the angles are congruent, then the lines are perpendicular.

      • If \(\angle ABD \cong \angle DBC\), then \(\overleftrightarrow{AC} \perp \overleftrightarrow{BD}\)
      Proving Perpendicular Lines

      Notes