B. Proving Angle Relationships

So far you have looked at angles that are formed when a transversal intersects a pair of parallel lines. You may have noticed that there appears to be a number of equal angle measures when this occurs. If you did, your observations are correct.

  • opposite angles are always equal
  • corresponding angles are equal
  • alternate interior angles are equal
  • alternate exterior angles are equal

These last three relationships are not true if the transversal intersects non-parallel lines. However, opposite angles are still equal in this case.

Corresponding Angles

Corresponding angles formed when a transversal intersects a pair of parallel lines are equal. One way to justify this is to think about what would happen if corresponding angles were not equal. If the corresponding angles were different, the "parallel lines" would eventually meet, which by definition would make them not parallel. As such, corresponding angles must be equal. The relationship that corresponding angles are equal will be used to prove the other angle relationships.

Example 1

Prove that when two lines intersect, the opposite angles are equal.

Begin with a labelled diagram. In this case a and c are opposite angles, so the goal is to show that a = c.

Angles a and b form a straight line so a + b = 180°.

Angles b and c form a straight line so b + c = 180°. Substituting 180° − a for b gives