Angles Formed by a Transversal with Parallel Lines

When a line cuts across two or more lines, it is called a transversal. If the lines being cut are parallel, many special angle pairs are formed. These angle pairs help us find unknown angles and understand geometric relationships easily.

 l1:  ---------
                 \
                  \  (Transversal t)
                 / 
 l2:  ---------

Here, line t is the transversal cutting the parallel lines l1 and l2.

Corresponding angles are angles that appear in the same relative position at each intersection where the transversal crosses the parallel lines. They are equal when the lines are parallel.

Think of corresponding angles as sitting in matching corners.

  (1)      (2)
  l1:  -----+-----
                \
                 \  t
                /
  l2:  -----+-----
  (3)      (4)

If angle (1) is corresponding to angle (3), then angle (1) = angle (3).

Alternate angles occur on opposite sides of the transversal. They also come in two types: alternate interior and alternate exterior. Both types are equal when the lines are parallel.

Co-interior angles are pairs of angles that lie on the same side of the transversal and between the parallel lines. They are supplementary, meaning their sum is \( 180^\circ \).

 l1:  ---------
                \
                 \ t
                /
 l2:  ---------

If angle X and angle Y are co-interior angles, then \( X + Y = 180^\circ \).

These angle relationships help us quickly identify unknown angles in diagrams:

• Corresponding angles → equal
• Alternate interior angles → equal
• Alternate exterior angles → equal
• Co-interior angles → supplementary (sum = \( 180^\circ \))

Whenever a transversal cuts parallel lines, spotting these pairs makes geometry problems much easier to solve.