Parallel Lines and Transversals

Parallel lines are lines in the same plane that never intersect, no matter how far they extend. The notation for "line AB is parallel to line CD" is AB ∥ CD.

On their own, parallel lines are straightforward. The interesting geometry happens when a third line — called a transversal — crosses both of them.

Angles Formed by a Transversal

When a transversal cuts two parallel lines, it creates 8 angles total (four at each intersection). These angles fall into named pairs based on their positions:

Alternate interior angles are between the two parallel lines, on opposite sides of the transversal. They look like a Z-shape. When the lines are parallel, alternate interior angles are equal.

Corresponding angles are in the same position at each intersection — one interior, one exterior, both on the same side of the transversal. They form an F-shape. When the lines are parallel, corresponding angles are equal.

Co-interior angles (also called same-side interior angles or consecutive interior angles) are between the parallel lines and on the same side of the transversal. These are supplementary — they add up to 180°.

A Quick Summary

If lines m and n are parallel, and a transversal crosses both:

• Alternate interior angles: equal
• Corresponding angles: equal
• Co-interior angles: supplementary (sum to 180°)

Worked Example

A transversal crosses two parallel lines. One angle at the upper intersection measures 65°. Find all other angles.

Since the lines are parallel:

• The alternate interior angle at the lower intersection also measures 65° (alternate interior = equal)
• The corresponding angle at the lower intersection also measures 65° (corresponding = equal)
• The supplementary angle at the upper intersection measures 180 − 65 = 115° (co-interior, or just a straight line)
• And by vertical angles, the angles opposite each 65° also measure 65°, and those opposite each 115° measure 115°

So all 8 angles come in two values: 65° and 115°.

Proving Lines Are Parallel

These angle relationships work in reverse too. If you can show any of the following about two lines cut by a transversal, the lines must be parallel:

• A pair of alternate interior angles have the same measure
• A pair of corresponding angles have the same measure
• A pair of co-interior angles sum to 180°

Worked Example with Variables

Two parallel lines are cut by a transversal. One angle measures \((3x + 15)°\) and its alternate interior angle measures \((5x - 7)°\). Find \(x\).

Since alternate interior angles are equal:

$$3x + 15 = 5x - 7$$ $$22 = 2x$$ $$x = 11$$

So both angles measure \(3(11) + 15 = 48°\). Check: \(5(11) - 7 = 48°\). ✓