EQN-04-08

Parallel Lines & Transversals

The equal-angle and supplementary-angle pairs that appear whenever a line crosses two parallels.

Difficulty 2/5

When a transversal crosses two parallel lines it makes 8 angles, but only two distinct sizes appear, and they add to 180°.

Corresponding angles — equal

Alternate interior — equal

Alternate exterior — equal

Vertically opposite — equal

Co-interior (same-side interior) angles add to 18 0^{\circ} .

Label one angle, then every other angle is either equal to it or supplementary $(18 0^{\circ} -$ it). One known angle fixes them all.

If one angle is 7 0^{\circ}, the other size is 11 0^{\circ}, alternating around both lines .

Equal corresponding (or alternate) angles also PROVE lines are parallel — useful in reverse.

Diagram

Parallel lines cut by a transversal: co-interior angles add to 180°.

Worked examples

Example 1 (difficulty 2/5)

Two parallel lines are cut by a transversal. One angle is 65°. What is its co-interior (same-side interior) partner?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

A transversal makes an alternate interior angle of 48° with two parallel lines. What is the matching alternate interior angle?

◎ Predict the answer first:

Only two angle sizes exist; they always sum to 180°.
Corresponding and alternate angles are EQUAL; co-interior angles are SUPPLEMENTARY.
Fix one angle, then propagate equal/supplementary around the figure.

Treating co-interior angles as equal

Why: Most other named pairs are equal.

Fix: Same-side interior angles add to 180°; they are not equal.

Applying these rules when the lines are NOT parallel

Why: The figure looks parallel.

Fix: The equalities require the lines to be parallel — check the given marks.

Worked examples

Example 1 (difficulty 2/5)

Two parallel lines are cut by a transversal. One angle is 65°. What is its co-interior (same-side interior) partner?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

A transversal makes an alternate interior angle of 48° with two parallel lines. What is the matching alternate interior angle?

◎ Predict the answer first:

Common mistakes

Treating co-interior angles as equal

Why: Most other named pairs are equal.

Fix: Same-side interior angles add to 180°; they are not equal.

Applying these rules when the lines are NOT parallel

Why: The figure looks parallel.

Fix: The equalities require the lines to be parallel — check the given marks.