When two lines intersect, vertical angles are and adjacent angles are
, that is, their measures sum to 180LaTeX: ^\circ∘. For example, in this figure angles 1 and 3 are equal, angles 2 and 4 are equal, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

Two intersecting lines.
When two parallel lines are cut by another line, called a
, two pairs of
are created. ("Interior" means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.

Two lines that do not intersect. A third line intersects with both lines.
Alternate interior angles are equal because a LaTeX: 180^\circ180 ∘ rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point LaTeX: MM halfway between the two intersections-can you see how rotating LaTeX: 180^\circ180 ∘ about LaTeX: MM takes angle 3 to angle 5?

Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is LaTeX: 70^\circ70 ∘ we use vertical angles to see that angle 3 is LaTeX: 70^\circ70 ∘, then we use alternate interior angles to see that angle 5 is LaTeX: 70^\circ70 ∘, then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is LaTeX: 110^\circ110 ∘ since LaTeX: 180 -70 = 110180 − 70 = 110. It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure LaTeX: 70^\circ70 ∘, and angles 2, 4, 6, and 8 measure LaTeX: 110^\circ110 ∘.