Lesson 14: Alternate Interior Angles

When two lines intersect, vertical angles are equal and adjacent angles are supplementary, that is, their measures sum to 180$^\circ$. 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.

When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles 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.

Alternate interior angles are equal because a $180^\circ$ rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point $M$ halfway between the two intersections—can you see how rotating $180^\circ$ about $M$ 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 $70^\circ$ we use vertical angles to see that angle 3 is $70^\circ$, then we use alternate interior angles to see that angle 5 is $70^\circ$, then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is  $110^\circ$ since $180 -70 = 110$. It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure $70^\circ$, and angles 2, 4, 6, and 8 measure $110^\circ$.