Alternate Interior Angles

Student Summary

When two lines intersect, vertical angles are congruent, and adjacent angles are supplementary, so their measures sum to 180. For example, in this figure angles 1 and 3 are congruent, angles 2 and 4 are congruent, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

Two intersecting lines.
Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.

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.

Two lines that do not intersect. A third line intersects with both lines.
Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.

Alternate interior angles are equal because a 180∘180^\circ rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point MM halfway between the two intersections. Can you see how rotating 180∘180^\circ about 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 70∘70^\circ we use vertical angles to see that angle 3 is 70∘70^\circ, then we use alternate interior angles to see that angle 5 is 70∘70^\circ, then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is  110∘110^\circ since 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 70∘70^\circ, and angles 2, 4, 6, and 8 measure 110∘110^\circ.

Visual / Anchor Chart

Standards

Building On
7.G.5

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Addressing
8.G.1

Verify experimentally the properties of rotations, reflections, and translations.

8.G.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.