Lesson 16

Earlier in this unit, students learned that when a \(180^\circ\) rotation is applied to a line \(\ell\), the resulting line is parallel to \(\ell\). Here is a picture students worked with earlier in the unit:

The picture was created by applying \(180^\circ\) rotations to \(\triangle ABC\) with centers at the midpoints of segments \(AC\) and \(AB\). Notice that \(E\), \(A\), and \(D\) all lie on the same grid line that is parallel to line \(BC\). In this case, we have the structure of the grid to help see why this is true. This argument exhibits one aspect of MP7, using the structure of the grid to help explain why the three angles in this triangle add to 180 degrees.

In this lesson, students begin by examining the argument using grid lines described above. Then they examine a triangle off of a grid, \(PQR\). Here an auxiliary line plays the role of the grid lines: the line parallel to line \(PQ\) through the opposite vertex \(R\). The three angles sharing vertex \(R\) make a line and so they add to 180 degrees. Using what they have learned earlier in this unit (either congruent alternate interior angles for parallel lines cut by a transverse or applying rigid transformations explicitly), students argue that the sum of the angles in triangle \(PQR\) is the same as the sum of the angles meeting at vertex \(R\). This shows that the sum of the angles in any triangle is 180 degrees. The idea of using an auxiliary line in a construction to solve a problem is explicitly called out in MP7.