Lesson 16

Parallel Lines and the Angles in a Triangle

Lesson Narrative

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:

Rotations of triangle ABC.

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.

Parallel lines and triangle PQR.

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. 


Learning Goals

Teacher Facing

  • Create diagrams using 180-degree rotations of triangles to justify (orally and in writing) that the measure of angles in a triangle sum up to 180 degrees.
  • Generalize the Triangle Sum Theorem using rigid transformations or the congruence of alternate interior angles of parallel lines cut by a transversal.

Student Facing

Let’s see why the angles in a triangle add to 180 degrees.

Required Materials

Learning Targets

Student Facing

  • I can explain using pictures why the sum of the angles in any triangle is 180 degrees.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • alternate interior angles

    Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

    This diagram shows two pairs of alternate interior angles. Angles \(a\) and \(d\) are one pair and angles \(b\) and \(c\) are another pair.

      Two horizontal parallel lines and a third diagonal line labeled transversal. Angles a b c and d.
  • straight angle

    A straight angle is an angle that forms a straight line. It measures 180 degrees.

    a 180 degree angle
  • transversal

    A transversal is a line that crosses parallel lines.

    This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(\ell\).

    Parallel lines l and m with transversal k.

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