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.
- 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.
Let’s see why the angles in a triangle add to 180 degrees.
- I can explain using pictures why the sum of the angles in any triangle is 180 degrees.
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.
A straight angle is an angle that forms a straight line. It measures 180 degrees.
A transversal is a line that crosses parallel lines.
This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(\ell\).
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