In this lesson, students justify that alternate interior angles are congruent, and use this and the vertical angle theorem, previously justified, to solve problems.
Thus far in this unit, students have studied different types of rigid motions, using them to examine and build different figures. This work continues here, with an emphasis on examining angles. In a previous lesson, 180 degree rotations were employed to show that vertical angles, made by intersecting lines, are congruent. The warm-up recalls previous facts about angles made by intersecting lines, including both vertical and adjacent angles. Next a third line is added, parallel to one of the two intersecting lines. There are now 8 angles, 4 each at the two intersection points of the lines. At each vertex, vertical and adjacent angles can be used to calculate all angle measures once one angle is known. But how do the angle measures compare at the two vertices? It turns out that each angle at one vertex is congruent to the corresponding angle (via translation) at the other vertex and this can be seen using rigid motions: translations and 180 degree rotations are particularly effective at revealing the relationships between the angle measures.
One mathematical practice that is particularly relevant for this lesson is MP8. Students will notice as they calculate angles that they are repeatedly using vertical and adjacent angles and that often they have a choice which method to apply. They will also notice that the angles made by parallel lines cut by a transversal are the same and this observation is the key structure in this lesson.
- Calculate angle measures using alternate interior, adjacent, vertical, and supplementary angles to solve problems.
- Justify (orally and in writing) that “alternate interior angles” made by a “transversal” connecting two parallel lines are congruent using properties of rigid motions.
Let’s explore why some angles are always equal.
Students need rulers and tracing paper from the geometry toolkits.
- If I have two parallel lines cut by a transversal, I can identify alternate interior angles and use that to find missing angle measurements.
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 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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