Perpendicular Transversals
Pacing:This lesson should take one class period
Goal: The goal of this lesson is to introduce students to the concept that parallel lines are equidistant from each
other and to prove lines parallel using the converse of the perpendicular to parallels theorem.
As students work through example 4, ask them to look at the slopes of the lines. Students should realize the slopes
are the same, thus they will never intersect. Have students create their own property describing this concept.
Additional Example:Have students place a ruler in any direction on a coordinate plane. Then, by tracing the top and
bottom of the ruler, the students will create parallel lines. Ask each student to find their equations for their personal
lines. Check with a partner to see if the equations are correct.
Non-Euclidean Geometry
Pacing:This lesson should take one class period
Goal:The purpose of this lesson is to extend students’ understanding of geometry beyond parallel and perpendicular
lines, angle pairs, and abstract drawings. Most students will enjoy this lesson due to the real life application.
However, even if students are unfamiliar with taxicabs, extend this lesson to rural areas with roads that intersect.
History Connection! Take time to discuss Euclid during the lesson. Show the following picture of his book,
Elements. Go through the first five postulates. Use the following website to gather additional information. Or,
have students write mini-reports of the impact Euclid had on present day Geometry. Offer “Euclid Day,” a day of
celebration on behalf of Euclid. The possibilities are endless!
Create a class discussion regarding Euclid’s 5thPostulate. “If two lines are cut by a transversal, and consecutive
interior angles have a total measure of less than 180 degrees, then the lines will intersect on that side of the
transversal.” Mathematicians tried to prove this true, thus making it a theorem as opposed to a postulate for 2000
years. Since many mathematicians did not regard this as truth, non-Euclidean geometries were founded.
Other types of non-Euclidean geometry are: spherical geometry, hyperbolic geometry and elliptic geometry. In
spherical geometry, straight lines are great spheres, so any two lines meet in two points. There are also no parallel
lines (think longitude lines meeting at the poles). Hyperbolic geometry satisfied all Euclid’s postulates except the
parallel postulate, replacing it with “For any infinite straight lineLand any pointPnot on it, there are many other
infinitely extending straight lines that pass throughPand which do not intersectL.” Elliptic geometry replaces
Euclid’s parallel postulate with “through any point in the plane, there exist no lines parallel to a give line.”
1.3. Parallel and Perpendicular Lines