Unit: Parallel and Perpendicular Lines (Geometry Level 3)
Unit Rationale
Prior to being taught this unit, the students should have learned about the basic elements of geometry: points, lines, planes, segments, and angles. They should have the ability to sketch points, lines, planes, and their intersections, measure segments and add their lengths, and measure and classify angles. Additionally, students should be able to analyze segment and angle bisectors, identify complementary, supplementary, vertical, and linear pairs of angles, and use properties of equality and congruence to support mathematical statements. It is necessary for students to have this background knowledge prior to learning this unit about parallel and perpendicular lines. Students must now apply their knowledge of basic elements of geometry in order to develop understanding about parallelism and perpendicularity, two essential relationships between pairs of lines.
Parallel and Perpendicular Lines is a fundamental unit in geometry because it serves as a basis for the remainder of the concepts studied throughout the course. This unit should be taught near the beginning of the course in order to provide the students with a solid foundation in geometry. The students will utilize the concepts and skills learned in this unit to identify relationships among and between various polygons and circles. As the students learn the process of making conjectures about all elements of geometry, they will need to justify their statements using ideas learned in this unit.
Toward the end of the unit, students are introduced the process of writing
two-column proofs.
Traditionally, teachers wait until after students
have learned about triangle relationships to introduce the idea of proofs.
However, since this topic can often be intially
confusing and overwhelming when seen by students, it is beneficial to provide a
basic introduction to proofs prior to being taught additional material.
After learning about parallelism and
perpendicularity, students are well-equipped with knowledge to form two-column
proofs about the relationships between lines and angles.
Allowing students to develop the skill of using
“reasoning and … proof to very or refute conjectures and theorems” (NJ Core
Curriculum Content Standard 4.2A) early in this course will ideally assist them
in becoming more adept when it is necessary to draw on more knowledge and write
longer, more complicated proofs later in the course.
Unit Overview
This unit will be introduced through the first lesson (3.1), which focuses on the concepts of parallel, perpendicular, and skew lines. Students will be taught how these lines are similar and how they are different. It should be emphasized that students cannot simply infer lines are parallel or perpendicular. Lines can only be categorized if these facts are stated, triangles are present to represent parallelism, or right angle symbols are present to represent perpendicularity. Students will also be introduced to parallel planes, and the relationship between parallel planes, parallel lines, and lines perpendicular to planes. The difference between lines and planes should be discussed in terms of two-dimensional and three-dimensional space. After learning these ideas, the students will be taught theorems about perpendicular lines (3.2). They will learn four theorems. These theorems will allow the students to draw conclusions about all perpendicular liens. Similarly, students will be able to identify perpendicular lines, given certain information about angles. It is important that students learn to identify given information, and support statements about perpendicular lines and/or angles using the theorems learned in the lesson.
The following two lessons focus on the angles formed when lines are cut by transversals (3.3/3.4). Students will learn to identify corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles. They will discover the properties of these angles, formed by parallel lines and transversals. The key concepts of these lessons are determining the relationships between pairs of each type of angles. Students should learn that corresponding angles, alternate interior angles, and alternate exterior angles are congruent when parallel lines are cut by a transversal, but same-side interior angles are supplementary when parallel lines are cut by a transversal. These ideas are summarized in the Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, and Same-Side Interior Angles Theorem. The next lesson introduces the students to the converse of an if-then statement, and the idea that converses of if-then statements are not always true (3.5). The converses of the postulates learned in the previous two lessons are presented. Students will now learn how to conclude whether two lines are parallel, based on the relationships between corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles. After these three lessons have been taught, students should learn how to make and support conjectures regarding angles formed by parallel lines cut by transversals, and parallel lines given specific information about various pairs of angles.
The students will then learn theorems that extend the concepts of parallelism and perpendicularity to more than two lines (3.6). Through the Parallel Postulate and Perpendicular Postulate, they will learn the uniqueness of a parallel line and perpendicular line to another line, drawn through a given point. Students will also understand the relationship between three parallel lines and the relationship between two lines perpendicular to the same line, in this lesson. They will use this information to draw conclusions about given information, and show how two lines are parallel.
Finally, students will be introduced to the process of writing two column proofs using their knowledge of parallel and perpendicular lines learned in this unit. Students will be taught the basic characteristics of two-column proofs. Based on their knowledge, they will then have to identify information given in diagrams, and draw conclusions about information that is not given. They will apply their knowledge in order to form logical arguments to prove various statements.
Unit Essential Questions
- How are parallel, perpendicular, and skew lines similar? How are they different?
- If two lines are perpendicular, what conclusions can be drawn?
- If parallel lines are cut by a transversal, what are the angle relationships?
- When is it appropriate to use the Corresponding (Alternate Interior, Alternate Exterior, Same-Side Interior) Angles Converse? How are these different from the original postulates and theorems about such angles?
- If given three parallel lines, what conclusions can be drawn?
- How could perpendicular lines be used to show that two lines are parallel?
- What must be true of every proof in geometry?