Notes — Chapter 6: Parallel Lines

To the Student. Lines are parallel if they lie in the same plane and don’t intersect. Parallel lines should have the same direction. In particular, through a point P not on a given line £ there should be exactly one line parallel to the line £. Why? Because through the point P there should be exactly one line ℓ’ having the same direction as line. The Parallel Postulate tells us there as exactly one such line ¢’. With the Parallel Postulate we can characterize parallel lines by the angles formed by lines cutting across them, transversal lines. We will also prove the Angle Sum Formula for triangles and establish the Angle Angle Side (AAS) and Hypotenuse Leg (HL) congruence theorems. _ “

Lesson 1. Line Symmetry. Paper folds, linkages, and symmetrical art and diagrams motivate the study of line symmetry in this lesson. Two points A and B are symmetric with respect to the line ¢ iff @ is the perpendicular bisector of the segment AB. The construction of perpendicular bisectors and the proof of the correctness of this procedure is background for the construction of parallels in the next lesson. It also sets the stage for the contrast made in Lesson 3 between the uniqueness of perpendiculars, which we can already prove, and the uniqueness of parallels, which can only be proved by introducing the Parallel Postulate. Line symmetry is treated as a property of figures. In Lesson 3 of Chapter 8, we will revisit line symmetry in terms of reflections and transformations.

The Set III exercise asks the students discover some properties of an amazing set of points in the plane. See the discussion given with Jacobs’s answers.

Lesson 2. Proving Lines Parallel. Parallel lines were defined earlier in Lesson 7 of Chapter 3. Here we see that the Exterior Angle Theorem allows us to conclude that two lines in a plane that make equal corresponding angles with a transversal cannot intersect. This gives us Theorem 17: Equal corresponding angles mean that lines are parallel. This statement is given in a simple and general form. Each application of this result requires students to think through the geometry and recognize the corresponding angles so that the basic idea and its uses are gone over repeatedly. The same is true for the corollaries given in this lesson. The exercises drive home these ideas. The converse of Theorem 17 is true, but its proof comes in Lesson 4 after the Parallel Postulate has been given in Lesson 3.

Lesson 3. The Parallel Postulate. Given a line é and a point P not on @ we know how to construct a line n through P parallel to. We also can construct an m which is perpendicular to @ and passes through P.

For perpendiculars, we can prove that there is exactly one such line m. This lesson asks the question: Can we prove that there is only one such line n, when n line is to be parallel to ℓ rather than perpendicular to ℓ2? The answer is: Not without a further assumption. This is further explored in Chapter 16, Non-Euclidean Geometries, but the assumption we need, the Parallel Postulate, is introduced here.

Set I. Exercises 11-18, Exactly One, focus on the meanings of “at least,” “no more than,” and “exactly.” Gateway (37-41) raises questions about what we know about figures and what is yet to be proven. Windshield wipers, gateways and paper folding are among the examples used in the exercises.

▶ About The Parallel Postulate. It took more than two thousand years for geometers to discover that the Parallel Postulate, or equivalent statements, could not be proven from the other axioms of Euclid’s system. The discovery of non-Euclidean geometries shook the faith people had in fixed systems of knowledge. The words of Morris Kline at the beginning of his chapter on non-Euclidean Geometries in Mathematical Thought from Ancient to Modern ‘Times, Oxford University Press, 1972, gives a sense of this. Kline wrote:

“Amidst all the complex and technical creations of the nineteenth century the most profound one, non- Euclidean geometry, was technically the simplest. This creation gave rise to important new branches of mathematics but its most significant implication is that it obliged mathematicians to revise radically their understanding of the nature of mathematics and its relation to the physical world.”

Where does this leave us? For one view, read the conclusion to Jan Gullberg’s section on non-Euclidean geometries in Mathematics: From the Birth of Numbers: “… Euclidean geometry is still the basis for most practical applications of geometry — it has taken human beings to the Moon and beyond.”

Lesson 4. Parallel Lines and Angles. The naturalist Jean Henri Fabre’s analysis of the geometry of a spider’s web leads in to this lesson on the converses of Theorem 17 and its corollaries. Now that we have the Parallel Postulate these converses can be proven. This lesson provides a natural opportunity to go over the distinction between a statement and its converse, both in the text and exercises.

The exercises include: proving that parallel lines are everywhere equidistant, an anticipation of the proof of the Angle Sum Theorem for triangles, and an exploration of the methods Kepler used in working out the path of Mars.

Lesson 5. The Angles of a Triangle. Over distances that are short compared to the earth’s radius, the sum of the angles in a survey triangle will be 180°. This is used as a check for measurement errors. This

idea opens our lesson. We prove the Angle Sum Theorem for triangles — a result announced in Lesson 6 of Chapter 2. We also prove Theorem 21: An exterior angle of a triangle is equal to the sum of the remote interior angles. This is a strengthening of Theorem 12, the. Exterior Angle Theorem found in Lesson 2 of Chapter 5. Useful corollaries include: If two angles of a triangle are equal to two angles of another triangle, then angles of the third pair are also equal and The acute angles of a right triangle are complementary.

The exercises include proofs of the corollaries in the lesson and an explanation of how the latitude of a town in the northern hemisphere can be determined by sighting the North Star.

Lesson 6. AAS and HL Congruence. A fractal construction using triangles introduces two new methods for proving that two triangles are congruent: the Angle Angle Side method (AAS, Theorem 22) and the Hypotenuse Leg method for right triangles (HL, Theorem 23). The proofs are immediate consequences of the Angle Sum Theorem of Lesson 5. This lesson gives an opportunity to review the methods of showing triangles congruent and to consider the questions of how many parts of two triangles can be equal without the triangles being congruent. This problem was looked at in the exercises for Lesson 6, SSS Congruence, of Chapter 4.

Set I. Triangle Problem (18-22) shows that a triangle is isosceles if its altitudes are equal. The Set II exercises begin by establishing the properties of mirror images, and Set III unveils the deceptions of the fallacious proof that all triangles are isosceles.