Notes — Chapter 3: Lines and Angles

To the Student. Intersecting pairs of lines form angles, and pairs of points define segments. We measure angles in degrees and intervals by their lengths, using various units. When you compare equal segments using a ruler, you could as well have copied the first segment onto the edge of the ruler and moved that copy over the other segment, matching it exactly. This matching is geometric equality, called congruence. A protractor lets you compare angles in the same way.

We begin with the algebraic tools used in geometry. Next come the Ruler and Protractor Postulates and the ideas about measuring distances and angles sketched above. We will cover complementary and supplementary angles, linear pairs, vertical angles, perpendicular and parallel lines, among other ideas. _ “

Lesson 1. Number Operations and Equality. This covers the transitive property of equality and “common notions of Euclid” such as “If equals be added to equals, the results will be equal.” These are part of algebra, and students will have seen them, but possibly in different forms. Here they are, with examples of their geometric uses. This lesson is an anchor for the algebra strand in this book. An algebraic puzzle from Lewis Carroll wraps up the lesson.

Lesson 2. The Ruler and Distance. Numbers and algebra do much of the work in our geometry. To move ahead we need the Ruler Postulate and the betweenness of points properties of this lesson. We have the theorem: If B lies between A and C on line /, then AB + BC = AC. The converse of this theorem follows from the triangle inequality: AB + BC > AC, unless B lies in the interval from A to B. We will prove this Chapter 5, on inequalities.

Lesson 3. The Protractor and Angle Measure. The Protractor Postulate does for angles what the Ruler Postulate did for the lengths of segments. There is no natural scale for lengths, but a half-rotation and a right angle are natural units for angles. When Euclid stated his Fourth Postulate, “All right angles are equal,” he was stating that angles that have the same measures in terms of halves or quarters of a circle are geometrically the same (congruent). Set III introduces the Tantalus problem. The key to the Tantalus problem lies in the figure for the Angles Problem on page 515.

Lesson 4. Bisection. Rulers and protractors allow us to divide segments or angles into any desired proportions. Such division can be done using algebra, with a calculator or computer software to carry out the details. Geometrical bisection is based on congruence, as we will see in Chapter 4. It also brings in ideas of reflection and symmetry that we will revisit in Chapter 5. A fold, not a ruler, gives you the quickest way to divide a piece of paper in half. Applications include origami, molecular

structures, carpentry, and the shapes of flowers. This lesson and its exercises revisit earlier ideas, including measurement, the use of coordinates, and constructions.

Lesson 5. Complementary and Supplementary Angles. These ideas are algebraic. Angles A and B are complementary if A+B = 90°, they are supplementary if A +B = 180°. Angles supplementary (or complementary) to the same angle are equal. The exercises give practice in algebra, logical reasoning, and seeing geometric configurations.

Lesson 6. Linear Pairs and Vertical Angles. Linear pairs are a natural geometric setting for applying the ideas of supplementary angles. The Protractor Postulate lets us prove that vertical angles are equal using algebra. Similar reasoning is used in the exercises to work out angles in folding chairs and in garden shears. In Set III, students figure out a proof from a Chinese geometry book.

Lesson 7. Perpendicular and Parallel Lines. Here we encounter two key concepts. If one of the four angles formed by intersecting lines is a right angle, then the lines are perpendicular and all four angles are right angles. Parallel lines lines lie in the same plane but do not intersect. Already, students should have the idea that parallel lines have the same direction. Indeed, this is what the Parallel Postulate of Chapter 6 will tell us.