Notes — Chapter 7: Quadrilaterals

To the Student. ‘This page is rectangular, so are the doors you walk through and the floors you walk on. Rectangles, squares, parallelograms, and other quadrilaterals are found everywhere. In this chapter you will see that what you know about triangles and parallel lines will let you understand the geometry of quadrilaterals. Squares, rhombuses, parallelograms, and other quadrilaterals are used to design buildings and machines and in art. The chapter is organized around the types of quadrilaterals and their properties.

The dozen theorems in the body of this chapter catalog various sorts of quadrilaterals and the properties they must have. For your overview of this chapter, run through the types of quadrilaterals, drawing pictures of each type and reviewing its properties. The final lesson of the chapter gives you a new result about triangles that can be proven using what you have just learned. This result is the Midsegment Theorem, which tells you that the segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side. t

Lesson 1. Quadrilaterals. The chapter opening is an unusual shape for a quadrilateral. Jacobs chose this to sharpen the student’s appreciation of what is common and what is unusual in quadrilaterals.

In Chapter 1, Lesson 3, Polygons and Polyhedra, we defined quadrilaterals. We return to quadrilaterals with the machinery needed to develop their properties. Learning to see and identify the various sorts of quadrilaterals and to instinctively know their properties is the key to this chapter.

First, we distinguish between convex polygons and concave ones. Second, we observe that the Angle Sum Theorem for triangles lets us conclude that the angle sum for quadrilaterals in 360°. Hence, an equiangular quadrilateral has four right angles. This gives us the rectangle, the most useful quadrilateral.

Notice the discovery exercises in Set I: Rectangles (16- 18), Rhombuses (19 and 20), Squares (20 and 21), and Parallelograms (23-25). Here students begin to learn the names of various special quadrilaterals, to recognize them, and to see what properties they have. See Set II, Gemstone Patterns (36-49) and Linkage Problems (52-55), which are also exploratory. The amazing dissection in the Set ITI exercise was seen before in Lesson 3 of Chapter 1. The dimensions and angles in this dissection can be worked out starting from the fact that the area of both figures must be the same. This might be suitable for motivated students: perhaps working as a team, using technology such as Geometer’s Sketchpad™, or looking material up in the library.

Lesson 2. Parallelograms and Point Symmetry. We define parallelograms in this lesson and develop their properties, including their point symmetry. In the next lesson we will look at converse results, seeing which sets of these properties imply that a quadrilateral is a parallelogram. We return to symmetry from the point of view of transformations in the next chapter.

Set I. Optical Illusions (4-10) involving parallelograms are seen here. The point symmetry of parallelograms is proven in Exercises 16-23. Set II exercises begin with the parallelogram law for the addition of vectors. The Set III problem asks students to determine the symmetries of playing cards.

Lesson 3. More on Parallelograms. Here we see that quadrilaterals whose opposite angles are equal are parallelograms and so on. The lesson opens with a powerful visual illusion based on a room in which seeming rectangles turn out to be trapezoids. Note the way algebra and geometry work together in this lesson and in its exercises.

Set II. Rope Trick (36-43) gives a beautiful African method for laying out the rectangular base for a house. Flexible Grid (47-55) is intriguing. A surprising construction using equilateral triangles is the basis for the Set III exercise. See Alternating Triangles on page 565

of Geometry, and see these notes for hints on how to proceed.

Lesson 4. Rectangles, Rhombuses, and Squares. These are the figures you encounter most frequently in construction. We know a great deal about quadrilaterals, and the student sees how this knowledge can be put to use. See especially Exercises 26-39, Checking a Wall, which walk the student through the practical aspects of measuring the sides and diagonals of a wall to see whether it is rectangular.

Lesson 5. Trapezoids. These are less constrained: exactly one pair of sides of a trapezoid is parallel. Structure remains, particularly for isosceles trapezoids, whose nonparallel sides are of equal length. Trapezoids show up Stepladders (34-42) and in geometric Trademark (9- 13). Set III features a surprising solution to a classic problem. This solution was found by high school student Andrew Miller, while he was was working from the First Edition of this book.

Lesson 6. The Midsegment Theorem. The segment connecting the midpoints of two sides of a triangle is called a midsegment, and it is parallel to the third side and half its length. This is a new fact about triangles that follows from methods developed in this chapter. The Midsegment Theorem anticipates some ideas related to similar triangles that will be developed in Chapter 10.

Set II. The three-dimensional models in Tetrahedra (37-41) are worth special attention. See the discussion of these in the Answer section. In Exercises 27−36, Midpoint Coordinates, students discover the relationship between the geometric and coordinate views of midpoints. Base Average (49-54) begins by asking the student what seems to be true. Then we see how the natural conjecture can be proved. Set III looks at a geometric progression of similar triangles and asks the student to estimate the sum of the lengths of all their sides. See the Answers for comments and links to other material in the book.