Mathematics
Quadrilaterals
Quadrilaterals are four-sided polygons with various properties and classifications. Common types of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and rhombuses. They are characterized by their angles, side lengths, and diagonal properties, making them a fundamental concept in geometry.
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9 Key excerpts on "Quadrilaterals"
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Now a rectangle has many more properties than four congruent angles. However, mathematicians have discovered that this information—quadrilateral, four congruent angles—is sufficient so that only rectangles can be drawn that meet that criteria. Thus the notion of “determines” is an important one in mathematics. In elementary school we do not get terribly technical, but that is not the same as saying we just have fun and play around. When we ask children to explore well-focused questions, their understanding of shapes and relationships between and among shapes and their ability to see and apply properties can grow tremendously. When this happens, high school mathematics makes much more sense! Quadrilaterals We found that we could describe different kinds of triangles by looking at their angles or by looking at relationships among their sides. With Quadrilaterals, which have one more side, new possibilities for categorization emerge: parallel sides, adjacent vs. opposite sides, relationships between diagonals, and the notion of concave and convex. Thus, how we go about naming and classifying Quadrilaterals is not the same as how we name and classify triangles. In this book, we will define the following kinds of Quadrilaterals: • A trapezoid (Figure 8.75) is defined as a quadrilateral with at least one pair of parallel sides. • A parallelogram (Figure 8.76) is defined as a quadrilateral in which both pairs of opposite sides are parallel. • A kite (Figure 8.77) is defined as a quad-rilateral in which two pairs of adjacent sides are congruent. • A rhombus (Figure 8.78) is defined as a quadrilateral in which all sides are congruent. • A rectangle (Figure 8.79) is defined as a quadrilateral in which all angles are congruent. • A square (Figure 8.80) is defined as a quadrilateral in which all four sides are congruent and all four angles are congruent. An active reader may have noted that there are many other possible categories of Quadrilaterals.
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
CHAPTER OUTLINE 4.1 Properties of a Parallelogram 4.2 The Parallelogram and Kite 4.3 The Rectangle, Square, and Rhombus 4.4 The Trapezoid ■ PERSPECTIVE ON HISTORY: Sketch of Thales ■ PERSPECTIVE ON APPLICATIONS: Square Numbers as Sums ■ SUMMARY Chapter 4 © Richard A. Cooke/CORBIS. 169 Quadrilaterals Comforting! Designed by architect Frank Lloyd Wright (1867–1959), this private home is nestled among the trees in the Bear Run Nature Preserve of southwestern Pennsylvania. Known as Fallingwater, this house was constructed in the 1930s. The geometric figure that dominates the homes designed by Wright is the quadrilat-eral. In this chapter, we consider numerous types of Quadrilaterals— among them the parallelogram, the rhombus, and the trapezoid. Also, the language and properties for each type of quadrilateral are developed. Each type of quadrilateral has its own properties and applications. Many of these real-world applications can be found in the examples and exercises of Chapter 4. Additional video explanations of concepts, sample problems, and applications are available on DVD. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 170 CHAPTER 4 ■ Quadrilaterals Unless otherwise noted, all content on this page is © Cengage Learning. A quadrilateral is a polygon that has exactly four sides. Unless otherwise stated, the term quadrilateral refers to a plane figure such as ABCD in Figure 4.1(a), in which the line seg-ment sides lie within a single plane.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The sides, which are line segments, are called sides of the quadrilateral. Also, a quadrilateral has four angles—these are called angles of the quadrilateral. Finally, the vertex of each of the four angles is called a vertex of the quadrilateral. It is interesting to note that triangles are composed of three line segments and yet they are named according to the number of angles. Perhaps trilateral, meaning “three sides,” would have been a more consistent name for a triangle. Section 12.2 Analyzing Geometric Shapes—Level 1 571 Fold line m l (a) (b) (c) (d) m l m l l m Figure 12.26 We now describe the names of some of the Quadrilaterals that can be seen in our surroundings through the examples shown in Table 12.3. Please note that although we are using three-dimensional objects like a railing and a jack as examples of a par- allelogram and a rhombus, we are only focusing on the two-dimensional outline as shown in the picture. TABLE 12.3 MODEL NAME ABSTRACTION DESCRIPTION Railing Parallelogram Quadrilateral with two pairs of parallel sides. Car jack Rhombus (plural: rhombi) Quadrilateral with four sides the same length. Door Rectangle Quadrilateral with four right angles. Floor tile Square Quadrilateral with four sides the same length and four right angles. Just as with triangles, we can analyze Quadrilaterals by looking at lengths of sides, angles formed, or whether sides are parallel or perpendicular. It is these properties that help one determine the type of a given quadrilateral. Tracing and folding can be used to determine whether two line segments are paral- lel or perpendicular or to locate the midpoint of a segment. Thus, these techniques can be used to determine properties of Quadrilaterals. To determine whether two line segments l and m are parallel, we use the following test (Figure 12.26). 572 Chapter 12 Geometric Shapes Determine whether a rhombus, rectangle, and square have parallel sides.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Can you think of a figure that has three sides and three angles and is not a triangle? (Hint: Try a open figure.) Quadrilaterals There are many special names for quadri- laterals. Common names are parallelograms, rectangles, squares, rhombuses, trapezoids, and kites. These classes of Quadrilaterals are not disjoint; one shape may fall into several categories depending on the definition. For example, a rectangle is also a parallelogram. This type of classifying process is more difficult for children than partitioning the whole set into disjoint classes, as is the case of classifying triangles by lengths of sides. It requires more than just recognizing examples of figures; it requires understanding the defining attributes (Level 3). For example, a parallelogram is a quadrilateral with two pairs of parallel sides. Assuming 316 Chapter 15 • Geometry different-shaped pentagons they see (the ends of many barns). Buildings in cities often use wonderful geometric shapes. The names heptagon, nonagon, and decagon are not widely used, so in doing activities you may have to remind children of these names. H uses the names as well as other attributes; students are asked to see shapes within other shapes, a task that is difficult for some. The geometric design paper in the Appendix found at this book’s Web site may be used for similar searches. Hidden figures Children often have difficulty seeing fig- ures within other figures. In the 2007 mathematics assessment of NAEP, the following two items were given. Students in the fourth-grade assessment were provided three copies of the grid in Figure 15-3. They were asked to outline a square in each grid and were told that the squares must not be the same size. The other item was the same except the figures were to be triangles. The squares that are turned gave the students more difficulty than the two squares that are not turned.
eBook - ePub- Mark Ryan(Author)
- 2016(Publication Date)
- For Dummies(Publisher)
Part 4Polygons of the Four-or-More-Sided Variety
IN THIS PART … Get to know the many types of Quadrilaterals. Work on proofs about Quadrilaterals. Solve real-word problems related to polygons. Work on problems involving similar shapes.
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Chapter 10
The Seven Wonders of the Quadrilateral World
IN THIS CHAPTER Crossing the road to get to the other side: Parallel lines and transversals Tracing the family tree of Quadrilaterals Plumbing the depths of parallelograms, rhombuses, rectangles, and squares Flying high with kites and trapezoidsIn Chapters 7 , 8 , and 9 , you deal with three-sided polygons — triangles. In this chapter and the next, you check out Quadrilaterals, polygons with four sides. Then, in Chapter 12 , you see polygons up to a gazillion sides. Totally exciting, right?The most familiar quadrilateral, the rectangle, is by far the most common shape in the everyday world around you. Look around. Wherever you are, there are surely rectangular shapes in sight: books, tabletops, picture frames, walls, ceilings, floors, laptops, and so on.Mathematicians have been studying Quadrilaterals for over 2,000 years. All sorts of fascinating things have been discovered about these four-sided figures, and that’s why I’ve devoted this chapter to their definitions, properties, and classifications. Most of these Quadrilaterals have parallel sides, so I introduce you to some parallel-line properties as well.Getting Started with Parallel-Line Properties
Parallel lines are important when you study Quadrilaterals because six of the seven types of Quadrilaterals (all of them except the kite) contain parallel lines. In this section, I show you some interesting parallel-line properties.Crossing the line with transversals: Definitions and theorems
Check out Figure 10-1 , which shows three lines that kind of resemble a giant not-equal sign. The two horizontal lines are parallel, and the third line that crosses them is called a transversal.
eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
Keep your quadrilateral definitions handy, and check to see if the diagram makes sense to you. Here are some of the things it should tell you. Some Quadrilaterals are kites, some are trapezoids, and some are scalene Quadrilaterals. Some trapezoids are parallelo- grams, some are isosceles, and some are neither. Parallelograms that are also isosceles trapezoids are rectangles; those that are both isosceles trapezoids and rhombuses are squares. Not only are all rectangles parallelograms, but all of the proper- ties of parallelograms are true for rectangles. Two properties of par- allelograms are that the opposite sides are parallel and the diagonals bisect each other. Since rectangles and rhombuses are parallelograms, then they also have opposite sides that are paral- lel and diagonals that bisect each other. Note that I am using the definition of a trapezoid that says that at least one pair of the sides must be parallel. If we have two sides that are parallel, then it’s also a parallelogram. Some math books use a different definition in which exactly one pair of sides is parallel. —Dr. Math, The Math Forum 46 Dr. Math Introduces Geometry R esources on the Web Learn more about two-dimensional geometric figures at these sites: Math Forum: Ask Dr. Math: Point and Line mathforum.org/library/drmath/view/55297.html A point has no dimension (I’m assuming), and a line, which has dimension, is a bunch of points strung together. How does some- thing without dimension create something with dimension? Math Forum: Problems of the Week: Middle School: Back Yard Trees mathforum.org/midpow/solutions/solution.ehtml?puzzle=35 How many different Quadrilaterals can be formed by joining any four of the nine trees in my backyard? Math Forum: Problems of the Week: Middle School: Picture-Perfect Geometry mathforum.org/midpow/solutions/solution.ehtml?puzzle=97 Graph four points and name the figure that you have drawn.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
For example, is it possible to make an isosceles, right-angle triangle? A scalene, obtuse triangle? An equilateral, right-angle triangle? Justify. Children need to consider what defines a triangle and the connections between their properties. Here are some definitions that children have provided: ‘A triangle has three sides.’ ‘A triangle has three sides and three angles.’ ‘A triangle is a polygon with three sides.’ Although all three statements are true, only the last one defines the triangle. Can you think of a figure that has three sides and is not a triangle? Refer back to In the classroom 16.5. Quadrilaterals There are many unique names for Quadrilaterals, which at first can appear overwhelming to children. Common names are rectangles, squares, the rhombus, parallelograms, trapeziums and kites. These classes of Quadrilaterals are not disjoint; one shape may fall into several categories depending on the definition. For example, a rectangle is also a special parallelogram. This type of classifying process is more difficult for children than partitioning the whole set into disjoint classes, as is the case of classifying triangles by lengths of sides. It requires more than recognising examples of figures but a deeper understanding of the defining properties (Level 3). For example, a parallelogram is a quadrilateral with two pairs of parallel sides and opposite angles equal. Assuming you know that a quadrilateral is a four-sided, closed, simple figure, can you identify which of the following are parallelograms? What other names do they have? Yes, they are all parallelograms. Thus, a square, a rhombus and a rectangle are all special types of parallelogram. Discuss these questions with a peer. A rhombus is a parallelogram with all sides congruent (equal in length). Does that mean that a square is a rhombus? Explain why or why not. A rectangle is a special parallelogram with all right angles. Does that mean a square is a rectangle? Explain why or why not.
eBook - PDFCollege Geometry
A Unified Development
- David C. Kay(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
161 4 Quadrilaterals,. Polygons,.and.Circles The fundamental properties of polygons and circles commonly studied in elementary geometry may be derived axiomatically with relative ease. The purpose of the present chapter is to explore that derivation briefly in the context of unified geometry. The polygon has an important spe-cial case in the quadrilateral—the object of discussion for the opening sections. 4 .1. Quadrilaterals Of great importance in geometry, topology, and analysis is the concept of a polygonal path —a set consisting of those segments ( sides ) joining a sequence of distinct points ( vertices ). Our focus at first is on closed polyg-onal paths of order four (four-sided polygonal paths where the first and last vertices coincide). We might note that, if we are given four points, there are precisely three distinct closed polygonal paths of order four hav-ing those points as vertices, and the location of those points relative to each other falls into two classes: (1) when the paths determined by the points are not self-intersecting (called simple ), as shown in Figure 4.1, and (2) when some paths self-intersect, as shown in Figure 4.2. Since a closed polygonal path can enclose a well-defined interior only if it is not self-intersecting, we are only interested in paths that do not self-intersect. In the two figures there are only four of these, which are called quadrilat-erals : all three in Figure 4.1 and one in Figure 4.2. Figure 4 .1 Simple closed paths of order four. 162 College Geometry: A Unified Development Figure 4 .2 Closed paths of order four. Definition 1 Let P 0 , P 1 , P 2 , P 3 , P 4 be 5 distinct points in the plane such that for each integer i and j in the range [0, 4], P i P j < α . A polygo-nal path of order four joining P 0 and P 4 is the set [ ] P PP P P P P PP P P P P 0 1 2 3 4 0 1 1 2 2 3 3 4 = ∪ ∪ ∪ with the given points called its vertices , and the segments joining them, its sides (Figure 4.3).
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
uni2220NMQ _ uni2220PQM 7. uni25B3NMQ _ uni25B3PQM 8. MP _ QN 1. Given 2. By definition, a rectangle is a uni25B1 with a right angle 3. Opposite sides of a uni25B1 are _ 4. Identity 5. By Corollary 4.3.1, the four uni2220s of a rectangle are right uni2220s 6. All right uni2220s are _ 7. SAS 8. CPCTC A B D C Square ABCD Figure 4.22 200 CHAPTER 4 ■ Quadrilaterals Because a square is a type of rectangle, it has four right angles and its diagonals are congruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular; see Exercise 34. In Chapter 8, we measure area in “square units.” For the calculation of area, we count the number of congruent squares (square units) that fit inside a geometric region. THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhombus is rhombi (pronounced rho ˘m-bi ¯ ). While this definition requires only two congruent sides, the following corollary deter- mines that a square actually has four congruent sides. Corollary 4.3.3 All sides of a square are congruent. Corollary 4.3.4 All sides of a rhombus are congruent. Corollary 4.3.5 The diagonals of a rhombus are perpendicular. SSG EXS. 5–7 A rhombus is a parallelogram with two congruent adjacent sides. (See Figure 4.23.) DEFINITION In Figure 4.23, the adjacent sides AB and AD of rhombus ABCD are marked congru- ent. Because a rhombus is a type of parallelogram, it is also necessary that AB _ DC and AD _ BC . Thus, we have Corollary 4.3.4. We will use Corollary 4.3.4 in the proof of the following corollary. To visualize Corollary 4.3.5, see Figure 4.24(a). Discover What type of quadrilateral is deter- mined when an isosceles triangle is reflected across its base? ANSWER Rhombus EXAMPLE 2 Study the picture proof of Corollary 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS.
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