Mathematics
Special quadrilaterals
Special quadrilaterals are specific types of four-sided shapes with unique properties. Examples include squares, rectangles, rhombuses, and trapezoids. Each type has distinct characteristics, such as equal sides and right angles for squares, opposite sides of equal length and parallel for rectangles, and opposite sides of equal length and parallel with adjacent sides of equal length for rhombuses.
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10 Key excerpts on "Special quadrilaterals"
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)
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. Through the centuries, definitions of shapes have changed. For example, once there were oblongs and squares, but the shape name rectangle is now used for a quadrilateral with four right angles, which is not a square. It is best not to use the term oblong in geometry, as it can be confused with other real-world concepts and applications which are mathematically different. Helping children learn spatial relationships and classifications takes time. Children first begin by verbalising properties of shapes. To build a deeper understanding, they need to visualise shapes and their properties according to Seah (2015). By engaging in paper-folding activities (of triangles, squares and polygons), children will be able to view two-dimensional shapes from different orientations. In particular, the properties of quadrilaterals according to their length of side and size of angles will become much clearer. For example, a square can be described as: • a closed, four-sided figure (property 1) • having opposite sides parallel (property 2) Pdf_Folio:566 566 Helping children learn mathematics • having all right angles (property 3) • having all sides congruent, equal in length (property 4). Properties 1 and 2 make it a parallelogram; properties 1, 2 and 3 make it a rectangle; properties 1, 2 and 4 make it a rhombus; properties 1, 2, 3 and 4 make it a square. Additionally, a square has the rare property that diagonals bisect each other at 90 degrees (also found in a kite and rhombus). In the classroom 16.12 helps children with this idea.
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 - 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 - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Society Publishing(Publisher)
It’s all of the four sides amount the equal length but, unsimilar as the case of a rectangle, any of all four angles measure 90 ° (Figure 4.7). Perimeter and Area 81 Figure 4.7: An example of a rhombus. Source: https://live.staticflickr.com/3889/14781242232_a5ca9113cb_b.jpg. 4.5.5. The Square The square is a type of rectangle, but also a type of rhombus. It has features of both of these. That is to say, all four angles are right angles, and all of the four sides are same in length (Figure 4.8). Figure 4.8: An example of a square. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/ Square_-_black_simple.svg/1024px-Square_-_black_simple.svg.png. A square is a four-sided figure which is formed with the help of linking all of the 4-line segments. The line segments in the square are all of the same lengths and they come together to create four right angles. 4.5.6. The Trapezoid The trapezoid also has four sides in totality. It has two sides that are parallel but the other two are not of the same length (Figure 4.9). Introduction to Mathematical Literacy 82 Figure 4.9: An example of Trapezoid. Source: https://live.staticflickr.com/8110/8533583517_c1772b0fb1.jpg. 4.5.7. The Polygon Another in the geometric shapes that any individual requires to study or understand about is a shape of the polygon. A polygon is created of only lines and these are having no curves. It may not have any open parts. In this case, a polygon is fundamentally a broader term to various numbers of types of shapes, for example, a square can be a polygon, can be a triangle, and can be a rectangle as well (Figure 4.10). Figure 4.10: An example of polygon. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Hexa-gon.svg/2000px-Hexagon.svg.png. Perimeter and Area 83 4.5.8. The Parallelogram A parallelogram is another shape in the field of geometric shapes in which the opposite side of the shape is parallel.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
All of the special types of quadrilaterals in this section must be convex. However, it is only in the definition of a kite that this needs to be specified. For each of the other quadrilaterals (rectangle, square, trapezoid, rhombus, and parallelogram), the given definition can be used to prove that the quadrilateral must be convex. Classroom Tip Definition of a Kite A kite is a convex quadrilateral that has exactly two pairs of adjacent sides that are congruent. Draw a quadrilateral that is not a kite, but still has exactly two pairs of adjacent sides that are congruent. SOLUTION Notice that both of the quadrilaterals at the right have exactly two pairs of adjacent sides that are congruent. However, the quadrilateral at the far right is not a kite because it is concave. Kite Not a kite Copyright 2014 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. Section 10.2 Quadrilaterals 381 There are many properties of quadrilaterals. The example below uses paper folding to investigate properties about the diagonals of special types of quadrilaterals. EXAMPLE 10 Exploring the Diagonals of Quadrilaterals Use paper folding to investigate whether the diagonals of rectangles, squares, parallelograms, and rhombuses intersect at right angles. SOLUTION Begin by drawing a representative quadrilateral and its two diagonals. Then cut out the quadrilateral and fold it along one of the diagonals. Determine whether the diagonals intersect at right angles. Quadrilateral Fold on Diagonal Rectangle Conclusion The diagonals of a rectangle do not necessarily intersect at right angles.
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 - PDFGeometry
A Self-Teaching Guide
- Steve Slavin, Ginny Crisonino(Authors)
- 2004(Publication Date)
- Wiley(Publisher)
Squares, rectangles, trapezoids, and parallelograms are all quadrilaterals. And a quadrilateral, as you may recall from chapter 1, is a four-sided polygon. Squares As you might well know, a square contains four right angles (angles of 90°) and has four sides of equal length. s s s s 126 GEOMETRY Perimeter of a square formula: P = 4s, where s stands for the length of a side. Here’s a nice easy one: Find the perimeter of the following square. P = 4s = 4(2) = 8. The perimeter is 8 feet. Now here’s a challenge. Find the area, A, of this same square. Area of a square formula: A = s 2 A = s 2 = 2 2 = 4. The area is 4 square feet. Rectangles A rectangle contains four right angles (90°), a pair of equal lengths, and a pair of equal widths. Perimeter of a rectangle formula: P = 2l + 2w Using this formula, find the perimeter of the following rectangle. P = 2l + 2w = 2(4) + 2(6) = 8 + 12 = 20. The perimeter is 20 inches. w = 4 inches = 6 inches w w s = 2 feet Perimeter and Area of Two-dimensional Polygons 127 Area of a rectangle formula: A = lw Go ahead and find the area of this rectangle. A = lw = 6(4) = 24. The area is 24 square inches. Example 1: Find the area of a rectangle if the length is 15 and the perimeter is 50. Solution: To find the area, first we have to find the width. We’ll use the formula for perime- ter to find the width. P = 2l + 2w Substitute 15 for l and 50 for P. 50 = 2(15) + 2w 50 = 30 + 2w Subtract 30 from both sides of the equation. 20 = 2w Divide both sides of the equation by 2. 10 = w The width is 10. Now that we know the width, we can substitute its value into the formula for area. A = lw A = 15(10) = 150 Example 2: Find the perimeter of a rectangle if its width is 9 inches and its area is 117 square inches. Solution: To find the perimeter, we have to find the length of the rectangle. We know the area, so we’ll start by using the area formula to find the length.
eBook - PDFPrimary Mathematics
Integrating Theory with Practice
- Penelope Serow, Rosemary Callingham, Tracey Muir(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
He very quickly told the teacher the name of all the quadrilaterals. When the teacher asked Peter to describe the shapes he replied: The rectangle is a stretched out square, the parallelogram is a pushed over rectangle, the rhombus is a pushed over square, the kite is a stretched diamond and the trapezium is a rectangle with bits chopped off. The use of the word ‘diamond’ is an example of the use of inappropriate language that hinders students’ progression in identifying the properties of figures. Teachers should take every opportunity to explore the same figures in different orientations to reinforce the notion that a change in position does not alter the properties of a figure. It is not appropriate to use ‘diamond’ for a square in a specific orientation. In fact, there is no need to use ‘diamond’ for the square or the rhombus. After exploration of the properties of 2D figures, it is useful to ask students to complete a table where they choose all the properties that belong to each class of figures. Teachers in the primary years may have the rewarding experience of teaching students who begin describing classes of quadrilateral figures with sub-sets. Figure 4.6 shows an actual higher-order student response to Activity 4.6. Note the inconsistent response where the student states that the ‘rhombus is a special square’. 86 PRIMARY MATHEMATICS ACTIVITY 4.6 Draw a tree diagram that shows how all the quadrilaterals are related to each other. You must justify the groups and links that you make. There are many interactive online games that enable manipulation of figures in an engaging environment. HOTmaths has a growing number of interactive games that can act as catalysts for further classroom discussion of geometric ideas. The first of these, named Peanut Bridge, requires the students to rotate and alter the sizes of shapes to complete the bridge so that a trio of elephants can cross it safely (see Figure 4.7).
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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