Mathematics
Similar and Congruent Shapes
Similar shapes have the same shape but can be different sizes, while congruent shapes are the same shape and size. In mathematics, similarity refers to having the same shape but not necessarily the same size, while congruence refers to having both the same shape and size. These concepts are important in geometry and are used to compare and classify shapes.
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10 Key excerpts on "Similar and Congruent Shapes"
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)
Note that the symbol for con-gruence combines the symbols for similarity and equality; that is, congruent polygons always have the same shape and the measures of corresponding parts are equal. ( ) ( ) ( ) . Similar Polygons Congruent Polygons Corresponding Vertices, Angles, and Sides KEY CONCEPTS Similar Polygons 5.2 (a) A C B (b) D F E Figure 5.3 Two congruent polygons are also similar polygons. While two-dimensional figures such as and in Figure 5.3 can be sim-ilar, it is also possible for three-dimensional figures to be similar. Similar orange juice con-tainers are shown in Figures 5.4(a) and 5.4(b) on page 218. Informally, two figures are “similar” if one is an enlargement of the other. Thus, a tuna fish can and an orange juice can are not similar, even if both are right-circular cylinders [see Figures 5.4(b) and 5.4(c) on page 218.] We will consider cylinders in greater detail in Chapter 9. DEF ABC 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. Figure 5.4 In this chapter, the discussion of similarity will generally be limited to plane figures. For two polygons to be similar, it is necessary that each angle of one polygon be con-gruent to the corresponding angle of the other. However, the congruence of angles is not sufficient to establish the similarity of polygons. The vertices of the congruent angles are corresponding vertices of the similar polygons. Consider Figure 5.5. If in one poly-gon is congruent to in the second polygon, then vertex A corresponds to vertex H , and this is symbolized ; we can indicate that corresponds to by writing .
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
A road map is similar to the territory that it represents and a photograph is similar to the object that is photographed. In plane geometry, proving or verifying that certain angles are equal and certain sides of two or more figures are equal or proportional is based on congruence or similarity. 23–1 CONGRUENT FIGURES Congruent figures have exactly the same size and shape. If congruent plane figures are placed on top of each other, the figures coincide or fit exactly. The symbol ù means congruent. Corresponding parts of congruent triangles are equal. The corresponding parts of triangles are corresponding sides and corresponding angles. If congruent triangles are placed one on the other (superimposed), the parts that fit exactly are corresponding. Corresponding Sides of Congruent Triangles The sides that lie opposite equal angles of congruent triangles are corresponding sides . EXAMPLE Refer to Figure 23–1. Triangles ABC and DEF are congruent triangles. Determine the corresponding equal sides. UNIT 23 CONGRUENT AND SIMILAR FIGURES Copyright 2019 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. Unit 23 CONGRUENT AND SIMILAR FIGURES 581 A C F E D 80 8 38 8 62 8 B 80 8 38 8 62 8 Figure 23–1 AB and DE both lie opposite 62 8 angles. AB 5 DE Ans BC and EF both lie opposite 38 8 angles. BC 5 EF Ans AC and DF both lie opposite 80 8 angles. AC 5 DF Ans Corresponding Angles of Congruent Triangles The angles that lie opposite equal sides of congruent triangles are corresponding angles . EXAMPLE Refer to Figure 23–2. Triangles MNP and RST are congruent triangles.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly stretching the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Here we assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar. Similar triangles To understand the concept of similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale. ________________________ WORLD TECHNOLOGIES ________________________ If you were to draw a map, you would probably try to preserve the shape of what you are mapping, while you would make your picture at a unit rate that is in proportion to the original size or value. In particular, similar triangles are triangles that have the same shape and are up to scale of one another. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserve their measures. Formally speaking, we say that two triangles and are similar if either of the following conditions holds: 1.
No longer available |Learn moreCollege Geometry
A Unified Approach
- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Congruence An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither similar nor congruent to any of the others. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The unchanged properties are called invariants. In geometry, two figures are congruent if they have the same shape and size. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. The related concept of similarity permits a change in size. ________________________ WORLD TECHNOLOGIES ________________________ Definition of congruence in analytic geometry In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping. A more formal definition: two subsets A and B of Euclidean space R n are called congruent if there exists an isometry f : R n → R n (an element of the Euclidean group E ( n )) with f ( A ) = B . Congruence is an equivalence relation. Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size. If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as: In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
We will consider cylinders in greater detail in Chapter 9. KEY CONCEPTS Similar Polygons Congruent Polygons Corresponding Vertices, Angles, and Sides 5.2 Similar Polygons Two congruent polygons are also similar polygons. Figure 5.3 (a) A C B (b) D F E Exercises 42, 43 230 CHAPTER 5 ■ SIMILAR TRIANGLES O . J . 6 o u n c e s O . J . 6 o u n c e s O . J . 1 6 o u n c e s O . J . 1 6 o u n c e s T U N A T U N A (a) (b) (c) Figure 5.4 In this chapter, the discussion of similarity will generally be limited to plane figures. For two polygons to be similar, it is necessary that each angle of one polygon be congruent to the corresponding angle of the other. However, the congruence of angles is not sufficient to establish the similarity of polygons. The vertices of the congruent angles are corresponding vertices of the similar polygons. Consider Figure 5.5. If uni2220A in one polygon is congruent to uni2220H in the second polygon, then vertex A corresponds to vertex H, and this is symbolized A 4 H; we can indicate that uni2220A corresponds to uni2220H by writing uni2220A 4 uni2220H. A pair of angles like uni2220A and uni2220H are corresponding angles, and the sides determined by consecutive and corresponding vertices are corresponding sides of the similar polygons. For instance, if A 4 H and B 4 J , then AB corresponds to HJ . EXAMPLE 1 Given similar quadrilaterals ABCD and HJKL with congruent angles as indicated in Figure 5.5, name the vertices, angles, and sides that correspond to each other. (a) B C A D (b) H L J K Figure 5.5 SOLUTION Because uni2220A _ uni2220H, it follows that A 4 H and uni2220A 4 uni2220H. Similarly, B 4 J and uni2220B 4 uni2220J C 4 K and uni2220C 4 uni2220K D 4 L and uni2220D 4 uni2220L Associating pairs of consecutive and corresponding vertices of similar polygons, we determine the endpoints of the corresponding sides.
- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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. CHAPTER 9 • An Introduction to Geometry 750 OBJECTIVE 1 Identify corresponding parts of congruent triangles. Simply put, two geometric figures are congruent if they have the same shape and size. For example, if ^ABC and ^DEF shown below are congruent, we can write ^ABC H11061 ^DEF Read as “Triangle ABC is congruent to triangle DEF.” A C B D F E One way to determine whether two triangles are congruent is to see if one triangle can be moved onto the other triangle in such a way that it fits exactly. When we write ^ABC H11061 ^DEF , we are showing how the vertices of one triangle are matched to the vertices of the other triangle to obtain a “perfect fit.” We call this matching of points a correspondence. ^ABC H11061 ^DEF A 4 D Read as “Point A corresponds to point D.” B 4 E Read as “Point B corresponds to point E.” C 4 F Read as “Point C corresponds to point F.” When we establish a correspondence between the vertices of two congruent triangles, we also establish a correspondence between the angles and the sides of the triangles. Corresponding angles and corresponding sides of congruent triangles are called corresponding parts. Corresponding parts of congruent triangles are always congruent. That is, corresponding parts of congruent triangles always have the same measure. For the congruent triangles shown above, we have m(H11028A) H11005 m(H11028D) m(H11028B ) H11005 m(H11028E ) m(H11028C ) H11005 m(H11028F ) m(BC ) H11005 m(EF ) m(AC ) H11005 m(DF ) m(AB ) H11005 m(DE ) Congruent Triangles Two triangles are congruent if and only if their vertices can be matched so that the corresponding sides and the corresponding angles are congruent.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Describe how the definition of similar differs when used in mathematics and when used in real life. Give an example of how two objects might be similar in real life but not in mathematics. 22. Writing Identify two careers that use similar figures. Explain how similar figures are used. 23. Grading Student Work On a diagnostic test, one of your students does the following work. Explain what the student did wrong. Which topics would you encourage the student to review? The two triangles are similar because they are both right triangles. 24. Grading Student Work On a diagnostic test, one of your students does the following work. Explain what the student did wrong. Which topics would you encourage the student to review? The two triangles are similar because of the SSS Similarity Theorem. 6 3 8 6 4 3 25. Activity: In Your Classroom Cut an isosceles right triangle with legs at least one foot long out of cardboard. Put the triangle on the floor in front of a desk and use a string to align the hypotenuse of the triangle to the top of the desk, as shown in the figure. 45° A B C a. Measure the legs of △ABC. b. Compare the ratio of the legs of the cardboard triangle with the ratio of the legs of △ABC. c. Describe the concepts your students will practice and discover by doing this activity. 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 13.2 Similarity 513 26. Activity: In Your Classroom To find the height of your classroom wall using the triangle and desk in Exercise 25, put the triangle on the desk.
eBook - PDF- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
a. 6 b. 11.25 7. 500 ft SECTION STUDY SET 9.5 VOCABULARY Fill in the blanks. 1. triangles are the same size and the same shape. 2. When we match the vertices of ^ABC with the vertices of ^DEF , as shown below, we call this matching of points a . A 4 D B 4 E C 4 F 3. Two angles or two line segments with the same measure are said to be . 4. Corresponding of congruent triangles are congruent. 5. If two triangles are , they have the same shape but not necessarily the same size. 6. A mathematical statement that two ratios (fractions) are equal, such as x 18 5 4 9 , is called a . CONCEPTS 7. Refer to the triangles below. a. Do these triangles appear to be congruent? Explain why or why not. b. Do these triangles appear to be similar? Explain why or why not. Copyright 2019 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. CHAPTER 9 • An Introduction to Geometry 854 14. Name the six corresponding parts of the congruent triangles shown below. S T E 3 in. 4 in. 4 in. 3 in. 5 in. 5 in. R F G Fill in the blanks. 15. Two triangles are if and only if their vertices can be matched so that the corresponding sides and the corresponding angles are congruent. 16. SSS property: If three of one triangle are congruent to three of a second triangle, the triangles are congruent. 17. SAS property: If two sides and the between them in one triangle are congruent, respectively, to two sides and the between them in a second triangle, the triangles are congruent.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
567 10 SECTION 10.1 Congruence Transformations SECTION 10.2 Symmetry and Tessellations SECTION 10.3 Similarity Geometry as Transforming Shapes In Chapter 9 we focused on understanding measurement of shapes, and in this chapter, we are going to look at what happens when we transform shapes. This will help us better understand the patterns that emerge when shapes are put together, as in quilts, floor patterns, art, and other situations. Look at the pictures in Figures 10.1 to 10.3. What do you see? What did the artist who created these have to think about? Note your ideas before reading on. Figure 10.1 Azat1976/Shutterstock.com Figure 10.2 Margrit Hirsch/Shutterstock.com Figure 10.3 OKing/Shutterstock.com 568 CHAPTER 10 Geometry as Transforming Shapes n n Transformations Before we get into the mathematical aspect of transformations, what do you think when you hear this word? The interesting examples offered by my students include the transformation of a frog into a prince in fairy tales and the transformation of a caterpillar to a butterfly. You are in the process of transforming from a student to a teacher. Virtually all examples of trans- formation give a sense of movement and change. In mathematics, there are many kinds of transformations. Explorations Manual 10.1, 10.2, 10.3 What do you think? l Where have you heard the words translation, reflection, and rotation in everyday life? l How might you describe the translation, reflection, or rotation of a three-dimensional figure in space? l How are the operations translation, reflection, and rotation like the operations addition, subtraction, multiplication, and division? Congruence Transformations SECTION 10.1 INVESTIGATION 10.1a How Do Things Move? A. Take an object such as your cell phone, a tangram triangle, an index card, and so on, and explore all the different ways you can move that object. Describe each unique type of movement and give it a name that makes sense to you.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
For example, many representations are geomet- ric in nature—models for fractions, area models for multiplication, and patterns that lead to algebraic expressions. • Geometry is closely connected to other subjects. The German Durer, the author of the opening quote, understood that knowing geometry was essential to him as an artist as do the two architects shown in Tech Connect 15.1. Tech Connect 15.1 Two female landscape architects show an award- winning project that uses many geometry terms men- tioned in this chapter. You can access this from www.the futureschannel.com/dockets/hands-on_math/landscape_ architects/index.php or from this book’s Web site. www.wiley.com/college/reys TABLE 15-1 • Overview of Standards Relating to Geometric Shape Concepts Geometry Summary Kindergarten Identify, describe, and name common three- and two-dimensional shapes. Describe relative position of objects. Compare, create, and compose shapes. Grade 1 Focus is on attributes: distinguish between defining attributes (e.g., triangles are closed and have three sides) versus non-defining attributes (e.g., orientation and color); build and draw shapes to possess defining attributes. Compare shapes and compose shapes to make other shapes (e.g., two triangles to make square). Grade 2 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of faces. Grade 3 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Grade 4 Emphasis is on identifying points, lines, line segments, rays, angles (right, acute, obtuse), and per- pendicular and parallel lines in two-dimensional figures.
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