Similar Triangles

11 Key excerpts on "Similar Triangles"

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  eBook - PDF

  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The formal definition of Similar Triangles requires that we establish a correspondence between the vertices of the triangles. The definition also involves the word proportional. Recall that a proportion is a mathematical statement that two ratios (fractions) are equal. An example of a proportion is 1 2 5 4 8 In this case, we say that 1 2 and 4 8 are proportional. Similar Triangles Two triangles are similar if and only if their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are proportional. Refer to the figure below. If ^PQR  ^CDE , name the congruent angles and the sides that are proportional. Q P R C D E Strategy We will establish the correspondence between the vertices of ^PQR and the vertices of ^CDE . WHY This will, in turn, establish a correspondence between the congruent corresponding angles and proportional sides of the triangles. Solution When we write ^PQR  ^CDE , a correspondence between the vertices of the triangles is established.       ^PQR  ^CDE Since the triangles are similar, corresponding angles are congruent: P > C Q > D R > E The lengths of the corresponding sides are proportional. To simplify the notation, we will now let PQ 5 m(PQ ) , CD 5 m(CD ) , QR 5 m(QR ) , and so on. PQ CD 5 QR DE QR DE 5 PR CE PQ CD 5 PR CE Written in a more compact way, we have PQ CD 5 QR DE 5 PR CE EXAMPLE 4 Self Check 4 If ^GEF  ^IJH , name the congruent angles and the sides that are proportional. F G E J I H Now Try Problem 39 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.

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  Basic Mathematics for College Students with Early Integers

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The formal definition of Similar Triangles requires that we establish a correspondence between the vertices of the triangles. The definition also involves the word proportional. Recall that a proportion is a mathematical statement that two ratios (fractions) are equal. An example of a proportion is 1 2 H11005 4 8 In this case, we say that 1 2 and 4 8 are proportional. Similar Triangles Two triangles are similar if and only if their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are proportional. Refer to the figure below. If ^PQR H11011 ^CDE , name the congruent angles and the sides that are proportional. Q P R C D E Strategy We will establish the correspondence between the vertices of ^PQR and the vertices of ^CDE . WHY This will, in turn, establish a correspondence between the congruent corresponding angles and proportional sides of the triangles. Solution When we write ^PQR H11011 ^CDE , a correspondence between the vertices of the triangles is established.       ^PQR H11011 ^CDE Since the triangles are similar, corresponding angles are congruent: H11028P H11061 H11028C H11028Q H11061 H11028D H11028R H11061 H11028E The lengths of the corresponding sides are proportional. To simplify the notation, we will now let PQ H11005 m(PQ ) , CD H11005 m(CD ) , QR H11005 m(QR ) , and so on. PQ CD H11005 QR DE QR DE H11005 PR CE PQ CD H11005 PR CE Written in a more compact way, we have PQ CD H11005 QR DE H11005 PR CE EXAMPLE 4 Self Check 4 If ^GEF H11011 ^IJH , name the congruent angles and the sides that are proportional. F G E J I H Now Try Problem 39 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.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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 505 13.2 Similarity Understand the definition of Similar Triangles. Use the AA Similarity Postulate. Use the SAS and SSS Similarity Theorems. Understand the definition of similarity as it applies to other polygons. Similar Triangles Similar figures have the same shape but not necessarily the same size. Standards Grades 3–5 Geometry Students should classify shapes by properties of their lines and angles. Grades 6–8 Geometry Students should understand congruence and similarity using physical models, transparencies, or geometry software. Definition of Similar Triangles Two triangles are Similar Triangles when their corresponding side lengths are proportional and their corresponding angles are congruent. In the figures below, triangle ABC is similar to triangle DEF. C B A E D F Triangles △ABC ~ △DEF The symbol ~ means “is similar to.” Side Lengths AB — DE = BC — EF = AC — DF Angles A ≅ D, B ≅ E, C ≅ F △ABC is similar to △JKL. Find the measure of A B C 42° J K L 48° (a) ∠B and (b) ∠L. SOLUTION a. Angle B and angle K are corresponding angles. So, the measure of angle B is 48º. b. Angle L and angle C are corresponding angles. So, the measure of angle L is 90º. EXAMPLE 1 Using the Definition of Similar Triangles △FGH is similar to △PQR. Find the value of x. 6 m 8 m F H G SOLUTION Use a proportion to find x. 9 m Q P R x m 6 — 9 = 8 — x Write a proportion. 6x = 72 Use the Cross Products Property. x = 12 Division Property of Equality So, x is 12 and the length of the side is 12 meters. EXAMPLE 2 Using the Definition of Similar Triangles Copyright 2014 Cengage Learning.

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  No longer available |Learn more

  First Course in Euclidean Geometry, A

  • (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.

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  Dr. Math Presents More Geometry

  Learning Geometry is Easy! Just Ask Dr. Math

  • (Author)
  • 2005(Publication Date)
  • Jossey-Bass

    (Publisher)

  As with con- gruence, there are several shortcuts you can use to determine if two triangles are similar. Hi, Qian, You can tell if two triangles are similar in a lot of different ways, but all of them involve comparing sides and angles. If all of the sides of each triangle are in proportion to the corresponding sides in the other triangle, or if all of the angles are equal, then the triangles are similar. Also, since equal angles are opposite similar-length sides, you can prove that two triangles are similar when any of the fol- lowing seven conditions are true: 1. If two sets of corresponding sides are in proportion and the angle between them is equal (SAS) 2. If two corresponding angles are equal and the side between them is in proportion (ASA) 3. If two angles are equal and another side is in proportion (AAS) 76 Dr. Math Presents More Geometry Dear Dr. Math, How do you prove two triangles to be similar? Yours truly, Qian Prove Triangles Are Similar 4. If, in a right triangle, the hypotenuse and one leg are both in proportion (HL) 5. If all three sides are in proportion (SSS) 6. If all three angles are equal (AAA) 7. If two sides are in proportion and one angle is equal (SSA) You should also know that any of these except items 6 and 7 can be used to prove triangles congruent if you change proportionate sides for congruent ones. —Dr. Math, The Math Forum Hi, Quentin, One way to find the height of a build- ing makes use of Similar Triangles. Put a vertical post in the ground (or have someone hold it vertical) and measure its height and the length of its shadow. Measure the length of the building’s shadow before the sun has time to move. Now you have two triangles with three known lengths: Triangles: Properties, Congruence, and Similarity 77 Dear Dr. Math, I have to write a shadow report for math, and I need to know what time of day is best to use a shadow to measure the height of a building or object with triangles.

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  Introductory Technical Mathematics

  • 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.

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Do you agree? Discuss. 24. Ricardo says these two triangles must be congruent by AAS. Do you agree? Why or why not? 25. Jose’s teacher showed that n n ABC ADC ≅ and then stated that this showed the diagonal bisected ∠A and ∠C. Jose asked if we could say the same thing about the diagonal FH in the parallelogram EFGH because n n EFH GHF ≅ . How would your respond? A E F H G C D B SIMILARITY OF TRIANGLES Triangle Similarity Informally, two geometric figures that have the same shape, but not necessarily the same size, are called similar. Children’s Literature www.wiley.com/college/musser See “Grandfather Tang’s Story” by Ann Tompert. Write a description of what is meant by “this object is similar to that object.” Use your description to determine whether the following pairs of objects are similar. 1. a volleyball and a basketball 2. a father and his son 3. two squares 4. you and your picture 5. a small soda cup and a medium soda cup from a fast-food restaurant 6. a piece of paper with writing on it and a clean piece of paper Discuss how the English definition of similar and your understanding of the mathematical definition of similar compare. 730 Chapter 14 Geometry Using Triangle Congruence and Similarity Similar Triangles Suppose that nABC and nDEF are such that under the correspondence A D B E C F ↔ ↔ ↔ , , , all corresponding sides are proportional and all corre- sponding vertex angles are congruent. Then nABC is similar to nDEF , and we write   ABC DEF ∼ . D E F I N I T I O N 1 4 . 2 Common Core – Grade 8 Understand that a two-dimen- sional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the simi- larity between them. Saying that corresponding sides in nABC and nDEF are proportional means that the ratios of corresponding sides are all equal.

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  eBook - ePub

  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  The Angle-Angle-Angle Similarity Theorem (AAA~): Given a one-to-one correspondence between the vertices of two triangles, or a triangle and itself. If the three angles of one of the triangles are congruent to the corresponding angles of the other, then the triangles are similar.

       In terms of the figure, we have

  Hypothesis: ∠A ≅ ∠D; ∠B ≅ ∠E; and ∠C ≅ ∠E

  Conclusion: ΔABC ~ ΔDEF

       Proof: Since the corresponding angles are congruent, we need only show that the corresponding sides are proportional. On and , respectively, there exist points G and H such that and . Since ∠C ≅ ∠F, we may conclude that ΔGCH ≅ ΔDFE. By CPCTC we have ∠CGH ≅ ∠D. But ∠D ≅ ∠A. Therefore, ∠CGH ≅ ∠A. Then it follows from T41 that . The Basic Proportionality Corollary can now be used to conclude that

  However, CG = FD and CH = FE. Thus we may write

  Note that these are ratios of corresponding sides. To show that we proceed analogously. Two easy corollaries follow.

       COROLLARY T58(a) The Angle-Angle Corollary (AA~): If two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.

       COROLLARY T58(b) A line parallel to one side of a triangle and intersecting the interiors of the other two sides determines a triangle similar to the given one.

       Example 1 In the figure, , AD = 2, DC = 3, and AB = 4.

       Find DE.

       implies

       Therefore,

       From the properties of congruence for angles and Lemma 6.1, we can derive a helpful lemma.

       LEMMA 6.2 Triangle similarity is reflexive, symmetric, and transitive.

       The proof of this lemma is left to the reader. It finds its application in the proofs of our next two theorems.

       THEOREM 59 The Side-Angle-Side Similarity Theorem

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  Maths: A Student's Survival Guide

  A Self-Help Workbook for Science and Engineering Students

  • Jenny Olive(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  We also see from this same diagram that, if we have a triangle with one side extended, then the exterior angle e is equal to a + b , the sum of the two interior opposite angles. This is shown drawn in on Figure 4.A.13 . 4.A. (d) Triangles with particular shapes Triangles can come in an infinite variety of shapes, but there are two particular types which have specific names. If a triangle has two sides equal then it is called isosceles (originally by the Greeks who were very keen on geometry – ‘iso’ means ‘equal’ and ‘sceles’ means ‘sides’. ‘Trigonometry’ also comes from the Greeks – ‘trigono’ is the Greek word for triangle.) 4.A Trigonometry in right-angled triangles 139 Figure 4.A.12 Figure 4.A.13 The two equal sides give these triangles a line of symmetry, so that one half folds exactly on to the other half, and the pair of angles opposite the equal sides are also equal. The line of symmetry divides the triangle into two equal right-angled triangles. (See Figure 4.A.14(a) .) The little dashes are there to mark the two equal sides. If a triangle has all three sides equal then it is called equilateral . Such a triangle is pictured in Figure 4.A.14(b) . It will have three lines of symmetry as shown, and will fit exactly onto itself three times in a complete turn. Therefore all its angles are equal, and so must be 60° each. All equilateral triangles can nest into each other, in any chosen corner. Some are shown here in Figure 4.A.15 . They are all similar to each other. (‘Similar’ in maths doesn’t just mean ‘more or less the same as’ but ‘an exact scale model of’ so that all the angles remain the same, and the pairs of sides are all in the same proportion.) 4.A. (e) Congruent triangles – what are they, and when? If two triangles are exactly the same size and shape so that they can be fitted onto each other exactly, they are called congruent . In this case, they will obviously have three equal pairs of angles and three equal pairs of sides.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Thus, the angles of each isosceles triangle measure 40°, 70°, and 70°. Applying Postulate 15 (AAA), the two triangles are similar. Corollary 5.3.1 of Postulate 15 follows from Corollary 2.4.4, which states “If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles must also be congruent.” S T H R J K nHJK nSRT Figure 5.11 238 CHAPTER 5 ■ Similar Triangles Corollary 5.3.1 If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar (AA). Rather than use AAA to prove triangles similar, we will use AA in Example 2 and later proofs because it requires fewer steps. STRATEGY FOR PROOF ■ Proving That Two Triangles Are Similar General Rule: Although there will be three methods of proof (AA, SAS, , and SSS, ) for Similar Triangles, we use AA whenever possible. This leads to a more efficient proof. Illustration: See statements 2 and 3 in the proof of Example 2. Notice that statement 4 follows by the reason AA. In some instances, we wish to prove a relationship that takes us beyond the similarity of triangles. The following consequences of the definition of similarity are often cited as reasons in a proof. CSSTP Corresponding sides of Similar Triangles are proportional. SSG EXS. 1–4 CASTC Corresponding angles of Similar Triangles are congruent. The first fact, abbreviated CSSTP, is used in Examples 3 and 4. Although the CSSTP statement involves triangles, the corresponding sides of any two similar polygons are pro- portional. That is, the ratio of the lengths of any pair of corresponding sides of one polygon equals the ratio of the lengths of another pair of corresponding sides of the second poly- gon. The second fact, abbreviated CASTC, used in Example 5, involves Similar Triangles. But it is also true that the corresponding angles of similar polygons are congruent. STRATEGY FOR PROOF ■ Proving a Proportion General Rule: First prove that triangles are similar.

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  Geometry: 1001 Practice Problems For Dummies (+ Free Online Practice)

  • Allen Ma, Amber Kuang(Authors)
  • 2022(Publication Date)
  • For Dummies

    (Publisher)

  The centroid and orthocenter of a triangle are 2 2 3 , and (4, 0) , respectively. What is the equation of the Euler line? CHAPTER 6 Similar Triangles 49 Similar Triangles A n important concept in geometry is the difference between congruence and similarity. This chapter focuses on the similarity of triangles. If two triangles are similar, their sides are in proportion. Here, you practice setting up proportions and solving them algebraically. You also work on geometric proofs dealing with Similar Triangles. The Problems You’ll Work On In this chapter, you see a variety of geometry problems: » Determining side lengths of Similar Triangles » Connecting the midpoints of two sides of a triangle to create a segment parallel to the third side » Finding the perimeter and area of Similar Triangles » Writing geometric proofs with Similar Triangles » Understanding that the product of the means equals the product of the extremes What to Watch Out For Don’t let common mistakes trip you up. Some of the following suggestions may be helpful: » Remember that if two triangles are similar, the corresponding sides are in proportion. » If you’re looking to prove that the sides of a triangle are in proportion, you must first prove the triangles similar by AA (angle-angle). » Similar Triangles are different only in size. The corresponding angles still have the same measure. » If two triangles are similar, the ratio of the perimeters of the triangles is the same as the ratio of the sides; however, the ratio of the area of the triangles is the square of the ratio of the sides of the triangles. » Make sure you know what the question is asking. Sometimes a question asks you to solve for a variable. In other cases, you have to find the value of a variable and then plug it in to find the measure of a segment or angle. Chapter 6 50 PART 1 The Questions Understanding Similar Triangles 261–267 Use the given information to answer each question regarding Similar Triangles  ABC and  DEF .

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