Congruent Triangles

7 Key excerpts on "Congruent Triangles"

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  eBook - ePub

  Solving Problems in Geometry

  Insights and Strategies for Mathematical Olympiad and Competitions

  • Kim Hoo Hang, Haibin Wang;;;(Authors)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  Chapter 1

  Congruent Triangles

  We assume the reader knows the following basic geometric concepts, which we will not define:

  • Points, lines, rays, line segments and lengths

  • Angles, right angles, acute angles, obtuse angles, parallel lines (//) and perpendicular lines (⊥)

  • Triangles, isosceles triangles, equilateral triangles, quadrilaterals, polygons

  • Height (altitudes) of a triangle, area of a triangle

  • Circles, radii, diameters, chords, arcs, minor arcs and major arcs

  1.1 Preliminaries We assume the reader is familiar with the fundamental results in geometry, especially the following, the illustration of which can be found in any reasonable secondary school textbook.

  (1) For any two fixed points, there exists a unique straight line passing through them (and hence, if two straight lines intersect more than once, they must coincide).

  (2) For any given straight line ℓ and point P, there exists a unique line passing through P and parallel to ℓ.

  (3) Opposing angles are equal to each other. (Refer to the diagram below. ∠1 and ∠2 are opposing angles. We have ∠1 = 180° – ∠3 = ∠2.)

  (4) In an isosceles triangle, the angles which correspond to equal sides are equal. (Refer to the diagram below.)

  The inverse is also true: if two angles in a triangle are the same, then they correspond to the sides which are equal.

  (5) Triangle Inequality: In any triangle ΔABC, AB + BC > AC.

  (A straight line segment gives the shortest path between two points.)

  (6) If two parallel lines intersect with a third, we have:

  • The corresponding angles are the same.

  • The alternate angles are the same.

  • The interior angles are supplementary (i.e., their sum is 180°). (Refer to the diagrams below.)

  Its inverse also holds: equal corresponding angles, equal alternate angles or supplementary interior angles imply parallel lines. One may use (6) to prove the following well-known results.

  Theorem 1.1.1 The sum of the interior angles of a triangle is 180°.

  Proof. Refer to the diagram below. Draw a line passing through A which is parallel to BC. We have ∠B = ∠1 and ∠C

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  The Mathematics That Every Secondary School Math Teacher Needs to Know

  • Alan Sultan, Alice F. Artzt(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  c of the three sides of a triangle ABC , then we immediately know cos A , cos B , and cos C and hence angles A, B , and C . This brings us to the topic of congruence.

  5.2.1 Congruence

  Recall that, in geometry, two triangles ABC and DEF are congruent if their corresponding sides and corresponding angles are congruent. That is, they have the same measure. So, if ABC is congruent to DEF , then AB = DE, BC = EF, AC = DF , and ∡A = ∡D , ∡B = ∡E , and ∡C = ∡F . (See Figure 5.2 .)

  Figure 5.2

  Under this correspondence, angles A and D are called corresponding angles, as are the angles B and E , and C and F . Sides

  A B

  and

  D E

  are called corresponding sides, as are the sides BC and EF , and the sides AC and DF . By definition of Congruent Triangles, corresponding parts have the same measure, which means corresponding sides have the same length and corresponding angles have the same degree measure. Notice that the order in which we write the letters tells us the angle correspondence and side correspondence. Had we written that triangle ACB was congruent to FDE , then it would mean that ∡A = ∡F , ∡C = ∡D , and ∡B = ∡E , and that AC = FD, CB = DE , and BA = EF .

  The first result we talk about is something we are all familiar with: If three sides of one triangle have the same lengths as three sides of another triangle, then the triangles are congruent. That is, all their corresponding parts match! This is quite remarkable since we have said nothing about the angles of these triangles. Yet, this follows immediately from the Law of Cosines.

  Theorem 5.2 (SSS = SSS) If the three sides of triangle ABC are congruent to the three sides of triangle DEF, then the triangles ABC and DEF are congruent .

  Proof . Let us assume that the sides that match are a and d, b and e , and c and f . (Refer to Figure 5.2 .) So a = d, b = e , and c = f . From equation (5.8)

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  Lapses in Mathematical Reasoning

  • V. M. Bradis, L. Minkovskii, A. K. Kharcheva(Authors)
  • 2016(Publication Date)
  • Dover Publications

    (Publisher)

  III. Stories, with Explanations, of Causes of Erroneous Reasoning

  63. Similar triangles with equal sides

  Take two similar scalene triangles and denote the sides of the first in the order of increasing size by the letters a, b, c (a < b < c), and the corresponding sides of the second by the letters a1 , b1 , c1 . By virtue of proportionality of the corresponding sides of similar polygons we have: a1 = aq, b1 = bq, c1 = cq, where q is the coefficient of proportionality, and therefore a1 < b1 < c1 .

  If q = 1, then all the sides of the two triangles are respectively equal, and the triangles are congruent. The congruence of the triangles is thus a special case of similarity.

  It may seem, that if the triangles are similar, but not congruent, then they have no equal sides. The erroneousness of such a conclusion is shown by a simple consideration of triangles with the sides 8, 12, 18 cm, and 12, 18, 27 cm. The sides of the second are one-and-one-half times as large as the corresponding sides of the first, and therefore the triangles are similar (but not congruent). As we see, these two triangles have two pairs of respectively equal sides.

  We shall establish the conditions which have to be satisfied by two similar, but not congruent, triangles having two pairs of respectively equal sides.

  Suppose that q > 1. The smallest side a of the first triangle (with the sides a, b, c) is less than the smallest side a1 of the second triangle (with sides a1 = aq, b1 = bq, c1 = cq), and cannot be equal to any of the sides of the latter.

  The middle side b of the first triangle may be equal only to the smallest side a1 of the second triangle (since b is less than b1 = bq and much smaller than c1 = cq), and the greatest side c of the first triangle is equal either to the smallest side a1 or to the middle side b1 of the second triangle. If two sides of the first triangle are equal to two sides of the second triangle, then b = a1 , c = b1 . Hence b = aq, c = bq = aq × q = aq2 . Thus the sides of the first triangle form a geometric progression a, aq, aq2 . The sides of the second triangle are equal to aq, aq2 , aq3

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  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Part 3

  Triangles: Polygons of the Three-Sided Variety

  IN THIS PART … Get familiar with triangle basics. Have fun with right triangles. Work on congruent triangle proofs. Passage contains an image Chapter 7

  Grasping Triangle Fundamentals

  IN THIS CHAPTER Looking at a triangle’s sides: Equal or unequal Uncovering the triangle inequality principle Classifying triangles by their angles Calculating the area of a triangle Finding the four “centers” of a triangle

  Considering that it’s the runt of the polygon family, the triangle sure does play a big role in geometry. Triangles are one of the most important components of geometry proofs (you see triangle proofs in Chapter 9 ). They also have a great number of interesting properties that you might not expect from the simplest possible polygon. Maybe Leonardo da Vinci (1452–1519) was on to something when he said, “Simplicity is the ultimate sophistication.”

  In this chapter, I take you through the triangle basics — their names, sides, angles, and area. I also show you how to find the four “centers” of a triangle.

  Taking In a Triangle’s Sides

  Triangles are classified according to the length of their sides or the measure of their angles. These classifications come in threes, just like the sides and angles themselves. That is, a triangle has three sides, and three terms describe triangles based on their sides; a triangle also has three angles, and three classifications of triangles are based on their angles. I talk about classifications based on angles in the upcoming section “Getting to Know Triangles by Their Angles .”

  The following are triangle classifications based on sides:

  • Scalene triangle: A scalene triangle is a triangle with no congruent sides
  • Isosceles triangle: An isosceles triangle is a triangle with at least two congruent sides
  • Equilateral triangle: A equilateral triangle is a triangle with three congruent sides

  Because an equilateral triangle is also isosceles, all triangles are either scalene or isosceles. But when people call a triangle isosceles, they’re usually referring to a triangle with only two equal sides, because if the triangle had three equal sides, they’d call it equilateral.

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  Exploring Geometry

  • Michael Hvidsten(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Elements .

  Definition 2.14 . Two triangles are congruent if and only if there is some way to match vertices of one to the other such that corresponding sides are congruent and corresponding angles are congruent .

  If ΔABC is congruent to ΔXYZ , we shall use the notation ΔABC ≅ ΔXYZ . We will use the symbol “≅” to denote congruence in general for segments, angles, and triangles. Thus, ΔABC ≅ ΔXYZ if and only if

  A B

  ¯

  ≅

  X Y

  ¯

  ,

  A C

  ¯

  ≅

  X Z

  ¯

  ,

  B C

  ¯

  ≅

  Y Z

  ¯

  and

  ∠ B A C ≅ ∠ Y X Z , ∠ C B A ≅ ∠ Z Y X , ∠ A C B ≅ ∠ X Z Y

  Let’s review a few triangle congruence theorems.

  Theorem 2.10 . (SAS: Side-Angle-Side ,

  Prop. 4 of Book I) If in two triangles there is a correspondence such that two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent

  .

  This proposition is one of the axioms in Hilbert’s axiomatic basis for Euclidean geometry. Hilbert chose to make this result an axiom rather than a theorem to avoid the trap that Euclid fell into in his proof of the SAS result. In Euclid’s proof, he moves points and segments so as to overlay one triangle on top of the other and thus prove the result. However, there is no axiomatic basis for such transformations in Euclid’s original set of five postulates. Most modern treatments of Euclidean geometry assume SAS congruence as an axiom. Birkhoff chooses a slightly different triangle comparison result, the SAS condition for triangles to be similar , as an axiom in his development of Euclidean geometry.

  Theorem 2.11 . (ASA: Angle-Side-Angle ,

  Prop. 26 of Book I) If in two triangles there is a correspondence in which two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent

  .

  Theorem 2.12 . (AAS: Angle-Angle-Side ,

  Prop. 26 of Book I) If in two triangles there is a correspondence in which two angles and the side subtending one of the angles are congruent to two angles and the side subtending the corresponding angle of another triangle, then the triangles are congruent

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  Making Sense of Mathematics for Teaching High School

  Understanding How to Use Functions

  • Edward C. Nolan, Juli K. Dixon, Farhsid Safi, Erhan Selcuk Haciomeroglu(Authors)
  • 2016(Publication Date)
  • Solution Tree Press

    (Publisher)

  Students use translations, rotations, and reflections to demonstrate understanding of congruence of geometric figures. Figures are shown to be congruent when one shape can be exactly superimposed onto another through rigid motion transformations. The use of dilation, a non-rigid motion transformation, supports the development of understanding similarity. The Pythagorean theorem is proved and applied to solve problems in the Cartesian coordinate plane and in the real world. Students solve mathematical and real-world problems involving volume.

  High School

  Students build on their understanding of translations, rotations, reflections, and dilations to develop richer understandings of congruence and similarity. Definitions of transformations are extended and described with precision. Transformations are also used to support the development of proving congruent and similar relationships between figures. Students represent their thinking geometrically through constructions and connect to algebraic concepts through coordinate representations using function notation.

  Students relate properties of geometric figures to the coordinate grid and algebraic representations. These connections are used to create geometric figures such as parabolas and circles as well as to prove theorems involving geometric figures. Circle relationships are used to make connections among chords, angles, area, and sectors.

  Students build on their work with right triangles and the Pythagorean theorem from grade 8 as they explore trigonometric relationships. Trigonometry is the study of the relationships of angle measures and lengths of sides of triangles. This connects student understanding of similarity to relationships within right triangles. Students solve problems involving trigonometry both in and out of context.

  The Mathematics

  Making sense of shape and space provides the tools to describe the world. Students in the middle grades are introduced to transformations, congruence, and similarity. High school students formalize these understandings, using them to explore transformations and demonstrate geometric relationships. Students will also use proof and constructions to investigate congruence and similarity.

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  The Handy Math Answer Book

  • Patricia Barnes-Svarney, Thomas E Svarney(Authors)
  • 2012(Publication Date)
  • Visible Ink Press

    (Publisher)

  The problem is an ancient one, and like many “Holy Grails” of mathematics, this problem was first stated centuries ago by astute mathematicians—this time, Persian mathematician al-Karaji (for more about al-Karaji, see “History of Mathematics”). In 1225, Fibonacci (for more about Fibonacci, see “History of Mathematics” and “Mathematics throughout History”) also tried to work on the problem. Many other mathematicians followed, but it took until around 2009 for an international team of mathematicians, using state-of-the-art computer techniques, to find the first trillion congruent numbers.

  In order, the first few congruent numbers known are 5, 6, 7, 13, 14, 15, 20, 21, and so on. But, as one can imagine, there are so many more such numbers. The researchers were able to figure out a computer compilation method that would allow them to not only find but also verify more congruent numbers. They found 3,148,379,694 congruent numbers up to a trillion. The mathematicians were no doubt grateful for the computers, too. According to some researchers, the numbers involved are so huge that if they were written by hand, the numbers would stretch to the Moon and back.

  SOLID GEOMETRY

  What is solid geometry?

  Solid geometry is the study of objects in three-dimensional Euclidean space. It deals with solids, as opposed to plane geometry, which deals with two dimensions. This part of geometry is concerned with entities such as polyhedra, spheres, cones, cylinders, and so on. (For more about Euclidean space and dimensions, see elsewhere in this chapter.)

  In geometry, solids are defined as closed three-dimensional figures, or any limited portion of space bounded by surfaces. They differ in subtle ways from what we perceive as solids: We see solids in terms of what surrounds us—three-dimensional figures with their surfaces the actual objects we perceive. Geometric solids are actually the union of the surface and regions of space; in a way, this adds another dimension to two-dimensional space.

  What is a polyhedron?

  The word polyhedron comes from the Greek poly (meaning “many”) and the Indo-European word hedron

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