DOCSLIB.ORG

Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from Textbook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from Textbook Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook Objectives • Identify congruent figures and corresponding parts of closed plane figures. • Prove that two triangles are congruent using definitions, properties, theorems, and postulates. Congruent Not Congruent *If two figures are congruent, then their corresponding parts are _______________________. In the diagram, ∆ABC ≅ ∆FED . Label the two triangles accordingly and mark all corresponding parts that are congruent. Congruence Statements: ________________________________________________________________ Example 1 Write a congruence statement for the triangles. Identify all pairs of corresponding congruent parts. Triangles _____________________________________ Corresponding Angles ____________________________________ Corresponding Sides _______________________________________ Example 2 In the diagram, DEFG ≅ SPQR . Find the value of x. _________________ Find the value of y. _________________ Example 3 In the diagram, a rectangular wall is divided into two sections. Are the sections congruent? Explain. _____________________________________________ _____________________________________________ Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangles, then the third angles are _____________________________________________ Example 4 Find m<BDC. Example 5 Graph the triangle with vertices D(1, 2), E(7, 2), and F(5, 4). Then, graph a triangle congruent to ∆DEF . Example 6 Given: AD ≅ CB , DC ≅ BA , ∠ACD ≅ ∠CAB , ∠CAD ≅ ∠ACB Prove: ∆ACD ≅ ∆CAB Statements Reasons 1. 1. Given 2. 2. 3. 3. Given 4. 4. 5. ∆ACD ≅ ∆CAB 5. Properties of Congruent Triangles Reflexive Property For any triangle ABC, ∆ABC ≅ ________________. Symmetric Property If ∆ABC ≅ ∆DEF , then _______________________. Transitive Property If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL , then _____________________. Unit 8: Congruent and Similar Triangles Lesson 8.2 Prove Triangles Congruent by SSS Lesson 4.3 from textbook Objectives • Use the Side-Side-Side (SSS) Congruence Postulate to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Prove that two triangles are congruent in the coordinate plane using the Distance Formula and the SSS postulate. Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then _________________________________________________ Example 1 Determine whether the congruence statement is true. Explain your reasoning. ∆DFG ≅ ∆HJK ∆ACB ≅ ∆CAD Example 2 Use the given coordinates to determine if ∆ABC ≅ ∆DEF . A(-3, -2), B(0, -2), C(-3, -8), D(10, 0), E(10, -3), F(4, 0) AB = __________ CA = __________ DE = __________ FD = __________ BC = __________ EF = __________ Example 3 Explanation: ____________________________________________________________________________________ ____________________________________________________________________________________ Example 4 Example 5 Statements Reasons 1. 1. 2. 2. 3. 3. Unit 8: Congruent and Similar Triangles Lesson 8.3 Prove Triangles Congruent by SAS and HL Lesson 4.4 from textbook Objectives • Use the Side-Angle-Side (SAS) and Hypotenuse-Leg (HL) Congruence Postulate to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Use two-column proofs to justify statements about congruent triangles. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to the corresponding to sides and corresponding and the corresponding included angle of a second triangle, then ___________________________________________________. Example 1 Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate. Hypotenuse-Leg Congruence Theorem If the leg and hypotenuse of a right triangle are congruent to the corresponding leg and hypotenuse of a second triangle, then ____________________________________________________. Example 2 State the third congruence that must be given to prove ∆ABC ≅ ∆DEF using indicated postulate. a) Given: AB ≅ DE ,CB ≅ FE , ______ ≅ ______ (SSS Congruence Postulate) b) Given: ∠A ≅ ∠D,CA ≅ FD , ______ ≅ _______ (SAS Congruence Postulate) c) Given: ∠B ≅ ∠E, AB ≅ DE , ______ ≅ _______ (SAS Congruence Postulate) Example 3 Statements Reasons 1. 1. 2. 2. Definition of perpendicular lines. 3. 3. Definition of a right triangle. 4. 4. 5. 5 6. 6. Example 4 Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Unit 8: Congruent and Similar Triangles Lesson 8.4 Prove Triangles Congruent by ASA and AAS Lesson 4.5 from textbook Objectives • Use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) Congruence Postulates to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Use two-column proofs to justify statements about congruent triangles. Angle-Side-Angle (ASA) Congruence Postulate Angle-Angle-Side (AAS) Congruence Theorem Example 1 Is it possible to prove that the two triangles are congruent? If so, state the postulate or theorem you would use. ________________________ _______________________ Example 2 State the third congruence that must be given to prove ∆ABC ≅ ∆DEF using indicated postulate. a) Given: AB ≅ DE ,∠A ≅ ∠D, ______ ≅ ______ (AAS Congruence Postulate) b) Given: ∠A ≅ ∠D,CA ≅ FD , ______ ≅ _______ (ASA Congruence Postulate) Example 3 Tell whether you can use the given information to determine whether ∆ABC ≅ ∆DEF . Explain your reasoning. ∠A ≅ ∠D,AB ≅ DE , AC ≅ DF __________________________ ∠B ≅ ∠E,∠C ≅ ∠F, AC ≅ DE __________________________ Example 4 Given: X is the midpoint of VY and WZ . Prove: ∆VWX ≅ ∆YZX Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Unit 8: Congruent and Similar Triangles Lesson 8.5 Using Congruent Triangles Lesson 4.6 from textbook Objectives • Use congruent triangles to plan and write proofs about their corresponding parts. Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC) If ___________________________ are congruent then the ____________________________ of the congruent triangles are also _____________________. Given congruent parts: ________________________ ∆ABC ≅ ∆DEF by the ________________ Other corresponding congruent parts: __________________________________________ Example 1 Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use? ∠A ≅ ∠D GK ≅ HJ ____________________ ____________________ ____________________ ____________________ Example 2 _____________________ ________________________ Example 3 Given: ∠Q ≅ ∠S,∠RTQ ≅ ∠RTS Prove: QT ≅ ST *FIRST PROVE TRIANGLES ARE CONGRUENT* Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Example 4 Given: NM ≅ KM Prove: ∠MLK ≅ ∠MPN Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Example 5 Use the diagram to write a plan for a proof: PLAN: Prove: ∠A ≅ ∠C Unit 8: Congruent and Similar Triangles Lesson 8.6 Prove Triangles Similar by AA Lesson 6.4 from textbook Objectives • Identify similar triangles using the Angle-Angle (AA) Similarity Postulate. • Find measures of similar triangles using proportional reasoning. ACTIVITY: Question: What can you conclude about two triangles if you know two pairs of corresponding angles are congruent? 1. Draw ∆EFG so that m<E = 40 o 2. Draw ∆RST so that <R = 40 o and m<T = 50 o, and m<G = 50 o. and is not congruent to ∆EFG . 3. Calculate m<F and m<S using Triangle Sum Theorem. ___________________________________ 4. Measure and record the side lengths of both triangles. (to the nearest mm). ___________________________________________ 5. Are the triangles similar? Explain. ____________________________________________________ 6. If all we know is that two angles in two different triangles are congruent, can we conclude that the triangles are similar? ______________________________ Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. _______________________________________________ Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. ______________________________________ ___________________________________ Example 2 Use the diagram to complete the information. MN ON MO ∆MON ~ _______ __ = = ___________________ ? ? ? 16 ? 12 ? = _________ = _______________ 12 10 16 y x = ____________ y = ____________ Example 3 The A-frame building shown in the figure has a balcony that is 16 feet long, 16 feet high, and parallel to the ground. The building is 28 feet wide at its base. How tall is the A-frame building? Height = ________________ Unit 8: Congruent and Similar Triangles Lesson 8.7 Prove Triangles Similar by SSS and SAS Lesson 6.5 from textbook Objectives • Use the similarity theorems such as the Side-Side-Side (SSS) Similarity Theorem and the Side- Angle-Side (SAS) Similarity Theorem to determine whether two triangles are similar. • Find measures of similar triangles using proportional reasoning. Side-Side-Side (SSS) Side-Angle-Side (SAS) Similarity Theorem Similarity Theorem If the corresponding side lengths of two If two sides of one triangle are proportiona l to triangles are proportional , then the triangles two sides of another triangle and their included are similar angles are congruent, then the triangles are similar. If ___________________________________ If ___________________________________ ___________________,
Load more

Recommended publications
  • Geometry Unit 4 Vocabulary Triangle Congruence
    Geometry Unit 4 Vocabulary Triangle Congruence Biconditional statement – A is a statement that contains the phrase “if and only if.” Writing a biconditional statement is equivalent to writing a conditional statement and its converse. Congruence Transformations–transformations that preserve distance, therefore, creating congruent figures Congruence – the same shape and the same size Corresponding Parts of Congruent Triangles are Congruent CPCTC – the angle made by two lines with a common vertex. (When two lines meet at a common point Included angle (vertex) the angle between them is called the included angle. The two lines define the angle.) Included side – the common side of two legs. (Usually found in triangles and other polygons, the included side is the one that links two angles together. Think of it as being “included” between 2 angles. Overlapping triangles – triangles lying on top of one another sharing some but not all sides. Theorems AAS Congruence Theorem – Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles. ASA Congruence Theorem -Triangles are congruent if any two angles and their included side are equal in both triangles. SAS Congruence Theorem -Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles. SAS SSS Congruence Theorem -Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other. Special congruence theorem for RIGHT TRIANGLES! .
    [Show full text]
  • A Congruence Problem for Polyhedra
    A congruence problem for polyhedra Alexander Borisov, Mark Dickinson, Stuart Hastings April 18, 2007 Abstract It is well known that to determine a triangle up to congruence requires 3 measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron P , how many measurements are required to determine P up to congruence? We show that in general the answer is that the number of measurements required is equal to the number of edges of the polyhedron. However, for many polyhedra fewer measurements suffice; in the case of the cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of n-gons requiring only n measurements. Finally, we show that one cannot do better: for any ordered set of n distinct points in the plane one needs at least n measurements to determine this set up to congruence. An appendix by David Allwright shows that the set of twelve face-diagonals of the cube fails to determine the cube up to conjugacy. Allwright gives a classification of all hexahedra with all face- diagonals of equal length. 1 Introduction We discuss a class of problems about the congruence or similarity of three dimensional polyhedra.
    [Show full text]
  • Understand the Principles and Properties of Axiomatic (Synthetic
    Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length
    [Show full text]
  • Geometry Course Outline
    GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid
    [Show full text]
  • Foundations of Geometry
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.
    [Show full text]
  • ∆ Congruence & Similarity
    www.MathEducationPage.org Henri Picciotto Triangle Congruence and Similarity A Common-Core-Compatible Approach Henri Picciotto The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry program in grades 8 to 10: geometric transformations, not congruence and similarity postulates, are to constitute the logical foundation of geometry at this level. This paper proposes an approach to triangle congruence and similarity that is compatible with this new vision. Transformational Geometry and the Common Core From the CCSSM1: The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.
    [Show full text]
  • Congruence Theory Explained
    UC Irvine CSD Working Papers Title Congruence Theory Explained Permalink https://escholarship.org/uc/item/2wb616g6 Author Eckstein, Harry Publication Date 1997-07-15 eScholarship.org Powered by the California Digital Library University of California CSD Center for the Study of Democracy An Organized Research Unit University of California, Irvine www.democ.uci.edu The authors of the essays that constitute this book have made a theory I first developed in the 1960s its unifying thread. This theory may be called "congruence theory." It has been elaborated several times, in numerous ways, since its first statement, but has not changed in essentials.1 I do not want to explain and justify the theory in all its ramifications here; for this, readers can consult the publications cited in the references. I do want to explain it sufficiently to clarify essentials. The Core of Congruence Theory Every theory has at its "core" certain fundamental hypotheses. These can be used as if they were "axioms" from which theorems (further hypotheses) may be deduced; the theory may then be tested directly through the axioms or indirectly through the theorems. A good deal of auxiliary theoretical matter is generally associated with the core-hypotheses, usually to make them amenable to appropriate empirical inquiry. These auxiliary matters are subsidiary to the core- postulates of a theory, but they are necessary for doing work with it and generally imply hypotheses themselves.2 Congruence theory has such an underlying core, with which a great deal of auxiliary material has become associated; the more important of these auxiliary ideas will be discussed later.
    [Show full text]
  • Complete the Following Statement of Congruence Xyz
    Complete The Following Statement Of Congruence Xyz IsRem Kraig still stateless chancing when solitarily Guy while legitimises audiovisual inferiorly? Abbott reminisces that geographers. Adjoining and darkened Henry dally: which Quinn is proposed enough? Learn how each section of rst in this for pqr pts by applying asa triangle is between the statement of the following congruence, along with their sides BSimilarly markedsegments are congruent. Graduate take the Intro Plan for unlimited engagement. Which unit is included between the sides DE and EF of ΔDEF? Easily assign quizzes to your students and track progress like a pro! One side of sorrow first pad is congruent to one side project the sideways triangle. Use a calculator to solve each equation. To reactivate your handbook, please sift the associated email address below. AB i and AC forms two obtuse triangles. Once field have clearly represented the corresponding parts, they bear more easily lament the questions. Find the volume remains the figure. QRSPPRQPRS In the diagram shown, lines and are parallel. However, to use this strong, all students in the class must doubt their invites. Drag questions to reorder. Which of attention following statements is meaningfully written? Plan to the volume of cookies and squares, cah stands for showing that qtrprove that is not the congruence. The schedule can navigate right. Why or enterprise not? Biscuits in column same packet. Houghton Mifflin Harcourt Publishing Company paid A student claims that support two congruent triangles must have won same perimeter. Determine the flat of each subordinate the three angles. Algebra Find the values of the variables.
    [Show full text]
  • The Axioms of Projective Geometry
    Cambridge Tracts In Mathematics and Mathematical Physics General Editors J. G. LEATHEM, M.A. E. T. WHITTAKER, M.A., F.R.S. No. 4 THE AXIOMS OF PROJECTIVE GEOMETRY by A. N. WHITEHEAD, Sc.D., F.R.S. Cambridge University Press Warehouse C. F. Clay, Manager London : Fetter Lane, E.G. Glasgow : 50, Wellington Street 1906 Prue 2s. 6d. net Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/axiomsofprojectiOOwhituoft Cambridge Tracts in Mathematics and Mathematical Physics General Editors J. G. LEATHEM, M.A. E. T. WHITTAKER, M.A., F.R.S. /X'U.C. /S^i-.-^ No. 4 The Axioms of Projective Geometry. CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, C. F. CLAY, Manager. EonUon: FETTER LANE, E.G. ©lasfloto: 50, WELLINGTON STREET. ILdpjtfl: F. A. BROCKHAUS. ^eia gorft: G. P. PUTNAM'S SONS. ffiombag aria (Calcutta : MACMILLAN AND CO., Ltd. [All rights reserved] THE AXIOMS OF PROJECTIVE GEOMETRY by A. N. WHITEHEAD, Sc.D., F.R.S. Fellow of Trinity College. Cambridge: at the University Press 1906 Camhrfttge: PRINTED BY JOHN CLAY, M.A. AT THE ONIVERSITY PRESS. PREFACE. "TN this tract only the outlines of the subject are dealt with. Accordingly I have endeavoured to avoid reasoning dependent upon the mere wording and on the exact forms of the axioms (which can be indefinitely varied), and have concentrated attention upon certain questions which demand consideration however the axioms are phrased. Every group of the axioms is designed to secure the deduction of a certain group of properties. For the most part I have stated without proof the leading immediate consequences of the various groups.
    [Show full text]
  • Chapter 9 the Geometrical Calculus
    Chapter 9 The Geometrical Calculus 1. At the beginning of the piece Erdmann entitled On the Universal Science or Philosophical Calculus, Leibniz, in the course of summing up his views on the importance of a good characteristic, indicates that algebra is not the true characteristic for geometry, and alludes to a “more profound analysis” that belongs to geometry alone, samples of which he claims to possess.1 What is this properly geometrical analysis, completely different from algebra? How can we represent geometrical facts directly, without the mediation of numbers? What, finally, are the samples of this new method that Leibniz has left us? The present chapter will attempt to answer these questions.2 An essay concerning this geometrical analysis is found attached to a letter to Huygens of 8 September 1679, which it accompanied. In this letter, Leibniz enumerates his various investigations of quadratures, the inverse method of tangents, the irrational roots of equations, and Diophantine arithmetical problems.3 He boasts of having perfected algebra with his discoveries—the principal of which was the infinitesimal calculus.4 He then adds: “But after all the progress I have made in these matters, I am no longer content with algebra, insofar as it gives neither the shortest nor the most elegant constructions in geometry. That is why... I think we still need another, properly geometrical linear analysis that will directly express for us situation, just as algebra expresses magnitude. I believe I have a method of doing this, and that we can represent figures and even 1 “Progress in the art of rational discovery depends for the most part on the completeness of the characteristic art.
    [Show full text]
  • 3 Attributes and Congruence-Shape Choreography
    ARTS IMPACT—ARTS-INFUSED INSTITUTE LESSON PLAN (YR2-AEMDD) LESSON TITLE: Polygons: Attributes and Congruence—Shape Choreography Dance and Math Lesson Artist-Mentor - Debbie Gilbert Grade Level: Third Grade____________________ Examples: Enduring UnderstAnding Number and type of angles, and number and length of sides can describe and identify shapes. Congruent polygons show the same shape and size. Target: Choreographs a three-part dance study using specific congruent shapes, movement, and a prop. CriteriA: 1) Selects a polygon (a parallelogram, rectangle, square, or a triangle); uses a stretchy band to make the shape; makes the shape matching with other dancers; 2) chooses and performs a locomotor or non-locomotor movement; 3) selects, and makes a different matching polygon (a parallelogram, rectangle, square, or a triangle) with other dancers with the stretchy band. Target: Draws and identifies polygons and movements made by the dancers. CriteriA: Sketches and labels two different shapes and determines if movement is locomotor or non-locomotor. TeAching And LeArning StrAtegies Introduction to Arts-Infused Concepts through ClAssroom Activities: Arts-Infused Concepts: Attributes of Polygons; Congruence · Do the BrainDance. If time is available, explore concepts in everyday life: · Draw the shapes of rectangles and parallelograms in the air with your chin. · Draw the shapes of squares, and triangles in the air with your elbows. · Make the shapes of polygons with your hands. 1. Leads students in BrAinDAnce warm-up. (Originally developed by Anne Green Gilbert, reference: Brain-Compatible Dance Education, video: BrainDance, Variations for Infants through Seniors). Music: “Geometry BrainDance (3rd grade)” #1, Geometry Dances. Prompts: The BrainDance is designed to warm up your body and make your brain work better at the same time.
    [Show full text]
  • A Diagrammatic Formal System for Euclidean Geometry
    A DIAGRAMMATIC FORMAL SYSTEM FOR EUCLIDEAN GEOMETRY A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Nathaniel Gregory Miller May 2001 c 2001 Nathaniel Gregory Miller ALL RIGHTS RESERVED A DIAGRAMMATIC FORMAL SYSTEM FOR EUCLIDEAN GEOMETRY Nathaniel Gregory Miller, Ph.D. Cornell University 2001 It has long been commonly assumed that geometric diagrams can only be used as aids to human intuition and cannot be used in rigorous proofs of theorems of Euclidean geometry. This work gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formalize most if not all of what is contained in the first several books of Euclid’s Elements.Thisformal system is much more natural than other formalizations of geometry have been. Most correct informal geometric proofs using diagrams can be translated fairly easily into this system, and formal proofs in this system are not significantly harder to understand than the corresponding informal proofs. It has also been adapted into a computer system called CDEG (Computerized Diagrammatic Euclidean Geometry) for giving formal geometric proofs using diagrams. The formal system FG is used here to prove meta-mathematical and complexity theoretic results about the logical structure of Euclidean geometry and the uses of diagrams in geometry. Biographical Sketch Nathaniel Miller was born on September 5, 1972 in Berkeley, California, and grew up in Guilford, CT, Evanston, IL, and Newtown, CT. After graduating from Newtown High School in 1990, he attended Princeton University and graduated cum laude in mathematics in 1994.
    [Show full text]