Classifying Shapes
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How Can You Put Shapes Together to Make Other Shapes?
Lesson 27 Understand Tiling in Rectangles Name: How can you put shapes together to make other shapes? Study the example showing how to put shapes together to make other shapes. Then solve Problems 1–4. Example How can you use smaller shapes to make a circle? Look at the shapes in the green box Draw lines to show how you could use the shapes Write how many of each shape you would need 2 4 1 Draw a line to show how you could use squares like the one in the green box to make a rectangle How many squares would you need? squares 2 Draw lines to show how you could use triangles like the one in the green box to make a rectangle How many triangles would you need? triangles ©Curriculum Associates, LLC Copying is not permitted. Lesson 27 Understand Tiling in Rectangles 283 Solve. 3 Draw lines to show how you could use hexagons to make the first shape below Then show how you could use trapezoids to make the second shape How many of each shape would you need? hexagons trapezoids 4 Draw lines to show how you could use triangles to make the first shape below Then show how you could use trapezoids to make the second shape Then show how you could use rhombuses to make the third shape How many of each shape would you need? triangles trapezoids rhombuses 284 Lesson 27 Understand Tiling in Rectangles ©Curriculum Associates, LLC Copying is not permitted. Lesson 27 Name: Draw and Count Squares Study the example showing how to draw and count squares. -
Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle
SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface. -
Quadrilateral Theorems
Quadrilateral Theorems Properties of Quadrilaterals: If a quadrilateral is a TRAPEZOID then, 1. at least one pair of opposite sides are parallel(bases) If a quadrilateral is an ISOSCELES TRAPEZOID then, 1. At least one pair of opposite sides are parallel (bases) 2. the non-parallel sides are congruent 3. both pairs of base angles are congruent 4. diagonals are congruent If a quadrilateral is a PARALLELOGRAM then, 1. opposite sides are congruent 2. opposite sides are parallel 3. opposite angles are congruent 4. consecutive angles are supplementary 5. the diagonals bisect each other If a quadrilateral is a RECTANGLE then, 1. All properties of Parallelogram PLUS 2. All the angles are right angles 3. The diagonals are congruent If a quadrilateral is a RHOMBUS then, 1. All properties of Parallelogram PLUS 2. the diagonals bisect the vertices 3. the diagonals are perpendicular to each other 4. all four sides are congruent If a quadrilateral is a SQUARE then, 1. All properties of Parallelogram PLUS 2. All properties of Rhombus PLUS 3. All properties of Rectangle Proving a Trapezoid: If a QUADRILATERAL has at least one pair of parallel sides, then it is a trapezoid. Proving an Isosceles Trapezoid: 1st prove it’s a TRAPEZOID If a TRAPEZOID has ____(insert choice from below) ______then it is an isosceles trapezoid. 1. congruent non-parallel sides 2. congruent diagonals 3. congruent base angles Proving a Parallelogram: If a quadrilateral has ____(insert choice from below) ______then it is a parallelogram. 1. both pairs of opposite sides parallel 2. both pairs of opposite sides ≅ 3. -
Properties of Equidiagonal Quadrilaterals (2014)
Forum Geometricorum Volume 14 (2014) 129–144. FORUM GEOM ISSN 1534-1178 Properties of Equidiagonal Quadrilaterals Martin Josefsson Abstract. We prove eight necessary and sufficient conditions for a convex quadri- lateral to have congruent diagonals, and one dual connection between equidiag- onal and orthodiagonal quadrilaterals. Quadrilaterals with both congruent and perpendicular diagonals are also discussed, including a proposal for what they may be called and how to calculate their area in several ways. Finally we derive a cubic equation for calculating the lengths of the congruent diagonals. 1. Introduction One class of quadrilaterals that have received little interest in the geometrical literature are the equidiagonal quadrilaterals. They are defined to be quadrilat- erals with congruent diagonals. Three well known special cases of them are the isosceles trapezoid, the rectangle and the square, but there are other as well. Fur- thermore, there exists many equidiagonal quadrilaterals that besides congruent di- agonals have no special properties. Take any convex quadrilateral ABCD and move the vertex D along the line BD into a position D such that AC = BD. Then ABCD is an equidiagonal quadrilateral (see Figure 1). C D D A B Figure 1. An equidiagonal quadrilateral ABCD Before we begin to study equidiagonal quadrilaterals, let us define our notations. In a convex quadrilateral ABCD, the sides are labeled a = AB, b = BC, c = CD and d = DA, and the diagonals are p = AC and q = BD. We use θ for the angle between the diagonals. The line segments connecting the midpoints of opposite sides of a quadrilateral are called the bimedians and are denoted m and n, where m connects the midpoints of the sides a and c. -
The Construction, by Euclid, of the Regular Pentagon
THE CONSTRUCTION, BY EUCLID, OF THE REGULAR PENTAGON Jo˜ao Bosco Pitombeira de CARVALHO Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Cidade Universit´aria, Ilha do Fund˜ao, Rio de Janeiro, Brazil. e-mail: [email protected] ABSTRACT We present a modern account of Ptolemy’s construction of the regular pentagon, as found in a well-known book on the history of ancient mathematics (Aaboe [1]), and discuss how anachronistic it is from a historical point of view. We then carefully present Euclid’s original construction of the regular pentagon, which shows the power of the method of equivalence of areas. We also propose how to use the ideas of this paper in several contexts. Key-words: Regular pentagon, regular constructible polygons, history of Greek mathe- matics, equivalence of areas in Greek mathematics. 1 Introduction This paper presents Euclid’s construction of the regular pentagon, a highlight of the Elements, comparing it with the widely known construction of Ptolemy, as presented by Aaboe [1]. This gives rise to a discussion on how to view Greek mathematics and shows the care on must have when adopting adapting ancient mathematics to modern styles of presentation, in order to preserve not only content but the very way ancient mathematicians thought and viewed mathematics. 1 The material here presented can be used for several purposes. First of all, in courses for prospective teachers interested in using historical sources in their classrooms. In several places, for example Brazil, the history of mathematics is becoming commonplace in the curricula of courses for prospective teachers, and so one needs materials that will awaken awareness of the need to approach ancient mathematics as much as possible in its own terms, and not in some pasteurized downgraded versions. -
Refer to the Figure. 1. If Name Two Congruent Angles. SOLUTION: Isosceles Triangle Theorem States That If Two Sides of T
4-6 Isosceles and Equilateral Triangles Refer to the figure. 1. If name two congruent angles. SOLUTION: Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite those sides are congruent. Therefore, in triangle ABC, ANSWER: BAC and BCA 2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC, ANSWER: Find each measure. 3. FH SOLUTION: By the Triangle Angle-Sum Theorem, Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. ANSWER: 12 eSolutions4. m ManualMRP - Powered by Cognero Page 1 SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: 60 SENSE-MAKING Find the value of each variable. 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem, That is, . ANSWER: 12 6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, . ANSWER: 16 7. PROOF Write a two-column proof. Given: is isosceles; bisects ABC. Prove: SOLUTION: ANSWER: 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown. a. If and are perpendicular to is isosceles with base , and prove that b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Quadrilaterals
Name: Quadrilaterals Match the quadrilateral with its definition. a. 1. All sides are the same length. There are four right angles. Parallelogram b. 2. There is only one pair of parallel sides. Rectangle c. 3. Opposite sides are parallel and the same length. There are four right angles. Trapezoid d. 4. There are two pairs of parallel sides. All sides are the same length. Rhombus 5. There are two pairs of opposite parallel e. sides. Square 6. List two ways a rectangle and square are alike and one way in which they are different. Super Teacher Worksheets - www.superteacherworksheets.com ANSWER KEY Quadrilaterals Match the quadrilateral with its definition. a. e 1. All sides are the same length. There are four right angles. Parallelogram b. c 2. There is only one pair of parallel sides. Rectangle c. b 3. Opposite sides are parallel and the same length. There are four right angles. Trapezoid d. d 4. There are two pairs of parallel sides. All sides are the same length. Rhombus a 5. There are two pairs of opposite parallel e. sides. Square 6. List two ways a rectangle and square are alike and one way in which they are different. Rectangles and squares are alike in that both have two pairs of parallel sides and four right angles. They are different because a square must have four equal sides lengths and a rectangle has two pairs of equal side lengths. Super Teacher Worksheets - www.superteacherworksheets.com. -
Similar Quadrilaterals Cui, Kadaveru, Lee, Maheshwari Page 1
Similar Quadrilaterals Cui, Kadaveru, Lee, Maheshwari Page 1 Similar Quadrilaterals Authors Guangqi Cui, Akshaj Kadaveru, Joshua Lee, Sagar Maheshwari Special thanks to Cosmin Pohoata and the AMSP Cornell 2014 Geometric Proofs Class B0 C0 B A A0 D0 C D Additional thanks to Justin Stevens and David Altizio for the LATEX Template Similar Quadrilaterals Cui, Kadaveru, Lee, Maheshwari Page 2 Contents 1 Introduction 3 2 Interesting Property 4 3 Example Problems 5 4 Practice Problems 11 Similar Quadrilaterals Cui, Kadaveru, Lee, Maheshwari Page 3 1 Introduction Similar quadrilaterals are a very useful but relatively unknown tool used to solve olympiad geometry problems. It usually goes unnoticed due to the confinement of geometric education to the geometry of the triangle and other conventional methods of problem solving. Also, it is only in very special cases where pairs of similar quadrilaterals exist, and proofs using these qualities usually shorten what would have otherwise been an unnecessarily long proof. The most common method of finding such quadrilaterals involves finding one pair of adjacent sides with identical ratios, and three pairs of congruent angles. We will call this SSAAA Similarity. 0 0 0 0 Example 1.1. (SSAAA Similarity) Two quadrilaterals ABCD and A B C D satisfy \A = AB BC A0, B = B0, C = C0, and = . Show that ABCD and A0B0C0D0 are similar. \ \ \ \ \ A0B0 B0C0 B0 C0 B A A0 D0 C D 0 0 0 0 0 0 Solution. Notice 4ABC and 4A B C are similar from SAS similarity. Therefore \C A D = 0 0 0 0 0 0 0 0 0 0 \A − \B A C = \A − \BAC = \CAD. -
13. Dodecagon-Hexagon-Square
In this chapter, we develop patterns on a 12-6-4 grid. To begin, we will first construct the grid of dodecagons, hexagons and squares. Next, we overlay patterns using the PIC method. Begin with a horizontal line and 3 circles as shown. This defines 6 points on the circumference of the center circle. Now several lines are drawn for reference. Begin with the diagonals connecting points on the center circle, then proceed to draw the inner segments making a 6-pointed star. Also, the central rectangle is bolded which will contain the final unit that can be tiled. The size of the hexagon can now be found as follows: First draw 2 circles about the midpoints of the two vertical edges of the unit. Then draw a circle, centered in the middle, that is tangent to these circles. Prepared by circleofsteve.com Then draw a circle of that radius at the 6 points shown below. Where these circles intersect the reference lines drawn, we can identify the sides of hexagons to inscribe. Prepared by circleofsteve.com Now by filing in the squares between the hexagons the dodecagons will appear. Prepared by circleofsteve.com The final grid appears below with the reference lines removed and the midpoints marked from which the incidence pattern can be drawn. Prepared by circleofsteve.com Several angles can naturally be identified based on the existing geometry. For example, 63.5 connects the midpoint of a square side with an opposing corner. Prepared by circleofsteve.com Continuing through all the polygons produces the following. Visible is the underlying grid and overlay design. -
Midsegment of a Trapezoid Parallelogram
Vocabulary Flash Cards base angles of a trapezoid bases of a trapezoid Chapter 7 (p. 402) Chapter 7 (p. 402) diagonal equiangular polygon Chapter 7 (p. 364) Chapter 7 (p. 365) equilateral polygon isosceles trapezoid Chapter 7 (p. 365) Chapter 7 (p. 402) kite legs of a trapezoid Chapter 7 (p. 405) Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards The parallel sides of a trapezoid Either pair of consecutive angles whose common side is a base of a trapezoid A polygon in which all angles are congruent A segment that joins two nonconsecutive vertices of a polygon A trapezoid with congruent legs A polygon in which all sides are congruent The nonparallel sides of a trapezoid A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards midsegment of a trapezoid parallelogram Chapter 7 (p. 404) Chapter 7 (p. 372) rectangle regular polygon Chapter 7 (p. 392) Chapter 7 (p. 365) rhombus square Chapter 7 (p. 392) Chapter 7 (p. 392) trapezoid Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards A quadrilateral with both pairs of opposite sides The segment that connects the midpoints of the parallel legs of a trapezoid PQRS A convex polygon that is both equilateral and A parallelogram with four right angles equiangular A parallelogram with four congruent sides and four A parallelogram with four congruent sides right angles A quadrilateral with exactly one pair of parallel sides Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. -
Day 7 Classwork for Constructing a Regular Hexagon
Classwork for Constructing a Regular Hexagon, Inscribed Shapes, and Isosceles Triangle I. Review 1) Construct a perpendicular bisector to 퐴퐵̅̅̅̅. 2) Construct a perpendicular line through point P. P • A B 3) Construct an equilateral triangle with length CD. (LT 1c) C D II. Construct a Regular Hexagon (LT 1c) Hexagon A six-sided polygon Regular Hexagon A six-sided polygon with all sides the same length and all interior angles the same measure Begin by constructing an equilateral with the segment below. We can construct a regular hexagon by observing that it is made up of Then construct 5 more equilateral triangles with anchor point on point R 6 adjacent (sharing a side) equilateral for one arc and anchor point on the most recent point of intersection for the second ( intersecting) arc. triangles. R III. Construct a Regular Hexagon Inscribed in a Circle Inscribed (in a Circle) All vertices of the inscribed shape are points on the circle Inscribed Hexagon Hexagon with all 6 vertex points on a circle Construct a regular hexagon with side length 퐴퐵̅̅̅̅: A B 1) Copy the length of 퐴퐵̅̅̅̅ onto the compass. 2) Place metal tip of compass on point C and construct a circle. 3) Keep the same length on the compass. 4) Mark any point (randomly) on the circle. C 5) Place the metal tip on the randomly marked • point and mark an arc on the circle. 6) Lift the compass and place the metal tip on the last arc mark, and mark new arc on the circle. 7) Repeat until the arc mark lands on the original point.