Geometric Shapes

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Chapter 12

• Geometric Shapes

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12.1 Recognizing Geometric Shapes

• The van Hiele Theory
• Level 0 Recognition
• Some relevant attributes of shape, straightness
  of lines, may be ignored.
• Some irrelevant attributes, such as orientation
  on a page, may be stressed.
• Recognizes shapes holistically without noticing
  component parts.

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The van Hiele Theory

• Level 1 Analysis
• Focus is on parts of figures, such as sides and
  angles.
• Relevant attributes are understood and
  differentiated from irrelevant ones.
• Analytical thinkers may not believe a figure can
  belong to several classes, and have several
  names.

A square is a rhombus and also a rectangle.
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• Level 2 Relationships
• A child does understand abstract relationships
  among figures.
• A child can also use deduction to justify
  observations made at level 1.

Level 3 Deduction Reasoning involves the study
of geometry as a formal mathematical system. A
child who reasons at level 3 understands the
notions of postulates and theorems and can write
formal proofs of theorems.
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• Level 4 Axiomatics
• Level 4 is highly abstract and does not
  necessarily involve concrete or pictorial models.
• Theorems and postulates are at the center of
  interest, and are the subject of intense
  scrutiny.
• This level of study is found typically at the
  college level.

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Recognizing Geometric Shapes

• Children are taught the basic shapes, and many
  achievement tests ask them to identify the
  square, triangle, etc.
• Too often, children have only seen the regular
  shapes, and thus, do not have a complete idea of
  the important attributes a shape must have to
  represent a general type.

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• Making new shapes by rearranging other shapes is
  a good way for students to develop visualization
  skills.

How many different rectangles are formed by the
heavy line segments?
Seven vertical rectangles, and two horizontal
rectangles, for a total of nine.
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Defining Common Geometric Shapes

• Line segment congruent line segments
• Angle congruent angles, equiangular
• Vertex (vertices)
• Quadrilateral parallelogram, rhombus, rectangle,
  square, kite
• Right angle
• Perpendicular
• Triangle Isosceles, equilateral, scalene
• Trapezoid isosceles trapezoid

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12.2 Analyzing Shapes

• Symmetry
• Reflection If a figure can be folded across a
  line so that one side of figure completely
  matches the other side, the figure has reflection
  symmetry. This line is called the line of
  symmetry.

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• Rotation Symmetry
• If there is a point around which the figure can
  be rotated, less than 360, and the image matches
  the original figure perfectly, the figure is said
  to have rotational symmetry.

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Perpendicular Line Segments Test

• Let P be the point of intersection of l and m.
  Fold l at point p so that l folds across P onto
  itself. Then l and m are perpendicular if and
  only if m lies along the fold line.

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Parallel Line Segments Test

• Fold so that l folds onto itself. Any fold line
  can be used except for l itself. Then l and m
  are parallel if and only if m folds onto itself
  or an extension of m.

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Regular Polygons

• A simple closed curve in the plane is a curve
  that does not cross itself, and has the same
  starting and stopping point.

A polygon is a simple closed curve comprised of
line segments.
A regular polygon is a polygon with all sides
congruent.
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Concave vs. Convex

• A convex shape is a simple closed curve which can
  completely contain a line segment within its
  interior.

A concave shape is a simple closed curve which
cannot contain a line segment within its interior.
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Pentagon
Exterior Angle
Central Angle
Vertex Angle
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Circles

• A circle is the set of all points in the plane
  that are a fixed distance away from a given
  point. The given point is called the center, and
  the fixed distance is called the radius.

Radius
Center
Diameter
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12.3 Properties of Geometric Shapes

• An infinitely large flat surface is called a
  plane.
• Points are locations within a plane.
• Connecting two points in a plane forms a line.
  Lines are straight, and extend infinitely long in
  each direction.
• Points that lie on the same line are collinear
  points.
• Two lines in the plane are parallel if they do
  not intersect.
• Three or more lines that contain the same point
  are concurrent lines.

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Angles
A
D
B
C
E
F
Obtuse
Acute
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Transversals
1
3
2
4
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Parallel Lines
1
3
5
2
4
6
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• Theorem
• Suppose that lines l and m are cut by a
  transversal t. Then l m if and only if
  alternate interior angles formed by l, m, and t
  are congruent.

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12.4 Regular Polygons and Tessellations
Angle Measures A regular n-gon is both
equilateral and equiangular.
What about vertex angles?
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Vertex Angles
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Tessellations

• A polygonal region is a polygon together with its
  interior.
• An arrangement of polygonal regions having only
  sides and vertices in common that completely
  covers the plane is a tessellation or tiling.

Theorem Only regular 3-gons, 4-gons and 6-gons
form tessellations of the plane by themselves.
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12.5 Describing Three-Dimensional Shapes

• A dihedral angle is formed by the union of
  polygonal regions in space that share an edge.
  The regions forming the angle are called faces.

Non-intersecting, non-parallel lines are called
skew lines. Two lines in 3-space can be parallel,
can intersect, or can be skew lines.
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Polyhedra

• A polyhedron is the union of polygonal regions,
  any two of which have at most a side in common,
  such that a connected finite region in space is
  enclosed without holes.

A polyhedron is convex if every line segment
joining two of its points is contained inside the
polyhedron or is on one of its faces
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Face
Edge
Vertex
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General Types of Polyhedra

• Prisms are polyhedra with one pair of opposite
  faces that are identical. These faces are called
  bases, and the lateral faces are parallelograms.

Pyramids are polyhedra formed by using a polygon
for the base and a point on the plane of the
base, called the apex, that is connected to each
vertex of the base with a line.
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• A regular polyhedron is one in which all faces
  are identical regular polygonal regions. There
  are exactly five regular convex polyhedra, called
  the Platonic Solids.

Cube
Tetrahedron
Octahedron
Dodecahedron
Icosahedron
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• Semiregular polyhedra have several different
  regular polygonal regions, with the same
  arrangement of polygons at each vertex.

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Curved Shapes in Three Dimensions

• Cylinders

Right Cylinder
Oblique Cylinder
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• Cones

Apex
Right circular cone
Oblique circular cone