Using Geogebra

1
Chapter 1

• Using Geogebra
• Exploration and Conjecture

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Questions, Questions

• What kind of figure did you observe when you
  connected midpoints of the quadrilateral

3
Questions, Questions

• You discovered the same thing as a man named
  Varigon in 1731
• Lets state precisely what we mean by
  parallelogram
• What could you add to the figure to help you
  verify parallel sides?

4
Questions, Questions

• What happened when the edges of the quadrilateral
  crossed each other?
• Is this still a quadrilateral?

5
Questions, Questions

• What conjecture(s) concerning the sum of the
  perpendicular segments?
• This illustrates a theorem from Viviani

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Questions, Questions

• What about when the point is outside the
  triangle?
• What aboutisoscelestriangles?
• Scalene?
• Squares, pentagons?
• Distance to vertices?

7
Questions, Questions

• Consider some vocabulary we are using
• When is a point
• on a triangle or circle
• interior
• exterior

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Language of Geometry

• Definitions
• Polygon
• Triangle, quadrilateral, hexagon, etc.
• Self-intersecting figures
• Convex, concave figures
• Special quadrilaterals
• rectangle, square, kite, rhombus, trapezoid,
  parallelogram, etc.

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Language of Geometry

• Definition of a geometric figure
• Use the smallest possible list of requirements
• Consider why we minimize the list of requirements

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Language of Geometry

• Definitions
• Transversal
• alternate interior/exterior angles
• Angle classifications
• right, acute, obtuse, (obese?)
• perpendicular, straight
• Angle measurement
• radians, degrees, grade (used in highway const.)

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Euclids Fifth Postulate

• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than the sum of two right angles, then
  the two straight lines, if produced indefinitely,
  meet on that side on which the angles are less
  than two right angles.

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Clavius Axiom

• The set of points equidistant from a given line
  on one side of it forms a straight line (
  Hartshorne, 2000, 299).

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Playfairs Postulate

• Given any line and any point P not on ,
  there is exactly one line through P that is
  parallel to .

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Euclids Postulates

1. Given two distinct points P and Q, there is a
   line ( that is, there is exactly one line) that
   passes through P and Q.
2. Any line segment can be extended indefinitely.
3. Given two distinct points P and Q, a circle
   centered at P with radius PQ can be drawn.
4. Any two right angles are congruent.

Accepted as axioms. We will not attempt to prove
them
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Euclids Postulates

• If two lines are intersected by a transversal in
  such a way that the sum of the degree measures of
  the two interior angles on one side of the
  transversal is less than the sum of two right
  angles, then the two lines meet on that side of
  the transversal.
• (Accepted as an axiom for now)

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Euclids Postulates
From Wikimedia Commons
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Euclids Postulates

• Recall results of activity 3

What was the relationship of these two angles?
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Euclids Postulates

• Result written as an if and only if statement
• Two lines are parallel iff the sum of the degree
  measures of the two interior angles formed on one
  side of a transversal is equal to the sum of two
  right angles.

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Congruence

• Intuitive meaning
• Two things agree in nature or quality
• In mathematics
• Two things are exactly the same size and shape
• What are two figures that are the same shape but
  different size?

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Ideas about Betweenness

• Euclid took this for granted
• The order of points on a line
• Given any three collinear points
• One will be between the other two

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Ideas about Betweenness

• When a line enters a triangle crossing side AB
• What are all the ways it can leave the triangle?

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Ideas about Betweenness

• Paschs theorem If A, B, and C are
  distinct, noncollinear points and is a line that
  intersects segment AB, then also intersects
  either segment AC or segment BC.
• Note proof on pg 16

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Ideas about Betweenness

• Crossbar Theorem
• Use Paschs theorem to prove

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Constructions

• Consider the distinction between
• Drawing a figure
• Constructing a figure
• For construction we will limit ourselves to
  straight edge and compass
• Available as a separate file
• View example

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Properties of Triangles

• Consider the exterior angle of a triangle (from
  Activity 5)
• What conjectures did you make?

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Properties of Triangles

• Conjecture 1 An exterior angle of a triangle will
  have a greater measure than either of the
  nonadjacent interior angles.
• Conjecture 2 The measure of an exterior angle of
  a triangle will be the sum of the measures of the
  two nonadjacent interior angles.
• How to prove these?

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Properties of Triangles

• Corollary to the Exterior Angle Theorem
• A perpendicular line from a point to a given line
  is unique. In other words, from a specified
  point, there is only one line perpendicular to a
  given line.
• How to prove?

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Properties of Quadrilaterals

• Recall convex quadrilateral from activity 2
• Consider how properties of diagonals can be a
  definition of convex

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Properties of Quadrilaterals

• Consider the cyclic quadrilateral of activity 9
• Cyclic means vertices lie on a common circle
• It is an inscribed quadrilateral
• What conjectures did you make?

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Properties of Quadrilaterals

• How would results of activity 7 help prove this?
• What if center of circle isexterior to
  quadrilateral
• What if quadrilateral is self intersecting?
• What if diagonals of a quadrilateral bisect each
  other what can be proven from this?

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Properties of Circles

• Definition a set of points equidistant from a
  fixed center
• circle does not include the center
• fixed distance from center is the radius
• Points closer than the fixed distance are
  interior
• Points farther are exterior

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Properties of Circles

• Consider given circle
• PR is a fixed chord
• Q is any other point
• ?PQR subtended by chord PR
• ?PQR inscribed in circle
• ?PCR is a central angle

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Properties of Circles

• In Activity 7, what happenswhen you move point
  Qaround the circle
• What was the relationshipbetween the central
  angleand the inscribed angle?
• What if the central angle is equal to or greater
  than 180??
• Prove your conjectures

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Properties of Circles

• Consider the circle from Activity 8
• ?PCR called a straightangle
• ?PQR is inscribed in a semi circle
• What was your conjectureabout an angle
  inscribedin a semi circle?
• Prove your conjecture

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Exploration and ConjectureInductive Reasoning

• A conjecture is expressed in the form If
  hypothesis then conclusion
• Hypothesis includes
• assumptions made
• facts or conditions given in problem
• Conclusion
• what you claim will always happen if conditions
  of hypothesis hold

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Exploration and ConjectureInductive Reasoning

• Process of
• making observations
• formulating conjectures
• This is called inductive reasoning
• Dynamic geometry software helpful to make
  observations, conjectures
• drag objects around to see if conjecture holds

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Exploration and ConjectureInductive Reasoning

• Next comes justifying the conjectures
• finding an explanation why conjecture is true
• This is the proof
• relies on deductive reasoning
• Chapter 2 investigates
• rules of logic
• deductive reasoning

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Chapter 1

• Using Geogebra
• Exploration and Conjecture