chapter 1 using geogebra exploration and conjecture

Chapter 1

Using Geogebra

Exploration and Conjecture

Questions, Questions

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


• You discovered the same thing as a man named Varigon in 1731

• Let’s state precisely what we mean by “parallelogram”

• What could you add to the figure to help you verify parallel sides?


• What happened when the edges of the quadrilateral crossed each other?

• Is this still a quadrilateral?


• What conjecture(s) concerning the sum of the perpendicular segments?

• This illustrates a theorem from Viviani


• What about when the point is outside the triangle?

• What aboutisoscelestriangles?

• Scalene?

• Squares, pentagons?

• Distance to vertices?


• Consider some vocabulary we are using

• When is a point on a triangle or circle interior exterior

Language of Geometry

Definitions:

• Polygon Triangle, quadrilateral, hexagon, etc.

• Self-intersecting figures

• Convex, concave figures

• Special quadrilaterals rectangle, square, kite, rhombus, trapezoid,

parallelogram, etc.


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

requirements


Definitions:• Transversal

alternate interior/exterior angles

• Angle classifications right, acute, obtuse, (obese?) perpendicular, straight

• Angle measurement radians, degrees, grade (used in highway

const.)

Euclid’s 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.

Clavius’ Axiom

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

Playfair’s Postulate

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

Euclid’s 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

Accepted as

axioms.

We will not

attempt to

prove them


5. 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)


FromWikimedia Commons


• Recall results of activity 3

What was the relationship of these two angles?


• 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.

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?

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


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

triangle?


• Pasch’s 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


• Crossbar Theorem:

• Use Pasch’s theorem to prove

If AD is between AC and AB ,

then AD intersects segment BC.

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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

Properties of Triangles

• Consider the exterior angle of a triangle (from Activity 5)

• What conjectures did you make?


• 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?


• 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?

Properties of Quadrilaterals

• Recall convex quadrilateral from activity 2

• Consider how properties of diagonals can be a definition of convex


• 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?


• 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?

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


• 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


• 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


• Consider the circle from Activity 8PCR called a straight

anglePQR is inscribed in a

semi circle• What was your conjecture

about an angle inscribedin a semi circle? Prove your conjecture

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


• 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


• 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

Chapter 1

Using Geogebra

Exploration and Conjecture

chapter 1 using geogebra exploration and conjecture

Documents

quadrilateral slide

conjecture slide

viviani slide

interior exterior slide

right angles

interior angles

given line

straight lines