Discovering Geometric Conjectures through Exploration: Chapter 1

1. Chapter 1 Using Geogebra Exploration and Conjecture

2. 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 • Let’s 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

6. 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

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

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

10. 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.)

11. 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.

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

13. Playfair’s Postulate • Given any line and any point P not on , there is exactly one line through P that is parallel to .

14. Euclid’s Postulates • Given two distinct points P and Q, there is a line ( that is, there is exactly one line) that passes through P and Q. • Any line segment can be extended indefinitely. • Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn. • Any two right angles are congruent. Accepted as axioms. We will not attempt to prove them

15. Euclid’s 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)

16. Euclid’s Postulates From Wikimedia Commons

17. Euclid’s Postulates • Recall results of activity 3 What was the relationship of these two angles?

18. Euclid’s 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.

19. 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?

20. 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

21. Ideas about “Betweenness” • When a line enters a triangle crossing side AB • What are all the ways it can leave the triangle?

22. Ideas about “Betweenness” • 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

23. Ideas about “Betweenness” • Crossbar Theorem: • Use Pasch’s theorem to prove

24. 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

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

26. 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?

27. 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?

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

29. 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?

30. 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?

31. 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

32. 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

33. 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

34. 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

35. Exploration and ConjectureInductive Reasoning • A conjecture is expressed in the formIf … [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

36. 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

37. 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

38. Chapter 1 Using Geogebra Exploration and Conjecture