1. Chapter 2 Construction  Proving

2. Historical Background Euclid’s Elements Greek mathematicians used  Straightedge  Compass – draw circles, copy distances  No measurement

3. 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

4. Euclid’s Postulates 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)

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

6. Euclid’s Postulates From Wikimedia Commons

7. Congruence Ordinary meaning:  Two things agree in nature or quality Mathematics:  Exactly same size and shape  Note: all circles have same shape, but not same size A C B

8. Congruence What does it take to guarantee two triangles congruent?  SSS?  ASA?  SAS?  SSA?  AAS?  AAA?

9. Congruence Criteria for Triangles SAS: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. We will accept this axiom without proof

10. Angle-Side-Angle Congruence State the Angle-Side-Angle criterion for triangle congruence (don’t look in the book) ASA: If two angles and the included side of one triangle are congruent respectively to two angles and the included angle of another triangle, then the two triangles are congruent

11. Angle-Side-Angle Congruence Proof Use negation Justify the steps in the proof on next slide

12. ASA Assume AB   DE

13. Similarity Definition  Exactly same shape, perhaps different size  Note: What if A and B are same height or same area?  What does it take to guarantee similar triangles? Any two polygons similar? C BA

14. Similarity Similar triangles can be used to prove the Pythagorean theorem  Note which triangles are similar  Note the resulting ratios

15. Constructions Be sure to use Geogebra to construct robust figures  If a triangle is meant to be equilateral, moving a vertex should keep it equilateral

16. Constructions Classic construction challenges  Doubling a cube  Squaring a circle  Trisecting an angle

17. Geometric Language Revisited Reminder  Constructions limited to straight edge & compass Straight edge for  Line, line segment, ray

18. Geometric Language Revisited Typical constructions  Finding midpoint  Finding “center” (actually centers) of different polygons  Tangent to a circle (must be  to radius)  Angle bisector Note Geogebra has tools to do some of these without limits of compass, straightedge … OK to use most of time

19. Conditional Statements Implication P implies Q if P then Q

20. Conditional Statements Viviani’s Theorem IF a point P is interior to an equilateral triangle THEN the sum of the lengths of the perpendiculars from P to the sides of the triangle is equal to the altitude.

21. Conditional Statements What would make the hypothesis false? With false hypothesis, it still might be possible for the lengths to equal the altitude

22. Conditional Statements Consider a false conditional statement  IF two segments are diagonals of a trapezoid THEN the diagonals bisect each other How can we rewrite this as a true statement

23. Conditional Statements Where is this on the truth table? We want the opposite  IF two segments are diagonals of a trapezoid THEN the diagonals do not bisect each other TRUE statement

24. Robust Constructions & Proofs Robust construction in Geogebra  Dynamic changes of vertices keep properties that were constructed Shows specified relationship holds even when some of points, lines moved  Note: robust sketch is technically not a proof Robust sketch will help formulate proof

25. Angles & Measuring Classifications of angles  Right  Acute  Obtuse  Straight Measured with  Degrees  Radians  Gradients

26. Constructing Perpendiculars, Parallels Geogebra has tools for doing this In certain situations the text asks for use of straight edge & compass only

27. Properties of Triangles Classifications  Equilateral  Isosceles  Scalene  Right  Obtuse  Acute  Similar

28. Properties of Triangles Consider relationships between interior angles and exterior angles. State your observations, conjectures

29. Properties of Triangles Conjecture 1  If an exterior angle is formed by extending one side of a triangle, then this exterior angle will be larger than the interior angles at each of the other two vertices.

30. Properties of Triangles Conjecture 2  If an exterior angle is formed by extending one side of a triangle, then the measure of this exterior angle will be the same as the sum of the measures of the two remote interior angles of the same triangle.

31. Properties of Triangles Corollary to 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 that is perpendicular to a given line. Proof by contradiction … assume two  ’s

32. 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.

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

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

35. Recall 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.

36. Use of Postulates for Constructions Use to prove possibility of construction  Then use that result to establish next Example: Equilateral triangles can be constructed with a straight edge and compass  Based on Proposition 1 in Elements

37. Use of Postulates for Constructions A line segment can be copied from one location to another with a straightedge and a compass.  Based on Propositions 2 and 3 in the Elements This figure specified a “floppy” compass for the construction

38. 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

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

40. Ideas about “Betweenness” Pasch’s theorem: If A, B, and C are distinct, non- collinear points and L is a line that intersects segment AB, then L also intersects either segment AC or segment BC. Note proof on pg 43

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

42. Chapter 2 Construction  Proving