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Remember the Isosceles Triangle Theorem?

Remember the Isosceles Triangle Theorem?. A. B. C. Do you remember how to prove it?. A. Construct an angle bisector and then prove the two resulting triangles congruent by SAS. . B. C. D. P. Sometime during class tonight we will prove the following theorem: Crossbar Theorem :

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Remember the Isosceles Triangle Theorem?

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  1. Remember the Isosceles Triangle Theorem? A B C

  2. Do you remember how to prove it? A Construct an angle bisector and then prove the two resulting triangles congruent by SAS.  B C D

  3. P Sometime during class tonight we will prove the following theorem: Crossbar Theorem: If is between and , then intersects segment BC. We’ve seen stranger lines than this, haven’t we !! Do you remember how to prove it? None of Euclid’s postulates or any of the incidence axioms says that the diagram can’t look like this. A Construct an angle bisector and then prove the two resulting triangles congruent by SAS. F C A G B C D D E B

  4. A * B * C is read “B is between A and C.” The Betweenness Axioms B-1 If A * B * C, then A, B, and C are three distinct points lying on the same line, and C * B * A. This axiom fills in the gap that prevented us from proving the statements in exercise 6, chapter 1. 6. (a) Given two points A and B and a third point C between them. Can you think of any way to prove from the postulates that C lies on ?

  5. A * B * C is read “B is between A and C.” The Betweenness Axioms B-1 If A * B * C, then A, B, and C are three distinct points lying on the same line, and C * B * A. This axiom fills in the gap that prevented us from proving the statements in exercise 6, chapter 1. 6. (a) Given two points A and B and a third point C between them. Can you think of any way to prove from the postulates that C lies on ? A * C * B

  6. A * B * C is read “B is between A and C.” The Betweenness Axioms B-1 If A * B * C, then A, B, and C are three distinct points lying on the same line, and C * B * A. B-2 Given any two distinct points B and D, there exist points A, C, and E lying on such that A * B * D, B * C * D, and B * D * E.      D B A C E This guarantees the “completeness” of a line. It guarantees that there can be no one-to-one correspondence between the points on a line and the set of natural numbers.

  7. A * B * C is read “B is between A and C.” The Betweenness Axioms B-1 If A * B * C, then A, B, and C are three distinct points lying on the same line, and C * B * A. B-2 Given any two distinct points B and D, there exist points A, C, and E lying on such that A * B * D, B * C * D, and B * D * E. B-3 If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two. F C A Here, D * F * E, F * E * D, and E * D * F. G D E B

  8. The Betweenness Axioms make some of our definitions more rigorous. Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line and are between A and B.

  9. The Betweenness Axioms make some of our definitions more rigorous. Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line and are between A and B. such that A * C * B.

  10. The Betweenness Axioms make some of our definitions more rigorous. Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line and are between A and B. such that A * C * B. The ray is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on such that B is between A and C.          A B C C C C C

  11. The Betweenness Axioms make some of our definitions more rigorous. :those such that A * B * C. Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line and are between A and B. such that A * C * B. The ray is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on such that B is between A and C. R and are said to be opposite rays if they lie on the same line and A is between B and C

  12. The Betweenness Axioms make some of our definitions more rigorous. :those such that A * B * C. Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line and are between A and B. such that A * C * B. The ray is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on such that B is between A and C. R and are said to be opposite rays if they lie on the same line and A is between B and C. B * A * C.

  13. P Unfortunately, these three betweenness axioms do not help us with this. We are going to need another axiom. But first… A B C D

  14. Proposition 3.1 For any two points A and B: (i)  = AB, and (ii)  = {} The ray is the following set of points: those points that belong to the segment AB and all points C such that A * B * C.

  15. Proposition 3.1 For any two points A and B: (i)  = AB, and (ii)  = {} Proof of (ii) Choose point D in  . Then, D is either on or on or both. If D is on , then either A*D*B or A*B*D. Either way, D is on line Similarly if D is on . definition of  definition of ray Axiom B-1 The ray is the following set of points: those points that belong to the segment AB and all points C such that A * B * C.

  16. Proposition 3.1 For any two points A and B: (i)  = AB, and (ii)  = {} Proof of (ii) 1. Choose point D in  . Then, by definitionof , D is either on or on or both. If D is on , then either A*D*B or A*B*D by definition of ray. Either way, D is on line by axiom B-1. Similarly if D is on . 2. Choose D on line . Either D*A*B, A*D*B, or A*B*D. Axiom B-3 The ray is the following set of points: those points that belong to the segment AB and all points C such that A * B * C.

  17. Proposition 3.1 For any two points A and B: (i)  = AB, and (ii)  = {} Proof of (ii) 1. Choose point D in  . Then, by definition of, D is either on or on or both. If D is on , then either A*D*B or A*B*D by definition of ray. Either way, D is on line by axiom B-1. Similarly if D is on . 2. Choose D on line . Either D*A*B, A*D*B, or A*B*D. Case 1. If D*A*B, then B*A*D. Therefore, D is on ray , and so D Axiom B-3 Axiom B-1 The ray is the following set of points: those points that belong to the segment AB and all points C such that A * B * C.

  18. Proposition 3.1 For any two points A and B: (i)  = AB, and (ii)  = {} Proof of (ii) 1. Choose point D in  . Then, by definition of, D is either on or on or both. If D is on , then either A*D*B or A*B*D by definition of ray. Either way, D is on line by axiom B-1. Similarly if D is on . 2. Choose D on line . Either D*A*B, A*D*B, or A*B*D. Case 1. If D*A*B, then B*A*D. Therefore, D is on ray , and so D Case 2. If A*D*B, then D is on segment AB and so it is on both rays (by part (i)), and, therefore, in their union. Case 3. If A*B*D, then D is on ray , and so D Axiom B-3, Axiom B-1 Therefore,  = {} The ray is the following set of points: those points that belong to the segment AB and all points C such that A * B * C.

  19. Definition: Let l be any line, and A and B any points that do not lie on l. If A = B or if segment AB contains no points lying on l, we say A and B are on the same side of l.whereas, if A ≠ B and segment AB does intersect l, we say that A and B are on opposite sides of l. l A   B

  20. Definition: Let l be any line, and A and B any points that do not lie on l. If A = B or if segment AB contains no points lying on l, we say A and B are on the same side of l,whereas, if A ≠ B and segment AB does intersect l, we say that A and B are on opposite sides of l. A   l  B

  21. Definition: Let l be any line, and A and B any points that do not lie on l. If A = B or if segment AB contains no points lying on l, we say A and B are on the same side of l, whereas, if A ≠ B and segment AB does intersect l, we say that A and B are on opposite sides of l. The fourth Betweenness Axiom (B-4) B-4 (Plane Separation) For every line l and for any three points A, B, and C not lying on l: (i) If A and B are on the same side of l and if B and C are on the same side of l, then A and C are on the same side of l. (ii) If A and B are on opposite sides of l and if B and C are opposite sides of l, then A and C are on same side of l. Corollary (iii) If A and B are on opposite sides of l and if B and C are on the same side as l, then A and C are on opposite sides of l.

  22. Definition A side of a linelis the set of all points that are on the same side of l as some particular point A not on l. We denote this side of l as HA (referred to as a half-plane bounded by l). Here HA = HC l   A C

  23. Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common. Proposition 3.3 Given A * B * C and A * C * D. Then B * C * D and A * B * D. Chapter 1, Exercise 6b Assuming you succeeded in proving C lies on , can you prove from the definition of “ray” and the postulates that = ? C  D1 Choose point D1 on ray B A * D1 * B

  24. Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common. Proposition 3.3 Given A * B * C and A * C * D. Then B * C * D and A * B * D. Chapter 1, Exercise 6b Assuming you succeeded in proving C lies on , can you prove from the definition of “ray” and the postulates that = ? C  D1 B A * D1 * B and A * B * C A B C A C D

  25. Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common. Proposition 3.3 Given A * B * C and A * C * D. Then B * C * D and A * B * D. Chapter 1, Exercise 6b Assuming you succeeded in proving C lies on , can you prove from the definition of “ray” and the postulates that = ? C  D1 B  A * D1 *C A * D1 * B and A * B * C Which proves D1is also on ray . A B C A C D

  26. Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common.  B 1. There is a point A not lying on l. O  Prop 2.3  A 2. There is a point O lying on l. Axiom I-2 l 3. There is a point B such that B * O * A. Axiom B-2 Def. of opp. sides 4. Then A and B are on opposite sides of l, so l bounds at least two half-planes. Let C be any point distinct from A and B and not lying on l. 5. If C and B are on the same side of l, then C is in HB. Definition of half plane 6. If C and B are on opposite sides of l, then C and A are on the same side of l (HA). Therefore, any point not on l is in either HA or HB, so l bounds exactly two half-planes (i.e. the set of points not on l is the union of the side HA of A and the side HB of B). Axiom B-4, part (ii) 7. Suppose a point P were on both sides of l. Then A and B would be on the same side of l. This contradicts step 4. Hence the two sides are disjoint (have no points in common). Axiom B-4, part (i)

  27. Proposition 3.3 Given A * B * C and A * C * D. Then B * C * D and A * B * D. D B B-1 1. A, B, and Care distinct collinear points.  A  E B-1 2. A, C, and D are distinct collinear points  B 3. Suppose points D and B were the same point. Then D is between A and C (i.e. A * D * C). But we are given A * C * D. This contradicts axiom B-3. Thus A, B, C, and D are four distinct collinear points.  C ?  D Prop 2.3 4. There exists a point E not on the line through A, B, C, and D. 5. Consider line . meets in point C and points A and D are on opposite sides of . Definition of opposite sides 6. Claim: A and B are on the same side of . Suppose A and B are on opposite sides of . Then meets in a point between A and B. That must be point C. But that means A * C * B. Yet we know A * B * C, which contradicts axiom B-3. Def of opposite sides Proposition 2.1 7. Therefore, B. But since B, C, and D are collinear, this means C is between B and D (i.e. B * C * D). Corollary to Axiom B-4 Corollary (iii) If A and B are on opposite sides of l and if B and C are on the same side as l, then A and C are on opposite sides of l. Similarly for

  28. Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common. Proposition 3.3 Given A * B * C and A * C * D. Then B * C * D and A * B * D. Corollary Given A * B * C and B * C * D. Then A * B * D and A * C * D. Proposition 3.4: If C * A * B andlis the line through A, B, and C (Axiom B-1), then for every point P lying onl, P lies either on ray or on opposite ray .

  29. Pasch’s Theorem: If A , B, and C are distinct, noncollinear points and l is any line intersecting AB in a point between A and B, then l also intersects either AC or BC. If C does not lie on l, then l does not intersect both AC and BC. l A  B   C Proposition 3.5: Given A * B * C. Then AC = AB  BC and B is the only point common to segments AB and BC. Proposition 3.6: Given A * B * C. Then B is the only point common to rays and , and = .

  30. Definition: Given an angle CAB, define point D to be in the interior of CAB if D is on the same side of as B and if D is also on the same side of as C. B  D A   C Definition: Ray is between rays and if and are not opposite rays and D is interior to CAB. Definition: The interior of a triangle is the intersection of the interiors of its three angles. A point is exterior to the triangle if it is not in the interior and does not lie on any side of the triangle.

  31. Proposition 3.7: Given an angle CAB and point D lying on line . Then D is in the interior of CAB if and only if B * D * C. B  D  A  E  C Proposition 3.8: If D is in the interior of CAB, then (a) so is every other point on ray except A; (b) no point on the opposite ray to is in the interior of CAB; and (c) if C * A * E, then B is in the interior of DAE. B   D A   C

  32. Proposition 3.8: If D is in the interior of CAB, then (a) so is every other point on ray except A; (b) no point on the opposite ray to is in the interior of CAB; and (c) if C * A * E, then B is in the interior of DAE. Definition of ray Def of same side of a line Def of interior of an angle Axiom B-4 (i)  Q of Def of opposite rays Def of opposite sides RAA hypothesis Def of interior of an angle Corollary to Axiom B-4 Def of interior of an angle Contradiction

  33. P Crossbar Theorem: If is between and , then intersects segment BC. The Crossbar Theoremwill eliminate this as a possibility. A B C D

  34. Crossbar Theorem: If is between and , then intersects segment BC. 1. D is in the interior of CAB. C Def of betweenness of rays  2. Let E be a point such that E * A * C. B-2 A   D  G 3. Since line intersects segment EC in point A between E and C, E and C are on opposite sides of line .  E  Def of opposite sides of a line B 4. B is in the interior of DAE. Step 1 and Proposition 3.8c 5. Hence B and E are on the same side of line. Def of interior of an angle 6. Therefore, B and C are on opposite sides of line. Step 3 and Corollary to B-4 7. Let G be the point between B and C that lies on line . Def of opposite sides of a line Step 7 and Proposition 3.7 8. Since B * G * C, G is in the interior of CAB. 9. G lies either on ray or on its opposite ray. Proposition 3.4 10. Suppose G lies on the opposite ray. RAA Hypothesis 11. Then G is not in the interior of CAB. Proposition 3.8b CONTRADICTION 12. Therefore, G lies on ray  ray intersects segment BC.

  35. Crossbar Theorem: If is between and , then intersects segment BC. C  A  D  A  B  Q   B D   C Corollary to the Crossbar Theorem: If point D is in the interior of CAB, then B and C are on opposite sides of . 1. intersects BC at a point Q. 2. B and C are on opposite sides of .

  36. Prove the following proposition: If point D is in the interior of CAB and a ray emanating from D intersects , then the ray does not intersect

  37. Prove the following proposition: If point D is in the interior of CAB and a ray emanating from D intersects , then the ray does not intersect 1. Points B and C are on opposite sides of . 2. Let the given ray from D intersect ray at point E. 3. Assume ray intersects at point R. A 4. Either D * R * E or D *E * R.  E  B  5. In either case, R and E are on the same side of . R  D   6. Points C and E are on opposite sides of . C 7. From steps 1 and 6, points B and E are on the same side of . 8. From steps 5 and 6, points R and C are on opposite sides of . 9. But both R and C are on ray and all points on are on the same side of . Contradiction

  38. Proposition 3.9: (a) If a ray r emanating from exterior point of ABC intersects side AB in a point between A and B, then r also intersects side AC or side BC. (b) If a ray emanates from an interior point of ABC, then it intersects one of the sides, and if it does not pass through a vertex, it intersects only one side.

  39. A  B • Proposition 3.8: If D is in the interior of CAB, then (a) so is every other point on ray • except A; (b) no point on the opposite ray to is in the interior of CAB; • and (c) if C * A * E, then B is in the interior of DAE. • (c) Since and are the same line, we already know that • B and D are on the same side of ( ) since D is in the • interior of CAB. All that remains is to prove that B and E • are on the same side of . • We know C * A * E, so C and E are on opposite sides of . • Since ray is between rays and , B and C are on opposite • sides of . Then by Axiom B-4, part ii, B and E are on the same • side of Therefore, B is in the interior of DAE.   D C  E  What is the flaw in this “proof” of part (c)?

  40. Proposition 3.5: Given A * B * C. Then AC = AB  BC and B is the only point common to segments AB and BC.

  41. Proposition 3.6: Given A * B * C. Then B is the only point common to rays and , and = .

  42. Proposition 3.7: Given and angle CAB and point D lying on line . Then D is in the interior of CAB if and only if B * D * C.

  43. Chapter 3 Homework Read Chapter 3 up to the top half of Page 113. Review Exercises 2 – 7 Exercises: 1, 2b,d, 4a, 8, 9, 15 Major Exercises: None

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