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Chapter Three

Chapter Three. Building Geometry Solidly. Incidence Axioms. I-1: For every point P and for every point Q not equal to P there exists a unique line l incident with P and Q. I-2: For every line l there exist at least two distinct points that are incident with l .

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Chapter Three

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  1. Chapter Three Building Geometry Solidly

  2. Incidence Axioms I-1: For every point P and for every point Q not equal to P there exists a unique line l incident with P and Q. I-2: For every line l there exist at least two distinct points that are incident with l. I-3: There exist three distinct points with the property that no line is incident with all three of them.

  3. Betweenness Axioms (1) B-1 If A*B*C, then A,B,and C are three distinct points all 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 BD 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.

  4. P-3.1: For any two points A and B: • Def: Let l be any line, A and B any points that do not lie on l. If A = B or if segment AB contains no point lying on l, we say A and Be are on the same sides of l. • Def: If A  B and segment AB does intersect l, we say that A and B are opposite sides of l.

  5. Betweenness Axioms (2) B-4:For every linel 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 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 B and C are on opposite sides of l, then A and C are on the same side of l. Corollary (iii) If A and B are on opposite sides of l and B and C are on the same side of l, then A and C are on opposite sides of l.

  6. P-3.2: Every line bounds exactly two half-planes and these half-planes have no point in common. P-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. P-3.4: Line Separation Property: If C*A*B and l is the line through A, B, and C (B-1), then for every point P lying on l, P lies either on ray or on the opposite ray .

  7. Pasch’s Theorem If A, B, 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 interesect both AC and BC.

  8. Def:Interior of an angle. Given an angle CAB, define a 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.P-3.5: Given A*B*C. Then AC = ABBC and B is the only point common to segments AB and BC.P-3.6: Given A*B*C. Then B is the only point common to rays and , and P-3.7: Given an angle CAB and point D lying on line . Then D is in the interior of CAB iff B*D*C.

  9. P3.8: 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. Crossbar Thm: If is between and , then intersects segment BC.

  10. A ray is between rays and if and are not opposite rays and D is interior to CAB. • The interior of a triangle is the intersection of the interiors of its three angles. • P-3.9: (a) If a ray r emanating from an ex-terior 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.

  11. Congruence Axioms .

  12. Congruence Axioms (1) • C-1: If A and B are distinct points and if A' is any point, then for each ray r emanating from A' there is a unique point B' on r such that B' ≠ A' and AB  A'B'. • C-2: If AB  CD and AB  EF, then CD  EF. Moreover, every segment is congruent to itself. • C-3: If A*B*C, A'*B'*C', AB  A'B', and BC  B'C', then AC  A'C'.

  13. Congruence Axioms (2) • C-4: Given any angle BAC (where by defini-tionof "angle” AB is not opposite to AC ), and given any ray emanating from a point A’, then there is a unique ray on a given side of line A'B' such that B'A'C' = BAC. • C-5: If A B and A C, then B C. Moreover, every angle is con-gruent to itself.

  14. Congruence Axioms (3) • C-6: (SAS). If two sides and the included angle of one triangle are congruent respec-tively to two sides and the included angle of another triangle, then the two triangles are congruent. • Cor. to SAS: Given ABC and segment DE  AB, there is a unique point F on a given side of line such that ABC  DEF.

  15. Propositions 3.10 - 12 • P3.10: If in ABC we have AB  AC, then  B  C. • P3.11:(Segment Substitution): If A*B*C, D*E*F, AB  DE, and AC  DF, then BC  EF. • P3.12: Given AC  DF, then for any point B between A and C, there is a unique point E between D and F such that AB  DE.

  16. Definition: • AB < CD (or CD > AB) means that there exists a point E between C and D such that AB  CE.

  17. Propositions 3.13 • P3.13:(Segment Ordering): • (a) (Trichotomy): Exactly one of the following conditions holds: AB < CD, AB  CD, or AC > CD; • (b) If AB < CD and CD  EF, then AB < EF; • (c) If AB > CD and CD  EF, then AB > EF; • (d) (Transitivity): If AB < CD and CD < EF, then AB < EF.

  18. Propositions 3.14 - 16 • P3.14: Supplements of congruent angles are congruent. • P3.15:(a) Vertical angles are congruent to each other. • (b) An angle congruent to a right angle is a right angle.  • P3.16: For every line l and every point P there exists a line through P perpen- dicular to l.

  19. Propositions 3.17 - 19 • P3.17:(ASA Criterion for Congruence): Given ABC and DEF with  A  D,  C  F, and AC  DF. Then ABC DEF. • P3.18: If in ABC we have  B  C, then AB  AC and ABC is isosceles. • P3.19:(Angle Addition): Given between and , between and ,  CBG  FEH, and  GBA  HED. Then  ABC  DEF.

  20. Proposition 3.20 • P3.20:(Angle Subtraction): Given between and , between and ,  CBG  FEH, and  ABC  DEF. Then  GBA  HED. • Definition: •  ABC <  DEF means there is a ray between and such that  ABC  GEF.

  21. Proposition 3.21 Ordering Angles • P3.21:(Ordering of Angles): • (a) (trichotomy): Exactly one of the following conditions holds: •  P <  Q,  P  Q, or  P >  Q • (b) If  P <  Q, and  Q  R, then  P <  R; • (c) If  P >  Q, and  Q  R, then  P >  R; • (d) If  P <  Q, and  Q <  R, then  P <  R.

  22. Propositions 3.22 - 23 • P3.22:(SSS Criterion for Congruence): Given ABC and DEF. If AB  DE, and BC  EF, and AC  DF, then ABC DEF. • P3.23:(Euclid's 4th Postulate): All right angles are congruent to each other.

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