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Chapter Three. Building Geometry Solid. 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
Chapter Three

Building Geometry Solid

incidence axioms
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.

betweenness axioms 1
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.

slide4
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.
betweenness axioms 2
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.

p 3 2 every line bounds exactly two half planes and these half planes have no point in common
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 .

pasch s theorem
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.

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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.

slide9

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 AD 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.

slide10
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.

slide11
C-1: If A and B are distinct points and if A' is any point, then for each ray r emana-ting 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'.
  • 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.
slide12
C-5: If A B and A C, then B C. Moreover, every angle is con-gruent to itself.
  • 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.