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Midterm Review - Definitions:
Consistent Axiom system
Model for Axiom system
Independent Axiom
Incidence Geometry
Euclidean Parallel postulate
Euclid’s 5th postulate
Segment, angle congruence
Isoceles, equilateral triangle
parallel, perpendicular, supplementary,
complementary
Saccheri, Legendre quadrilaterals
3.2 Neutral Geometry
Thm: Pasch’s Axiom
The Crossbar Thm:
Isoceles Triangle Thm: If two sides of a
triangle are congruent, then the angles opposite
those sides are also congruent.
Thm: A point is on the perpendicular bisector of
a line segment iff it is equidistant from the
endpoints of that segment.
**Exterior Angle Theorem: An exterior angle of
a triangle is greater in measure than either of
the nonadjacent interior angles.
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2
3.3
***Thm: ASA Triangle 
Converse of Isosceles Triangle theorem:
***AAS Triangle  Theorem:
Inverse of the isosceles triangle theorem:
If two sides of a triangle are not , then the
angles opposite those sides are not , and the
larger angle is opposite the larger side.
Thm: If 2 angles of a triangle are not , then
the sides opposite them are not , and the larger
side is opposite the larger angle.
Triangle Inequality:
Hinge Thm:
***SSS  Theorem
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3.4, Parallels
***Alt. Interior angle theorem:
to the same line are .
Corollary: 2 lines
Corollary: Given a line  and a point P not on ,
there is at least one line through P  to .
Corollary: If 2 lines are intersected by a
transversal such that a pair of corresponding
s are , then the lines are .
Corollary: If 2 lines are intersected by a
transversal such that a pair of interior s on the
same side of the transversal is supplementary,
then the lines are .

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Theorem: Euclid’s 5th postulate is equivalent
to the Euclidean  postulate.
Thm: The Euclidean  postulate is equivalent
to the converse of the alternate interior angle
theorem.
Thm: The Euclidean  postulate is equivalent
to one
to the following statement: If a line is
of 2  lines, it is
to the other.
Thm: The Euclidean  postulate is equivalent
to the following statement: Given any PQR and
any line segment AB,  a ABC such that PQR
is similar to ABC.
Thm: the Euclidean  postulate is equivalent
to: The angle sum of every  is 180 


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3.5
***Lemma: The sum of the measures of any two
angles of a triangle is less than 180  .
Lemma: For any ABC,  A 1 B 1 C 1 having the
same angle sum with mA 1  1 mA
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Saccheri-Legendre Theorem The angle sum
of any triangle is less than or equal to 180  .
Theorem: The angle sum of any convex
quadrilateral is less than or equal to 360 
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Know the implications:
3.6
In Saccheri quadrilaterals:





***The diagonals are congruent
***The summit s are congruent
The summit s are not obtuse
The line jointing the midpoints of the summit and
to both
base is

The summit and base are 
Thm: In any Saccheri quadrilateral, the length of
the summit is  the length of the base.
For Lambert quadrilaterals:


the 4th  is not obtuse.
***The measure of a side between 2 right angles
is  the measure of the opposite side.
Thm: The measure of the line joining the
midpoints of the base and summit of a Saccheri
quadrilateral is  the measure of its sides.
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1 rectangle exists,
  a rectangle with 2 arbitrarily large sides 
  a rectangle with 2 adjacent sides  to two
given line segments
 every right triangle has angle sum 180  .
 every  has angle sum 180  .
 1  has angle sum 180 
 1 right  has angle sum 180 
 a rectangle exists.
Thm: the Euclidean  postulate is equivalent
to: The angle sum of every  is 180 
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