geo postulates theorems

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mandyg233
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61795
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geo postulates theorems
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2011-01-25 01:36:44
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  1. Through any two points there is exactly one line
    postulate
  2. through any three noncollinear points there is exactly one plane containing them
    postulate
  3. If two points lie in a plane, then the line containing those points lies in the plane
    postulate
  4. if two lines intersect, then they intersect in exactly one point
    postulate
  5. if two planes intersect, then they intersect in exactly one line
    postulate
  6. the points on a line can be put into a one-to-one correspondence with the real numbers
    ruler postulate
  7. If B is between A and C, then AB+BC=AC
    Segment Addition Postulate
  8. given line AB and a point O on line AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180
    protractor postulate help classify angles by their measure
  9. If S is in the interior of <PQR, then m<PQS + m<SQR= m<PQR

    < Add. Post.
    angle addition postulate
  10. if p --> q is a true statement and p is true, then q is true
    law of detachment
  11. if p --> and q --> r are true statements, then p --> r is a true statement
    law of syllogism
  12. addition property of equality
    if a=b, then a+c=b+c
  13. subtraction property of equality
    if a=b, then a-c =b-c
  14. multiplication property of equality
    if a=b, then ac=bc
  15. division property of equality
    if a=b, and c DOES NOT = 0, then a/c = b/c
  16. reflexive property of equality
    a=a

    duhhh
  17. symmetric property of equality
    if a=b then b=a
  18. transitive property of equality
    if a=b and b=c, then a=c
  19. substitution property of equality
    if a=b, then b can be substituted for a in any expression
  20. reflexive property of congruence
    figure a ~= figure a

    line EF =~ line EF

    Reflex. Prop. of =~
  21. Symmetric property of congruence
    if figure a ~= figure B, then figure B ~= figure A

    sym. prop. of ~=
  22. Transitive Property of Congruence
    If figure A~= figure B and figure B ~= figure C, than figure A~= figure C

    trans. prop. of ~=
  23. linear pair theorem
    if two angles form a linear pair, then they are supplementary.

    hypothesis: <A and <B form a linear pair

    Conclusion: <a and <b are supplementary
  24. congruent supplements theorem
    if two angles are supplementary to the same angle *or to two congruent sides), then the two angles are congruent

    hypothesis: <1 and <2 are supplementary. <2 and <3 are supplementary

    conclusion: <1 ~= <3
  25. right angle congruence theorem
    all right angles are congruent

    Hypothesis: <a and <b are right angles

    conclusionL <a~=<b
  26. congruent complements theorem
    if two angles are complementary to the same angle (or to 2 congruent <'s) then the 2 <'s are congruent

    Hypothesis: <1 and <2 are complementary. <2 and <3 are complementary

    conclusion: <1~=<3
  27. common segments theorem
    given collinear points A, B, C, and D arranged, if line AB ~= line CD, then line AC~= line BD

    Hypothesis: line AB ~= line CD

    Conclusion: line AC ~= line BD
  28. Vertical Angles Theorem
    Vertical angles are congruent

    Hypothesis: <a and <b are vertical angles

    conclusion: <a ~= <b
  29. if two congruent angles are supplementary, then each angle is a right angle

    ~= <'s suppl. --> right <'s
    • hypothesis:
    • <1~= <2 <1 and <2 are supplementary

    conclusion: <1 and <2 are right angles
  30. corresponding angles postulate
    if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
  31. alternate interior angles theorem
    if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent
  32. alternate exterior angles theorem
    if 2 || lines are cut by a transversal, then the pairs of alternate exterior angles are congruent
  33. same-side interior angles theorem
    if 2 || lines are cut by a transversal, then the 2 pairs of same-side interior angles are supplementary
  34. converse of the corresponding angles postulate
    if 2 coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the 2 lines = ||

    Hypothesis: <1 ~= <2

    conclusion: m || n
  35. parallel postulate
    through a point P not on line l, there is exactly one line parallel to l
  36. converse of the alternate interior angles theorem
    if 2 coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the 2 lines are parallel
  37. converse of the alternate exterior angles theorem
    if 2 coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the 2 lines are parallel
  38. converse of the same side interior angles theorem
    if 2 coplanar lines are cut by a transversal so that a pair of same side interior angles are supplementary, the the 2 lines are parallel
  39. 2 intersecting lines form a linear pair of congruent angles --> the lines are perpendicular
    theorem

    conclusion: l is perpendicular to m
  40. perpendicular transversal theorem
    in a plane, if a transversal is perpendicular to one of 2 parallel lines, then it is perpendicular to the other line

    hypothesis: transversal perpendicular to one parallel line

    conclusion: transversal perpendicular to the other parallel line
  41. 2 lines perpendicular to the same line --> the 2 lines are ||
    theorem
  42. parallel lines theorem
    in a coordinate plane, 2 nonvertical lines are parallel iff they have the same slope.

    any 2 vertical lines are parallel
  43. perpendicular lines theorem
    in a coordinate plane, 2 non vertical lines are perpendicular iff the product of their slopes is -1

    vertical and horizontal lines are perpendicular
  44. triangle sum theorem
    the sum of the angle measures of a triangle is 180 degrees
  45. the acute angles of a right triangle are complementary
    corollary
  46. the measure of each angle of an equiangular triangle is 60 degrees
    corollary

    m<a = m<b = m<c = 60 degrees
  47. exterior angle theorem
    the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles
  48. third angles theorem
    if 2 angles of one triangle are congruent to 2 angles of another triangle ---> third pair of angles are congruent
  49. side side side (SSS) congruence postulate
    if 3 sides of 1 triangle are congruent to 3 sides of another triangle, then the triangles are congruent
  50. side angle side (SAS) congruence
    if 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle --> the triangles are congruent
  51. angle side angle (ASA) congruence
    2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle --> congruent triangles
  52. angle angle side (AAS) congruence theorem
    2 angles and non included side of 1 triangle are ~= to the corresponding angles and non included side of another triangle --> triangles are congruent
  53. Hypotenuse-leg (HL) congruence
    the hypotenuse and a leg of a right triangle are ~= to the hypotenuse and a leg of another right triangle --> the triangles are congruent
  54. CPCTC
    corresponding parts of congruent triangles are congruent
  55. isosceles triangle theorem
    2 sides of a triangle are congruent --> angles opposite those angles are congruent
  56. equilateral triangle corollary
    if a triangle is equilateral --> it's equiangular
  57. equiangular triangle corollary
    equiangular triangle --> equilateral triangle
  58. perpendicular bisector theorem
    point is on the perpendicular bisector of a segment --> it is equidistant from the endpoints of the segment
  59. converse of the perpendicular bisector theorem
    a point is equidistant from the endpoints of a segment ---> perpendicular bisector of the segment
  60. angle bisector theorem
    if a point is on the bisector of an angle --> equidistant from the sides of the angle
  61. converse of the angle bisector theorem
    if a point in the interior of an angle is equidistant from the sides of the angle --> on the bisector of the angle
  62. circumcenter theorem
    the circumcenter of a triangle is equidistant from the vertices of the triangle

    can be inside the triangle, outside the triangle, or on the triangle
  63. incenter theorem
    the incenter of a triangle is equidistant from the sides of the triangle
  64. centroid theorem
    the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side
  65. triangle midsegment theorem
    a midsegment of a triangle is || to a side of the triangle, and it's length is half the length of that side.
  66. if 2 sides of a triangle are not congruent --> larger angle is opposite the longer side
    theorem
  67. in triangle, the longer side is oposite the larger angle
    theorem
  68. triangle inequality theorem
    the sum of any 2 side lengths of a triangle is greater than the 3rd side length

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