# geo postulates theorems

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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
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

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
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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 Author: mandyg233 ID: 61795 Filename: geo postulates theorems Updated: 2011-01-25 06:36:44 Tags: homework Folders: Description: no description. im most likely going to still fail. Show Answers:

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