Download Definitions Synthetic Geometry- the study of description of points

Definitions
1. Synthetic Geometry- the study of description of points and lines
2. Description of a point- A point is an exact location
3. Description of a line- A line is a set of points extending in both directions containing the
shortest path between any two points on it.
4. Distance- Between two points on a coordinatized line is the absolute value of the difference
of their coordinates. │x-y│
5. Plane- Set of points thought of as something flat
6. Ordered plane- location of a point. Coordinated determined by x and y axis
7. Figure- Set of points
8. Space- the set of all points in the geometry
9. Collinear- three or more points are collinear if and only if they are on the same line.
10. Coplanar- Four or more points are coplanar if and only if they are in the same plane.
11. Segment or Line Segment- set consisting of the distinct points A and B and all points
between A and B written as
12. Endpoints- a point or value that marks the end of a ray or one of the ends of a line segment
or interval.
13. Ray- A line with a start point but no end point (it goes to infinity)
14. Opposite Rays- two rays that both start from a common point and go off in exactly opposite
directions.
15. Convex Set- Set in which every segment that connects points of the set lies entirely in the
set.
16. Instance of a Conditional- Satisfies the hypothesis and conclusion
17. Counterexample to a Conditional- Satisfies the hypothesis but NOT the conclusion
18. Converse- p→q is q→p
19. Midpoint- Point halfway between two points
20. Circle- set of all points in a plane at a certain distance, its radius, from a certain point , its
center
21. Union of Two Sets- A and B is the set of elements which are in A, in B, OR IN BOTH
22. Intersection of Two Sets- A and B is the set of elements which are in both A and B
23. Polygon- Plane shape with straight sides
24. Sides- The segments that make a polygon
25. Vertices- Endpoints of the sides
26. Consecutive (or adjacent) vertices- End points of a side
27. Consecutive (or adjacent) Sides- Sides that share an endpoint
28. Angle- Union of two rays that have the same endpoint
29. Bisector- Line that divided one thing into two equal parts
30. Degree of a minor arc or semicircle- Measure of its central angle
31. Degree measure of a major- is a 360 degree
32. Right Angle- Angle with a measure of 90 degrees
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Obtuse Angle- Angle with a measure of 91+
Acute Angle- Angle with a measure of 89 or lower
Straight Angle- Angle with the measure of 180
Angle Bisector- Middle angle between 2 angles
Complementary Angles- Angles that combine to equal 90 degrees
Supplementary Angles- Angles that add up to equal 180 degrees
Adjacent Angles- Two non straight and nonzero angles
Linear Pair- Two adjacent angles
Vertical Angles- Two non-straight angles
Proof Argument- A conditional is a sequence of justified conclusions starting with the
antecedent and ending with the consequent
Slope- How steep a line is
Perpendicular- Two segments, rays, or lines are perpendicular iff the lines have an angle of
90 degrees
Transformation- Correspondence between two sets of points such that each point in the
preimage set has a unique image and each point in the image set has exactly one preimage.
Translation/Slide- The composite of two reflections over parallel lines.
Rotation- Composite of two reflections over intersecting lines
Magnitude- Distance traveled
Direction- Way it travels
Vector- characterized by distance and magnitude
Isometry- Reflection or composition of reflections AKA movements in the plan that
preserve ABCD
Congruent Figures- Two figures F and G are congruent figures iff G is the image of F under
isometry
Auxiliary Figures- Objects we add to a diagram to help prove something
Postulates/Assumption
54. Postulates- True statement; assumption
55. Unique Line Assumption- Through any two points, there is exactly one line
56. Number Line Assumption- Every line is a set of points that can be put into one-to-one
correspondence with the real numbers, with any point on it corresponding to 0 and any
other point corresponding to 1.
57. Dimension Assumption- Given a line in a plane, there is at least one point in the plane that
is not on the line; Given a plane in space, there is at least one point in space that is not in
the plane.
58. Uniqueness Property- On a line, there is a unique distance between two points
59. Additive Property- If B is on line segment AC then AB+BC=AC
60. Unique Measure Assumption- Every angle has a unique measure from 0 degrees to 180
degrees
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61. Unique Angle Assumption- Given any ray and any real number between 0 and 180, there is a
unique angle in each half-plane
62. Zero Angle Assumption- If to rays are the same hem the angle measure is zero
63. Straight Angle Assumption- If they are opposite rays then the angle measure is 180
64. Angle Addition Property- If a ray is in the interior of an angle then the angle measure plus
the other angle equals the interior angle.
65. Reflexive Property of Equality- a=a
66. Symmetric Property of Equality- If a=b, then b=a
67. Transitive Property of Equality- If a=b and b=c then a=c
68. Substitution Property- If a=b you can replace a with b
69. Addition Property- Add/subtract a number from both sides of an equation
70. Multiplication Property- Multiply/divide an equation by the same number on both sides
71. Transitive Property of Inequality- If a ‹ b and b ‹ c then a ‹ c
72. Addition Property of Inequality- If a ‹ b then a+c ‹ b+c
73. Multiplication Properties of Inequality- If a ‹ b and c › 0 then ac ‹ bc; If a ‹ b and c › 0 then ac ›
bc
74. Equation to Inequality Property- If a and b are positive numbers an a+b=c then c › a and c › b
75. Corresponding Angles Postulate- Two corresponding angles have same measure, then the
lines are parallel; If the lines are parallel, then corresponding angles have the same
measure
76. Reflection Postulate- Angle measure, distance, betweeness, Colinearity, orientation
Theorems
77. Theorem- A statement which follows from postulates, definitions, or previously proved
theorems.
78. Line Intersection Theorem- Two different lines intersect in at most one point
79. Parallel Lines- Two coplanar lines m and n are parallel lines, written m // n if and only if they
have no points in common or they are identical
80. Linear Pair Theorem- If two angles form a linear pair then they are supplementary
81. Vertical Angles Theorem- If two angles are vertical angles then they are equal
82. Transitivity of Parallelism Theorem- In a plane, if line l is parallel to line m and line m is
parallel to line n, then line l is parallel to line n
83. Two Perpendiculars Theorem- Two coplanar lines l and m are each perpendicular to the
same line then they are parallel to each other
84. Perpendicular to Parallels Theorem- In a plane if a line is perpendicular to one of two
parallel lines, then it is also perpendicular to the other.
85. Perpendicular Lines and Slopes Theorem- Two nonvertical lines are perpendicular iff the
product of their slopes is -1
86. Parallel Lines and Slope Theorem- Two nonvertical lines are parallel iff they have same
slope
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87. Figure Reflection Theorem- If a figure is determined by certain points, then its reflection
image is the corresponding figure determined by the reflection images of those points.
88. Two Reflection Theorem for Translations- If m // l the translation rm ◦ rl has magnitude two
times the distance between l and m in the direction from l perpendicular to m
89. Two Reflection Theorem for Rotations- If m intersects n then the rotation rm ◦ rl has a
magnitude twice the measure of the smaller angle between the 2 lines and a center that
intersect points in both lines
90. Corresponding Parts of Congruent Figures (CPCF) Theorem- Two figures are congruent
then any
91. A-B-C-D Theorem- Every isometry preserves Angle measure, Betweeness, Collinearity, and
Distance
92. Reflexive Property of Congruence- F is congruent to F
93. Symmetric Property of Congruence- If F is congruent to G then G is congruent to F
94. Transitive Property of Congruence- If F is congruent to G then G is congruent to H then F is
congruent to H
95. Segment/Angle Congruence Theorem- Two segments/angles are congruent iff they have the
same length/measure
96. Alternate Interior Angles Theorem- Two // lines are intersected by a transversal iff alternate
interior angles are congruent
97. AIA Congruent → // Lines Theorem- If two lines are cut by a transversal and form congruent
alternate interior angles, then the lines are parallel.
98. Perpendicular Bisector Theorem- If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
99. Uniqueness of Parallels Theorem- Through a point not on a line, there is exactly one line
parallel to the given line
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Triangle Sum Theorem- Sum of the measures of the angles of a triangle is 180
degrees
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Quadrilateral Sum Theorem- Sum of the measures of the angles of a convex
quadrilateral is 360
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Polygon Sum Theorem- Sum of the measures of the angles of a convex n-gon is (n2)x180
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