Geometry Honors Midterm

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Geometry Honors Midterm
2011-01-25 01:36:53
Geometry Midterm Theorms

Cramming for the midterm is never fun.
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  1. undefined terms
    can not be defined using other figures.

    • Point
    • Line
    • Plane
  2. Point
    names a location and has no size. it is represented by a dot.

    A capital letter
  3. Line
    a straight path that has no thickness and extends forever

    A lower case letter or two points on the line
  4. Plane
    • a flat surface that has no thickness and extends forever.
    • a script capital letter
    • or
    • three points not on a line.
  5. collinear
    points that lie on the same line
  6. noncollinear
    points that do not lie on the same line
  7. line segment
    the part of a line consisting of two points and all points between them
  8. endpoint
    a point at one end of a segment or the starting point of a ray

    A capital letter
  9. ray
    part of a line that starts at an endpoint and extends forever in one direction

    It's endpoint and any other point
  10. Opposite Rays
    two rays that have a common endpoint and form a line
  11. postulate
    a statement that is accepted as true without proof
  12. intersection
    the set of all points the two or more figures have in common
  13. system of equations
    a set of two or more equations containing two or more of the same variables
  14. coordinate
    a number used to identify the location of a point
  15. distance between any two points
    the absolute value of the difference of the coordinates
  16. length of AB
    the distance between A and B
  17. Finding the Length of a Segment
    | a-b | or | b-a |
  18. congruent segments
    segments that have the same length

    uses tick marks
  19. midpoint
    the point that bisects, or divides the segment into two congruent segments
  20. segment bisector
    any ray, segment, or line that intersects a segment at its midpoint
  21. angle
    a figure formed by two rays, or sides, with a common endpoint
  22. vertex
    a common endpoint
  23. How to name an angle
    • Vertex
    • a point on each ray and the vertex
    • number

    <R <SRT <1
  24. interior of an angle
    the set of all points between the sides of the angle
  25. exterior of an angle
    the set of all points outside the angle
  26. the measure of an angle
    the absolute value of the difference of the real numbers that the rays correspond with on a protractor
  27. Acute Angle
    measures greater than 0 degrees and less than 90 degrees
  28. Right Angle
    • Measures 90 degrees
    • must have box
  29. obtuse angle
    measures greater than 90 degrees and less than 180 degrees
  30. straight angle
    • formed by two opposite rays
    • measures 180 degrees
  31. congruent angles
    angles that have the same measure
  32. angle bisector
    a ray that divides an angle into two congruent angles

    the (particular points) of all points in the interior of the angle that are equidistant from the sides of the angle
  33. Adjacent Angles
    two angles in the same plane with a common vertex and a common side, but no common interior points
  34. linear pair of angles
    a pair of adjacent angles whose noncommon sides are opposite rays
  35. complementary angles
    two angles whose measures have a sum of 90 degrees
  36. supplementary angles
    two angles whose measures have a sum of 180 degrees
  37. Finding the Complementary angle
  38. finding the supplementary angke
  39. vertical angles
    two nonadjacent angles formed by two intersecting lines
  40. perimeter
    the sum of the side lengths of the figure

    • rectangle- 2l+2w
    • square- 4s
    • triangle a+b+c
  41. area
    the number of non overlapping square units of a given size that cover the figure

    • rectangle- l*w
    • Square side squared
    • Triangle- 1/2bh
    • circle- pi(r) squared
  42. base of a triangle
    any side of a triangle
  43. height
    a segment from a vertex that forms a right angle with a line containing the base

    • may be:
    • inside the triangle (interior)
    • outside (exterior)
    • a side of the triangle
  44. diameter
    a segment that passes through the center of the circle and whose endpoints are on the circle
  45. radius
    a segment whose endpoints are the center of the circle and a point on the circle
  46. circumference
    the distance around a circle

    c=pi(d) or 2pi(r)
  47. pi

    3.14 or 22/7
  48. coordinate plane
    a plane that is divided into 4 regions by the x and y axis
  49. Midpoint formula
    m= (x1+x2 / 2, y1+y2 /2)
  50. distance formula
    d= square root of (x2-x1)2 + (y2-y1)2
  51. pythagorean theorem
    • ONLY in a right triangle
    • used to find distance
    • a2+b2=c2
  52. legs of a right triangle
    the 2 sides that form the right angle
  53. hypotenuse
    the side across from the right triangle that stretches from one leg to the other
  54. transformation
    a change in the position, size, or shape of a figure
  55. preimage
    the original figure
  56. image
    the resulting figure
  57. ---> arrow notation
    used to describe a transformation
  58. ' primes
    used to label the image
  59. reflection (flip)
    a transformation across a line, called the line of reflection.

    each points and its image are the same distance from the line of reflection
  60. Rotation (turn)
    a transformation about a point P, called the center of rotation.

    Each point and its image are the same distance from P
  61. Translation (slide)
    a transformation in which all the points of a figure move the same distance in the same direction
  62. inductive reasoning
    the process of reasoning that a rule or statement is true because specific cases are true

    used to draw a conclusion from a pattern

    • 1) look for a pattern
    • 2) Make a conjecture
    • 3) prove the conjecture or find a counterexample
  63. conjecture
    a statement believed to be true based on inductive reasoning
  64. counterexample
    an example that proves a conjecture false
  65. conditional statement
    a statement that can be written in the form "if p, then q"

    p--> q
  66. hypothesis
    the part of p of a conditional statement following the word "if"
  67. conclusion
    the part q of a conditional statement following the word, "then"
  68. truth value
    true or false of a conditional statement

    false when the hypothesis is true and the conclusion if false
  69. negation of statement p (opposite)

    the negation of a true statement is false, the negation of a false statement is true
  70. converse
    statement formed by exchanging the hypothesis and conclusion

  71. inverse
    statement formed by negating the hypothesis and the conclusion

    ~p--> ~q
  72. contrapositive
    statement formed by both exchanging and negating the hypothesis and conclusion

    ~q --> ~p
  73. logically equivalent statements
    related conditional statements that have the same truth value
  74. deductive reasoning
    process of using logic to draw conclusions from given facts, definitions, and properties
  75. biconditional statement
    a statement that can be written in the form, "p if and only if q" (iff)

    if p, then q

    if q, then p

    used to write defintions
  76. definition
    a statement that describes a mathematical object and can be written as a true biconditional

  77. polygon
    closed plane figure formed by 3 or more line segments

    each segment intersects exactly two other segments only at their endpoints

    no two segments with a common endpoint are collinear
  78. triangle
    a three sided polygon
  79. quadrilateral
    a 4 sided polygon
  80. proof
    an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true
  81. theorem
    any statement that you can prove
  82. two column proof
    proof that lists the steps of the proof in the left column and the matching reason in the right column.
  83. proof process
    1) write the conjecture to be proven

    2) draw a diagram to represent the hypothesis of the conjecture

    3) state the given information and mark it on the diagram

    4) state the conclusion of the conjecture in terms of the diagram

    5) plan your argument and prove the conjecture
  84. flow chart proof
    uses boxes and arrows to show the structures of the proof

    moves from left to right or top to bottom

    justification written below the box
  85. paragraph proof
    presents the steps of the proof and their matching reasons as sentences in a paragraph
  86. parallel lines (||)
    coplanar and do not intersect
  87. perpendicular lines (upside down T)
    intersect at 90 degree angles.
  88. skew lines
    are not coplanar, are not parallel, and do not intersect
  89. parallel planes
    planes that do not intersect
  90. transversal
    a line that intersects two coplanar lines at two different points


    other 2 lines = r and s
  91. corresponding angles
    lie on the same side of the transversal, on the same sides of lines r and s.
  92. alternate interior angles
    nonadjacent angles

    lie on opposite sides of the transversal t, between lines r and s
  93. alternate exterior angles
    lie on opposite sides of the transversal t

    outside lines r and s
  94. same-side interior angles (consecutive interior angles)
    lie on the same side of the transversal

    between lines r and s
  95. perpendicular bisector of a segment
    a line perpendicular to a segment at the segment's midpoint
  96. distance from a point to a line
    the length of the perpendicular segment from the point to the line
  97. rise
    the difference in the y values of the 2 points on a line
  98. run
    the difference in the x values of 2 points on a line
  99. slope
    the ratio of rise to run,

    m= y2-y1 / x2-x1
  100. positive slope
  101. negative slope
  102. zero slope
  103. undefined slope
  104. opposite reciprocals
    a/b and -b/a
  105. point slope form of a line

    x1, y1 is a given point on the line
  106. slope intercept form


    b=y intercept
  107. the equation of a vertical line

    a= x intercept
  108. equation of horizontal line

    b= y intercept
  109. y=5x+8

    same slope, different y intercept
  110. acute triangle
    three acute sides
  111. equiangular triangle
    3 congruent acute angles
  112. right triangle
    one right angle
  113. obtuse triangle
    one obtuse angle
  114. equilateral triangle
    three congruent sides
  115. isosceles triangle
    at least 2 congruent sides
  116. scalene triangle
    no congruent sides
  117. auxiliary line
    a line that is added to a figure to aid in a proof

  118. corollary
    a theorem whose proof follows directly from another theorem
  119. interior
    set of all points inside the figure
  120. exterior
    the set of all points outside the figure
  121. interior angle
    formed by 2 sides of a triangle
  122. exterior angle
    formed by one side of the triangle and the extension of an adjacent side
  123. remote interior angle
    an interior angle that is not adjacent to the exterior angle
  124. triangle rigidity
    if the side lengths of a triangle are given, the triangle can have only one shape.
  125. included angle
    an angle formed by two adjacent sides of a polygon
  126. included side
    the common side of 2 consecutive angles in a polygon
  127. coordinate proof
    uses coordinate geometry and algebra
  128. strategies for positioning figures in the coordinate plane
    use the origin as a vertex, keeping the figure in Quad I

    center the figure at the origin

    center a side of the figure at the origin

    use one or both axes as sides of the figure
  129. vertex angle
    angle formed by the legs
  130. base
    side opposite the vertex angle
  131. base angles
    2 angles that have the base as a side
  132. equidistant
    a point is the same distance from 2 or more objects
  133. concurrent
    three or more lines intersect at one point
  134. point of concurrency
    the point where the lines intersect
  135. circumcenter of a triangle
    the point of concurrency in a triangle
  136. circumscribed
    a circle that contains all the vertices of a polygon
  137. incenter of the triangle
    the point of concurrency where the angle bisectors meet? (unsure)
  138. inscribed
    intersects each line that contains a side of the polygon at exactly one point
  139. incenter
    the center of the triangle's inscribed triangle

    always inside the triangle
  140. median of a triangle
    a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

    each triangle has 3 medians, which are concurrent
  141. centroid of the triangle
    the point of concurrency of the medians of a triangle

    always inside the triangle

    center of gravity
  142. altitude of a triangle
    perpendicular segment from a vertex to the line containing the opposite side

    each triangle has 3 altitudes

    can be inside, out, or on the triangle
  143. the height of a triangle is the length of an altitude
    miscellaneous shit
  144. orthocenter of the triangle
    the point of intersection of the 3 altitudes of a triangle
  145. midsegment of a triangle
    segment that joins the midpoints if two sides of the triangle

    every triangle has 3 midsegments, qhich form the midsegment triangle
  146. indirect proof
    1) identify the conjecture

    2) assume the opposite of the conclusion is true

    3) use direct reasoning to show that the assumption leads to a contradiction

    4) conclude that since the assumption is false, the original conjecture is true