673: GEOMETRY

673: GEOMETRY

1. Points, Lines, Planes, Angles

   • Distance
     • Using coordinates (x1, y1) and (x2, y2) d =√(x2 - x1)^2 + (y2-y1)^2
   • Midpoint
     • Line seg AM = Line seg MB 4x - 6 = 2x + 10 2x = 16 x = 8
     • Line seg AM = 4(8) - 6 = 26 Line seg AB = 26(2) = 52
   • Types of Angles
     • complementary
       • 2x + 25 + x + 5 = 90 3x + 30 = 90 3x = 60 x = 20
     • supplementary
     • Vertical
     • Adjacent

2. Perpendicular and Parallel Lines

   • Parallel lines cut by a transversal
     • alternate interior angles
     • same-side interior angles
     • alternate exterior angles
       • 2x -10 = 110 2x = 120 x = 60 
   • Relationship of slopes
     • if lines are parallel, slopes are same
     • if lines are perpendicular, slopes are opposite reciprocals

3. Logical Reasoning

   • deductive reasoning
     • formal proofs
   • counter-example
     • use when proving a conjecture false!
   • inductive reasoning
     • find patterns, make conjectures
     • hypothesis
   • conjecture
   • proofs
     • writing proofs, paragraphs proofs, two-column proofs
   • conditional statement
     • use to draw valid conclusions
     • Law of detachment
       • Law of detachment is used when you have a conditional statement and another statement that matches the hypothesis (the part following if) of the conditional. The conclusion is the conclusion (second part) of the conditional.  1. If p, then q 2. p -- Therefore, q
     • Syllogism
       • Law of syllogism is used when you have two conditionals and the hypothesis of one matches the conclusion of the other.  If p, then q If q, then r -- Therefore if p, then r
   • Properties!
     • reflexive
       • The reflexive property of equality simply states that a value is equal to itself. Further, this property states that for all real numbers, x = x.
     • symmetric
       • The Symmetric Property states that for all real numbers x and y, if x=y , then y=x .
     • transitive
       • The Transitive Property states that for all real numbers x, y, and z, if x=y and y=z , then x=z .
   • converse, contrapositive, inverse
     • determine if true or false

4. Attributes/Properties of Triangles

   • triangle congruency with SSS, SAD, ASA, AAS
   • equilateral, isoscele, scalene, right, obtuse, acute, equiangular
     • 180 degrees
       • x + 102 + 51 = 180 x + 153 = 180 x = 27 degrees
   • Pythagorean theorem with right triangles!
     • trig!!!
       • sine, cosine, tangents (SOH CAH TOA)
     • 30, 60, 90 and 45, 45, 90

5. Quadrilaterals

   • polygon
     • parallelogram, rhombus, rectangle, square, trapezoid, kite

6. Transformations

   • reflection (isometry), translation (isometry), rotation (isometry) (angle of rotation), dilation (scale factor)
     • line of reflection, center of rotation, center of dilation

7. Similarity

   • cross product property
     • 5 (x) = 7 (x - 2) 5x = 7x - 14 -2x = -14 x = 7
   • geometric means
   • proportionality theorems

8. Circles

   • chord, secant, trangent, minor arc, major arc
   • diameter, radius
   • inscribed angle, intercepted arc, circumscribed

9. 3D Figures

   • volume
     • base
     • slant height/height
   • face
   • lateral area
   • prism, pyramid, cylinder, cone, sphere
   • nets!!
     • Flattened out 3D firgure
     • The net to the left is a triangular pyramid (A)

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