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Geometry

Explore the basic assumptions in geometry and how postulates guide measurements and relationships of points and angles. Learn about structure and foundational principles.

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Geometry

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  1. Geometry Section 2.2 – The Structure of Geometry

  2. Basic assumptions or statements that are accepted without proof Postulates (axioms) -

  3. The points on a line can be matched, one-to-one with the set of real numbers such that the distance between any two points is equal to the absolute value of the difference of their coordinates Find AB. Ex: A C B -2 0 2 OR The Ruler Postulate

  4. On , if B is between A and C, then AB + BC = AC Ex: Given that AC = 10 • Find the value of x • Find BC x x+6 A B C ANSWERS: a) x = 2 b) BC = 8 Segment Addition Postulate:

  5. Ex: Given that SG = 20, find the following • If SE = 5, EG = • If SE = EG, SE = • If SE = 4 and EG = 2x, EG = • If SE = y and EG = 2y+11, • then SE= and EG = 15 S E G 10 16 3 17

  6. Ex: F A V B R C S D E 0 4 5 • Find BD • Find the length of • Find the distance between B and E • Find the coordinate of the midpoint of • Find thepoint on whose distance from B is 3 • Find the midpoint of 5 6 -4 -2 2 1 F -1/2

  7. On in a given plane, P, choose any point O between A and B. • -consider and (opposite rays) and all the rays that can be drawn from O on the same side of • -these rays can be paired with real numbers from 0 to 180 inclusive The Protractor Postulate A O B

  8. D If C and D are in plane P, then the measure of <COD is equal to the absolute value of the difference between the real numbers for and C A O B R 110ᵒ Ex: find m<PAR P 40ᵒ A ANSWER: m<PAR = = 70°

  9. If C is in the interior of <AOD, then m<AOC +m<COD = m<AOD A C Also: D O C D A O B m<AOD + m<DOC + m<COB = m<AOB = 180° Angle Addition Postulate:

  10. 4. 90° 5. 142°

  11. D Ex: E C 7 1 6 5 A 2 4 3 B • <DEC and are adjacent angles • m<1 + m<2 = • If m<5 = 3x-6 and m<6 = 4x+4, then • x = • m<5 = • m<6 = <5 or <7 m<DAB 26 72° 108°

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