Geometry Section 2 2 The Structure of Geometry

Geometry Section 2. 2 – The Structure of Geometry

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

The Ruler Postulate 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 Ex: Find AB. A -2 C 0 2 OR B

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

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

Ex: F -4 A V -2 B R 0 C 2 S D E 4 Find BD 5 5 Find the length of Find the distance between B and E 6 Find the coordinate of the midpoint of 1 e) Find the point on whose distance from B is 3 F f) Find the midpoint of -1/2 a) b) c) d)

The Protractor Postulate 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 A O B

D C A O B 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 Ex: find m<PAR R 110ᵒ P 40ᵒ A ANSWER: m<PAR = = 70°

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

4. 90° 5. 142°

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