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Chapter 2.5 Postulates and Paragraph Proofs Objective: Introduce Proofs Check.1.14 CLE 3108.4.3 Spi.1.4 Identify and explain the necessity of postulates, theorems, and corollaries in a mathematical system. Develop an understanding of the tools of logic and proof, including aspects of formal logic as well as construction of proofs. Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. Postulates (points, lines and planes) Copy the following on a clean sheet of paper 2.1 Through any two points, there is exactly one line. 2.2 Through any three points, not on the same line, there is exactly one plane. 2.3 A line contains at least two points 2.4 A plane contains at least three points, not on the same line 2.5 If two points lie in a plane, then the entire line containing those points lies in that plane. 2.6 If two lines intersect, then their intersection is exactly one point. 2.7 If two planes intersect, then their intersection is a line. Glossary Section Postulate (Axiom) Statement that describes basic relationship of Geometry Postulates and Proofs Determine if the following is always, sometimes or never true. If points, A, B, and C lie in a plane M, then they are collinear. Sometimes There is exactly one plane that contains noncollinear points P, Q and R. Always, Postulate 2.2 There are at least two lines through points M and N. Never, Postulate 2.1 Use the postulates Determine if the following is always, sometimes or never true. If a plane T contains EF and EF contains point G, then plane T contains point G. Always, Postulate 2.5 For XY, if X lines in plane Q and Y lies in plane R, then plane Q intersects plane R. Sometimes, if R and Q are parallel they would not intersect GH contains three noncollinear points. Never, by definition of non-collinear Proofs A logical argument in which each statement is supported by a statement accepted as true. Paragraph Proof or Informal Proof 1. 2. 3. 4. 5. Include the theorem or conjecture to be proven List the given information Draw a diagram (if possible) State what is to be proven Develop a system of deductive reasoning Proof 1. 2. 3. 4. 5. Conjecture to prove Given information Diagram Statement Deductive reasoning Given that M is the midpoint of PQ, write a paragraph proof to show that PM MQ. M Q P Prove PM MQ. Given: M is the midpoint of PQ From the definition of a midpoint, PM = MQ. This means that PM and MQ have the same measure. By the definition of congruence, if two segments have the same measure then they are congruent, thus PM MQ. Proof 1. 2. 3. 4. 5. Conjecture to prove Given information Diagram Statement Deductive reasoning Given AC intersecting CD, show that A, C, and D determine a plane. D C Prove ACD is a plane. Given: AC intersects CD A AC and CD must intersect a C because 2.6 if two lines intersect then their intersection is exactly one point. Point A is on AC and D is on CD. Therefore A and D are not collinear. Therefore ACD is a plane as 2.2 states that any three noncollinear points define a line. Proof: ? A. B. C. D. Definition of midpoint Segment Addition Postulate Definition of congruent segments Substitution Practice Assignment Page 129, 24-44 even