Presentation on theme: "Postulates and Paragraph Proofs"— Presentation transcript:
1Postulates and Paragraph Proofs Chapter 2.5Postulates and Paragraph ProofsObjective: Introduce ProofsCheck.1.14Identify and explain the necessity of postulates, theorems, and corollaries in a mathematical system.CLEDevelop an understanding of the tools of logic and proof, including aspects of formal logic as well as construction of proofs.Spi.1.4Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems.
2Postulates (points, lines and planes) Copy the following on a clean sheet of paperPostulates (points, lines and planes)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 points2.4 A plane contains at least three points, not on the same line2.5 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.
3Statement that describes basic relationship of Geometry Glossary SectionPostulate (Axiom)Statement that describes basic relationship of Geometry
4Postulates and ProofsDetermine if the following is always, sometimes or never true.If points, A, B, and C lie in a plane M, then they are collinear.There is exactly one plane that contains noncollinear points P, Q and R.There are at least two lines through points M and N.SometimesAlways, Postulate 2.2Never, Postulate 2.1
5Use the postulatesDetermine 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.For XY, if X lines in plane Q and Y lies in plane R, then plane Q intersects plane R.GH contains three noncollinear points.Always, Postulate 2.5Sometimes, if R and Q are parallel they would not intersectNever, by definition of non-collinear
6ProofsA logical argument in which each statement is supported by a statement accepted as true.Paragraph Proof or Informal ProofInclude the theorem or conjecture to be provenList the given informationDraw a diagram (if possible)State what is to be provenDevelop a system of deductive reasoning
7Conjecture to proveGiven informationDiagramStatementDeductive reasoningProofGiven that M is the midpoint of PQ, write a paragraph proof to show that PM MQ.Prove PM MQ.Given: M is the midpoint of PQFrom 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.PQM
8Conjecture to proveGiven informationDiagramStatementDeductive reasoningProofGiven AC intersecting CD, show that A, C, and D determine a plane.Prove ACD is a plane.Given: AC intersects CDAC 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.ADC
9Proof:?Definition of midpointSegment Addition PostulateC. Definition of congruent segmentsD. Substitution