Lesson 2-5Lesson 2-5Postulates

and

Paragraph Proofs

Ohio Content Ohio Content Standards:Standards:

Ohio Content Ohio Content Standards:Standards:

Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.

Ohio Content Ohio Content Standards:Standards:

Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

Ohio Content Ohio Content Standards:Standards:

Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof.

PostulatePostulate

PostulatePostulateStatement that describes a fundamental relationship between the basic terms of geometry.

Postulate 2.1Postulate 2.1

Postulate 2.1Postulate 2.1Through any two points, there is exactly one line.

Postulate 2.2Postulate 2.2

Postulate 2.2Postulate 2.2Through any three points not on the same line, there is exactly one plane.

Postulate 2.3Postulate 2.3

Postulate 2.3Postulate 2.3A line contains at least two points.

Postulate 2.4Postulate 2.4

Postulate 2.4Postulate 2.4A plane contains at least three points not on the same line.

Postulate 2.5Postulate 2.5

Postulate 2.5Postulate 2.5If two points lie in a plane, then the entire line containing those points lies in that plane.

Postulate 2.6Postulate 2.6

Postulate 2.6Postulate 2.6If two lines intersect, then their intersection is exactly one point.

Postulate 2.7Postulate 2.7

Postulate 2.7Postulate 2.7If two planes intersect, then their intersection is a line.

Determine whether each Determine whether each statement is statement is always, always,

sometimes, sometimes, or or never never true. true. Explain.Explain.

Determine whether each Determine whether each statement is statement is always, always,

sometimes, sometimes, or or never never true. true. Explain.Explain.

If plane If plane T T contains contains EFEF and and EF EF contains contains point point GG, then , then plane plane TT contains contains point point GG..

Determine whether each Determine whether each statement is statement is always, always,

sometimes, sometimes, or or never never true. true. Explain.Explain.

For For XYXY, if , if XX lies lies in plane in plane QQ and and YY lies in plane lies in plane RR, , then plane then plane QQ intersects intersects RR. .

Determine whether each Determine whether each statement is statement is always, always,

sometimes, sometimes, or or never never true. true. Explain.Explain.

GH GH contains contains three three noncollinear noncollinear points. points.

TheoremTheorem

TheoremTheoremA statement or conjecture that has been shown to be true.

ProofsProofs

ProofsProofsA logical argument in which each statement you make is supported by a statement that is accepted as true.

Paragraph ProofParagraph Proof

Paragraph ProofParagraph ProofYou write a paragraph to explain why a conjecture for a given situation is true.

Five essential parts of a good

proof:

Five essential parts of a good

proof:

1) State the 1) State the theorem of theorem of conjecture to conjecture to be proven.be proven.

Five essential parts of a good

proof:

2) List the 2) List the given given information.information.

Five essential parts of a good

proof:

3) If possible, 3) If possible, draw a draw a diagram to diagram to illustrate the illustrate the given given information.information.

Five essential parts of a good

proof:

4) State what 4) State what is to be is to be proved.proved.

Five essential parts of a good

proof:

5) Develop a 5) Develop a system of system of deductive deductive reasoning.reasoning.

Theorem 2.8Theorem 2.8

Theorem 2.8Theorem 2.8

If M is the midpoint of AB then AM MB.

Given Given ACAC intersecting intersecting CDCD, , write a paragraph proof to write a paragraph proof to

show that show that A, C, A, C, and and D D determine a plane.determine a plane.

Assignment:Assignment:

Pgs. 92 - 93 Pgs. 92 - 93 16-28 all16-28 all