2-5 Postulates and Paragraph Proofs

• Identify and use basic postulates about points, lines, and planes.

Definition

A postulate or axion is a statement that is accepted as true without proof.

Basic ideas about points, lines, and planes can be stated as postulates.

Euclid• Greek mathematician• Wrote The Elements

around 300BC• Organized geometry

& other maths into a deductive system.

• Called basic assumptions—self-evident truths

• Self-evident truths=postulates

Questions????What is the smallest number of points through

which only one line can be drawn?

Questions????

Three points, A, B, and C are not all on the same line. How many planes contain all three of the points?

A

B

CP

Questions????

What does the intersection of two planes look like?

Questions????

If a plane contains two points of a line, must it contain the whole line?

Questions????

What is the smallest number of points that cannot be contained in just one plane?

AB

C

P

Lines & Points Postulate 2.1

Every line contains at least two points.

Plane Postulate 2.2

Three noncollinear points are contained in one and only one plane. (Three noncollinear points determine a plane.)

1

2

3

Straight-Line Postulate 2.3

Two points are contained in one and only one line. (Two points determine a line.)

Points-Existence Postulate 2.4

Every plane contains at least three noncollinear points.

Flat-Plane Postulate 2.5

If two points are in a plane, then the line containing the points is in the same plane.

1

2

Intersecting Lines 2.6

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

s

t

P

Plane-Intersection Postulate 2.7

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

ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.

A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q.

Answer: Points F and G lie on line m, and the line lies in plane Q. Postulate 2.5, which states that if two points lie in a plane, the entire line containing the points lies in that plane, shows that this is true.

ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.

B. Points A and C determine a line.

Answer: Points A and C lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true.

A. Through any two points there is exactly one line.

B. A line contains at least two points.

C. A plane contains at least three noncollinear points.

D. A plane contains at least two noncollinear points.

ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true.

A. Plane P contains points E, B, and G.

A. Through any two points there is exactly one line.

B. A line contains at least two points.

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

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

ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true.

B. Line AB and line BC intersect at point B.

Answer: Never; noncollinear points do not lie on the same line by definition.

B. Determine whether the following statement is always, sometimes, or never true. Explain.

contains three noncollinear points.

A. Determine whether the statement is always, sometimes, or never true.

Plane A and plane B intersect in exactly one point.

A. always

B. sometimes

C. never

Found on page 127 in your book.

2-5 Assignment day 1

Page 131, 16-29