2.5 Postulates & Paragraph Proofs

Objectives

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

Write paragraph proofs.

Postulates

In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry.

Postulates are always accepted as true.

Postulates

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

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

SNOW CRYSTALS Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal?

Explore The snow crystal has six vertices since a regular hexagon has six vertices.

Plan Draw a diagram of a hexagon to illustrate the solution.

Example 1:

Solve Label the vertices of the hexagon A, B, C, D, E, and F. Connect each point with every other point. Then, count the number of segments. Between every two points there is exactly one segment. Be sure to include the sides of the hexagon. For the six points, fifteen segments can be drawn.

Example 1:

Answer: 15

Examine In the figure, are all

segments that connect the vertices of the snow crystal.

Example 1:

ART Jodi is making a string art design. She has positioned ten nails, similar to the vertices of a decagon, onto a board. How many strings will she need to interconnect all vertices of the design?

Answer: 45

Your Turn:

More Postulates

Postulate 2.3 – A line contains at least two points.

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

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

And Even More Postulates

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

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

Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane.

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

If plane T contains contains point G, then plane T contains point G.

Example 2a:

Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes.

For , if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R.

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

Example 2b:

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

contains three noncollinear points.

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

Example 2c:

Determine whether each statement is always, sometimes, or never true. Explain.

a. Plane A and plane B intersect in one point.

b. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R.

Answer: Never; Postulate 2.7 states that if two planes intersect, then their intersection is a line.

Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line.

Your Turn:

c. Two planes will always intersect a line.

Answer: Sometimes; Postulate 2.7 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect.

Determine whether each statement is always, sometimes, or never true. Explain.

Your Turn:

Theorems

We use undefined terms, definitions, postulates, and algebraic properties of equality to prove that other statements or conjectures are true. Once a statement or conjecture has been shown to be true, it is called a theorem.

Once proven true, a theorem can be used like a definition or postulate to justify other statements or conjectures.

Paragraph Proofs

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

Paragraph Proof – a type of proof in which you write a paragraph to explain why a conjecture for a given situation is true (also, referred to as an “informal proof ”)

Paragraph Proofs

There are 5 essential parts of a good proof:

State the theorem or conjecture to be proven.

List the given information.If possible, draw a diagram to

illustrate the given information.State what is to be proved.Develop a system of deductive

reasoning.

Paragraph Proofs

Hint:

Before writing a proof, you should have a plan. One strategy is to work backwards. Start with what you want to prove, and work backwards step by step until you reach the given information.

Theorems

Theorem 2.8 (Midpoint Theorem)

If M is the midpoint of AB, then AM MB.

Prove: ACD is a plane.

Given intersecting , write a paragraph proof to show that A, C, and D determine a plane.

Given: intersects

Proof: must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on Therefore, points A and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line.

Example 3:

Given is the midpoint of and X is the midpoint of write a paragraph proof to show that

Your Turn:

Proof: We are given that S is the midpoint of and

X is the midpoint of By the definition of midpoint,

Using the definition

of congruent segments, Also

using the given statement and the definition of

congruent segments, If then

Since S and X are midpoints,

By substitution, and by definition of

congruence,

Your Turn:

Assignment

Geometry:Pg. 92

#12 - 27

Pre-AP Geometry: Pg. 92

# 12 - 28