SECTION 2.5Postulates and Paragraph Proofs

Example 1: 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.

Postulate 2.5: If 2 points lie in a plane, then the entireplane containing those points lies in that plane. In other words, since points F and G are on line m and arealso in plane Q, the line must be in plane Q.

Example 1: 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.

Postulate 2.1: Exactly 2 points determine a line.In the diagram, A and C are points, and a line goesthrough them.

You can use postulates to explain your reasoning when analyzing statements.

Example 2: Determine whether the following statement is always, sometimes, or never true. Explain.  a) If plane T contains and contains point G, then plane T contains point G.

b) contains three noncollinear points.

EF�������������� �

EF�������������� �

GH�������������� �

Never true. If a point is on the same line as , it is impossible to benoncollinear, since noncollinear means not on the same line.

Always true. When a line is in a plane, every point on that linewill be in the plane.

To prove a conjecture, you use deductive reasoning to move from a hypothesis to the conclusion of the conjecture you are trying to prove. This is done by writing a proof, which is a logical argument in which each statement you make is supported by a statement that is accepted as true. Once a statement or conjecture has been proven, it is called a theorem, and it can be use as a reason to justify statements in other proofs. 

Tiffany’s Jewelry Uggs Air Jordans Rolex Watch

As a group, choose one of the items below.

Use the internet to find how to prove your item is authentic (real vs fake).Find at least 2 characteristics to prove it is the real thing.

Proofs…why do we need them?

Proofs are similar to comic strips. Comic strips have sequential order.

One method of proving statements and conjectures, a paragraph proof, involves writing a paragraph to explain why a conjecture for a given situation is true. Paragraph proofs are also called informal proofs, although the term informal is not meant to imply that this form of proof is any less valid than any other type of proof.

Example 3: Given intersects write a paragraph proof to show that A, C, and D determine a plane.

AC�������������� �

,CD�������������� �

(Sketch a picture that shows this intersection).

ANSWER: Since intersects , the intersection is point C. ThisIntersection creates 3 noncollinear points. According to Postulate 2.2, 3 noncollinear points determine a plane. Therefore, points A, C, and D determine a plane.

The conjecture in Example 3 is known as the Midpoint Theorem.

Extra Example 4: Complete the following proof.

Given: Point B is the midpoint of . Point C is the midpoint of .Prove:

Statements Reasons

1. B is the midpoint of 1. Given (it is the 1st set of directions that are given to us)

2. (if B cuts the segment in half, the left segment is to the right segment)

2. Midpoint Theorem

3. C is the midpoint of (this is the 2nd set of directions given to us.)

3. Given

4. 4. Midpoint Theorem (the logical order of knowing a midpoint exists (from Step 3) is to find which two segments become congruent (already typed into step 4)).

5. (this is from the directions. We are told to prove this, so it must be the last step.)

5. (we will learn in Section 2.7)

A

D

C

B