geometry section 2-5 1112

Section 2-5Postulates and Paragraph Proofs

Thursday, November 6, 14

Essential Questions

• How do you identify and use basic postulates about points, lines, and planes?

• How do you write paragraph proofs?

Vocabulary1. Postulate:

2. Axiom:

3. Proof:

4. Theorem:

5. Deductive Argument:

Vocabulary1. Postulate: A statement that is accepted to be true

without proof

2. Axiom:

3. Proof:

4. Theorem:

without proof

2. Axiom: Another name for a postulate

3. Proof:

4. Theorem:

without proof

3. Proof: A logical argument made up of statements that are supported by another statement that is accepted as true

4. Theorem:

without proof

4. Theorem: A statement or conjecture that has been proven true

without proof

4. Theorem:

A logical chain of statements that link the given to what you are trying to prove

A statement or conjecture that has been proven true

Vocabulary6. Paragraph Proof:

7. Informal Proof:

Vocabulary6. Paragraph Proof: When a paragraph is written to

logically explain why a given conjecture is true

7. Informal Proof:

Vocabulary6. Paragraph Proof: When a paragraph is written to

logically explain why a given conjecture is true

7. Informal Proof: Another name for a paragraph proof as it allows for free writing to provide the logical explanation

Harkening back to Chapter 1

Old ideas about points, lines, and planes are now postulates!

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

2.1: Through any two points, there is exactly one line.2.2: Through any three noncollinear points, there is exactly

one plane.

one plane.2.3: A line contains at least two points.

2.4: A plane contains at least three noncollinear points.

2.4: A plane contains at least three noncollinear points.2.5: If two points lie in a plane, then the entire line

containing those points lies in the plane.Thursday, November 6, 14

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.

Example 1Determine whether the statement is always, sometimes, or

never true.a. Points E and F are contained by exactly one line.

b. There is exactly one plane that contains points A, B, and C.

Always true

Always trueOnly one line can be drawn through any two points

Sometimes true

Sometimes trueIf the three points are collinear, then an infinite number

planes can be drawn. If they are noncollinear, then it is true.Thursday, November 6, 14

never true.

c. Planes R and T intersect at point P.

never true.

c. Planes R and T intersect at point P.Never true

never true.

c. Planes R and T intersect at point P.Never true

Two planes intersect in a line

Example 2Given that AC intersects CD, write a paragraph proof to

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

Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point.

Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point.Points A and D are on different lines, so A, C, and D

are noncollinear by definition of noncollinear.

Since the two lines intersect, they must intersect at point C as two lines intersect in exactly one point.Points A and D are on different lines, so A, C, and D

are noncollinear by definition of noncollinear.Since three noncollinear points determine exactly one

plane, points A, C, and D determine a plane.

Example 3Given that M is the midpoint of XY, write a paragraph

proof to show that XM ≅ MY.

If M is the midpoint of XY, then by the definition of midpoint, XM = MY. Since they have the same measure, we

know that, by the definition of congruence, XM ≅ MY.

Theorem 2.1 (Midpoint Theorem):

Theorem 2.1 (Midpoint Theorem): If M is the midpoint of XY, then XM ≅ MY.

Problem Set

p. 128 #1-41 odd

“The first precept was never to accept a thing as true until I knew it as such without a single doubt.” - Rene DescartesThursday, November 6, 14

geometry section 2-5 1112

p h p r o o f

p o s t u

t h e o r e

deductive argument

noncollinear points

logical argument

old ideas

truewithout proof2

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