Geometry Section 2-5 1112

Section 2-5
Postulates 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?

Vocabulary
1. Postulate:
2. Axiom:
3. Proof:
4. Theorem:
5. Deductive Argument:

Vocabulary
1. P o s t u la t e : A statement that is accepted to be true
without proof
2. Axiom:
3. Proof:
4. Theorem:

Vocabulary
without proof
2. Axiom: Another name for a postulate
3. Proof:
4. Theorem:

Vocabulary
without proof
3. P r o o f: A logical argument made up of statements
that are supported by another statement that is
accepted as true
4. Theorem:

Vocabulary
without proof
accepted as true
4. T h e o r e m : A statement or conjecture that has been
proven true

Vocabulary
without proof
accepted as true
4. Theorem:
A statement or conjecture that has been
A logical chain of statements
proven true
that link the given to what you are trying to prove

Vocabulary
6. Paragraph Proof:
7. Informal Proof:

Vocabulary
6. P a r a g r a p h P r o o f : When a paragraph is written to
logically explain why a given conjecture is true
7. Informal Proof:

Vocabulary
6. P a r a g r a p h P r o o f : When a paragraph is written to
logically explain why a given conjecture is true
7. I n fo r m a l P r o o f : 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!

Harkening back to
Chapter 1
postulates!
2.1: Through any two points, there is exactly one line.

Harkening back to
Chapter 1
postulates!
2.2: Through any three noncollinear points, there is exactly
one plane.

Harkening back to
Chapter 1
postulates!
one plane.
2.3: A line contains at least two points.

Harkening back to
Chapter 1
postulates!
one plane.
2.4: A plane contains at least three noncollinear points.

Harkening back to
Chapter 1
postulates!
one plane.
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.

Harkening back to
Chapter 1
postulates!
2.6: If two lines intersect, then their intersection is exactly
one point.

Harkening back to
Chapter 1
postulates!
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 1
Determine 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.

Example 1
never true.
Always true
C.

Example 1
never true.
Always true
Only one line can be drawn through any two points
C.

Example 1
never true.
Always true
C.
Sometimes true

Example 1
never true.
Always true
C.
Sometimes true
If the three points are collinear, then an infinite number
planes can be drawn. If they are noncollinear, then it is true.

Example 1
never true.
c. Planes R and T intersect at point P.

Example 1
never true.
Never true

Example 1
never true.
Never true
Two planes intersect in a line

Example 2
Given that AC intersects CD, write a paragraph proof to
show that A, C, and D determine a plane.

Example 2
Since the two lines intersect, they must intersect at
point C as two lines intersect in exactly one point.

Example 2
Points A and D are on different lines, so A, C, and D
are noncollinear by definition of noncollinear.

Example 2
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 3
Given that M is the midpoint of XY, write a paragraph
proof to show that XM ≅ MY.

Example 3
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.

Example 3
Theorem 2.1 (Midpoint Theorem):

Example 3
Theo r e m 2 . 1 ( M id p o i n t T h e o r e m ) : If M is the midpoint of
XY, then XM ≅ MY.

Problem Set

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 Descartes

Geometry Section 2-5 1112

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