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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?
Thursday, November 6, 14
Vocabulary1. Postulate:
2. Axiom:
3. Proof:
4. Theorem:
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary1. Postulate: A statement that is accepted to be true
without proof
2. Axiom:
3. Proof:
4. Theorem:
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof:
4. Theorem:
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that are supported by another statement that is accepted as true
4. Theorem:
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that are supported by another statement that is accepted as true
4. Theorem: A statement or conjecture that has been proven true
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that are supported by another statement that is accepted as true
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
5. Deductive Argument:
Thursday, November 6, 14
Vocabulary6. Paragraph Proof:
7. Informal Proof:
Thursday, November 6, 14
Vocabulary6. Paragraph Proof: When a paragraph is written to
logically explain why a given conjecture is true
7. Informal Proof:
Thursday, November 6, 14
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
Thursday, November 6, 14
Harkening back to Chapter 1
Old ideas about points, lines, and planes are now postulates!
Thursday, November 6, 14
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.
Thursday, November 6, 14
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.2.2: Through any three noncollinear points, there is exactly
one plane.
Thursday, November 6, 14
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.2.2: Through any three noncollinear points, there is exactly
one plane.2.3: A line contains at least two points.
Thursday, November 6, 14
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.2.2: Through any three noncollinear points, there is exactly
one plane.2.3: A line contains at least two points.
2.4: A plane contains at least three noncollinear points.
Thursday, November 6, 14
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.2.2: Through any three noncollinear points, there is exactly
one plane.2.3: A line contains at least two 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
Harkening back to Chapter 1
Old ideas about points, lines, and planes are now postulates!
Thursday, November 6, 14
Harkening back to Chapter 1
Old ideas about points, lines, and planes are now postulates!
2.6: If two lines intersect, then their intersection is exactly one point.
Thursday, November 6, 14
Harkening back to Chapter 1
Old ideas about points, lines, and planes are now 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.
Thursday, November 6, 14
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.
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.a. Points E and F are contained by exactly one line.
Always true
b. There is exactly one plane that contains points A, B, and C.
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.a. Points E and F are contained by exactly one line.
Always trueOnly one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.a. Points E and F are contained by exactly one line.
Always trueOnly one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.
Sometimes true
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.a. Points E and F are contained by exactly one line.
Always trueOnly one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.
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
Example 1Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.Never true
Thursday, November 6, 14
Example 1Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.Never true
Two planes intersect in a line
Thursday, November 6, 14
Example 2Given that AC intersects CD, write a paragraph proof to
show that A, C, and D determine a plane.
Thursday, November 6, 14
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.
Thursday, November 6, 14
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.Points A and D are on different lines, so A, C, and D
are noncollinear by definition of noncollinear.
Thursday, November 6, 14
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.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.
Thursday, November 6, 14
Example 3Given that M is the midpoint of XY, write a paragraph
proof to show that XM ≅ MY.
Thursday, November 6, 14
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.
Thursday, November 6, 14
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):
Thursday, November 6, 14
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): If M is the midpoint of XY, then XM ≅ MY.
Thursday, November 6, 14
Problem Set
Thursday, November 6, 14
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