1-4 If then statements and postulates
Download
Report
Transcript 1-4 If then statements and postulates
Section 1.4
If-Then Statements
and Postulates
4/8/2015
Geometry
1
Objectives-What we’ll
learn
4/8/2015
Recognize and analyze a
conditional statement
Write postulates about points,
lines, and planes using conditional
statements
Geometry
2
Postulate vs. Theorem
A postulate is a statement that is
assumed true without proof.
A theorem is a true statement that
can be proven.
Conditional Statement
4/8/2015
A conditional statement has two
parts, a hypothesis and a
conclusion.
When conditional statements are
written in if-then form, the part after
the “if” is the hypothesis, and the
part after the “then” is the
conclusion.
p → q represents “if p then q”
Geometry
4
Examples
If you are 13 years old, then you
are a teenager.
Hypothesis:
Conclusion:
4/8/2015
You are 13 years old
You are a teenager
Geometry
5
Rewrite in the if-then form
(Conditional Statement)
All mammals breathe oxygen
A number divisible by 9 is also
divisible by 3
4/8/2015
If an animal is a mammal, then it
breathes oxygen.
If a number is divisible by 9, then it
is divisible by 3.
Geometry
6
Rewrite in the if-then form
(Conditional Statement)
Two lines intersect at a point.
Three non-collinear points
determine a plane.
4/8/2015
If two lines intersect, then they
intersect at a point.
If there are three non-collinear
points, then they determine a
plane.
Geometry
7
Writing a
Counterexample
Write a counterexample to show that the
following conditional statement is false
If x2 = 16, then x = 4.
As a counterexample, let x = -4.
The
hypothesis is true, but the conclusion is
false. Therefore the conditional statement is
false.
4/8/2015
Geometry
8
Converse
4/8/2015
The converse of a conditional
statement is formed by switching
the hypothesis and the conclusion.
The converse of p → q is q → p
Geometry
9
Rewrite in the Converse
form.
If you are 13 years old, then you
are a teenager.
If you are a teenager, then you are
13 years old.
If a number divisible by 9, then it is
also divisible by 3
If a number is divisible by 3, then it
is divisible by 9.
Rewrite in the Converse
form.
If two angles are vertical angles,
then they are congruent.
If two angles are congruent, then
they are vertical angles.
If a quadrilateral has 4 right angles,
then it is a rectangle.
If a quadrilateral is a rectangle,
then it has 4 right angles.
Point, Line, and Plane
Postulates
4/8/2015
Postulate 1-1: Through any two points there
exists exactly one line
Postulate 1-2: Through any three
noncollinear points there exists exactly one
plane
Postulate 1-3: A line contains at least two
points
Postulate 1-4: A plane contains at least three
points not on the same line
Geometry
12
4/8/2015
Postulate 2-5: If two points lie in a plane, then
the line containing them lies in the plane
Postulate 2-6: If two planes intersect, then their
intersection is a line
Geometry
13