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