Download Conditional Statement

Section 2.1
Conditional
Statements
5/3/2017
Geometry
1
Goals


5/3/2017
Recognize and analyze a
conditional statement
Write postulates about points,
lines, and planes using conditional
statements
Geometry
2
Conditional Statement



5/3/2017
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
Geometry
3
Examples


If you are 13 years old, then you
are a teenager.
Hypothesis:


Conclusion:

5/3/2017
You are 13 years old
You are a teenager
Geometry
4
Rewrite in the if-then
form

All mammals breathe oxygen


A number divisible by 9 is also
divisible by 3

5/3/2017
If an animal is a mammal, then it
breathes oxygen.
If a number s divisible by 9, then it
is divisible by 3.
Geometry
5
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.
5/3/2017
Geometry
6
Converse


5/3/2017
The converse of a conditional is
formed by switching the hypothesis
and the conclusion.
The converse of p → q is q → p
Geometry
7
Negation


The negative of the statement
Example: Write the negative of the
statement
A is acute
 A is not acute


5/3/2017
~p represents “not p” or the
negation of p
Geometry
8
Inverse and
Contrapositive

Inverse
Negate the hypothesis and the
conclusion
 The inverse of p → q, is ~p → ~q


Contrapositive
Negate the hypothesis and the
conclusion of the converse
 The contrapositive of p → q, is
~q → ~p.

5/3/2017
Geometry
9
Example

Write the (a) inverse, (b) converse, and (c)
contrapositive of the statement.




5/3/2017
If two angles are vertical, then the angles are
congruent.
(a) Inverse: If 2 angles are not vertical, then
they are not congruent.
(b) Converse: If 2 angles are congruent, then
they are vertical.
(c) Contrapositive: If 2 angles are not
congruent, then they are not vertical.
Geometry
10
Equivalent Statements



5/3/2017
When 2 statements are both true
or both false
A conditional statement is
equivalent to its contrapositive.
The inverse and the converse of
any conditional are equivalent.
Geometry
11
Point, Line, and Plane
Postulates




5/3/2017
Postulate 5: Through any two points there
exists exactly one line
Postulate 6: A line contains at least two
points
Postulate 7: If 2 lines intersect, then their
intersection is exactly one point
Postulate 8: Through any three noncollinear
points there exists exactly one plane
Geometry
12



5/3/2017
Postulate 9: A plane contains at least three
noncollinear points
Postulate 10: If two points lie in a plane, then
the line containing them lies in the plane
Postulate 11: If two planes intersect, then their
intersection is a line
Geometry
13