Download 2.2 Analyze Conditional Statements

2.2 Analyze Conditional
Statements
Definitions:
conditional statement: a logical statement that has
two parts, a hypothesis and a conclusion.
if-then form: a conditional statement written where
“if” contains the hypothesis and “then” contains the
conclusion.
Example:
On Sundays, there is football on TV.
“If it is Sunday, then there is football on TV.”
hypothesis
conclusion
Example 1
Rewrite
the conditional statement in if-then form.
(a) All 90° angles are right angles.
If an angle measures 90°, then it is a right angle.
(b) When n = 9, n2 = 81.
If n = 9, then n2 = 81.
(c) Three points are collinear if there is a line containing
them.
If there is a line containing three points, then the
three points are collinear.
Definitions:
Negation: the opposite of the original statement
Statement: The ball is red.
Negation: The ball is not red.
Statement: The cat is not black. Negation: The cat is black.
Related Conditionals:
Converse: exchange the hypothesis and conclusion
Inverse: negate both the hypothesis and conclusion
Contrapositive: exchange both the hypothesis and
conclusion, then negate them both.
Example 2
Rewrite the conditional statement in if-then form, then
write the converse, inverse, and contra-positive.
“Since m  A = 99º it is an obtuse angle.”
If-Then: If m A = 99º, then  A an obtuse angle.
Converse: If  A is an obtuse angle, then m  A = 99º.
Inverse: If m  A ≠ 99º, then  A is not obtuse.
Contrapositive: If  A is not obtuse, then m  A ≠ 99º.
Which of the above statements are true?
You try!
Rewrite the conditional statement in if-then form, then
write the converse, inverse, and contra-positive.
“The supplementary angles add up to 180º”
If-Then: If two angles are complementary, then they
add up to 180º
Converse: If two angles add up to 180º, then they are
supplementary.
Inverse: If two angles are not supplementary, then they do
not add up to 180º
Contrapositive: If two angles do not add up to 180º, then they are
not supplementary.
*With any definition, both the conditional statement
and its converse are true.
m
Perpendicular lines: If two lines
intersect to form a right angle, then
they are perpendicular lines.
Or...If two lines are perpendicular,
then they intersect to form a right
angle.
l
l⊥m
Example 3
Determine whether the statement is a valid definition.
a. If a polygon is a square, then the polygon has four
congruent sides.
b. If a polygon is both equilateral and equiangular, then the
polygon is a regular polygon.
c. If two angles have the same measure, then they are
congruent.
Biconditional Statement: written when a conditional
statement and its converse are true; contains the
phrase “if and only if.”
m
Definition: If two lines intersect to form a right
angle, then they are perpendicular.
Converse: If two lines are perpendicular, then
they intersect to form a right angle.
Biconditional: Two lines are perpendicular if and only if
they intersect to form a right angle. .
*All definitions can be written as biconditional statements.
l
Example 4
Write the definition of a right angle as (a) an if-then
statement, (b) the converse of your if-then statement, and (c)
a biconditional statement.
(a) If an angle is a right angle, then
the measure of the angle is 90º.
(b)
If the measure of an angle is 90º,
then it is a right angle.
(c) An angle is a right angle if and only
if the measure of the angle is 90º.
Example 3
Determine whether the statement about the diagram is true.
(a) AC ⊥ BD
Yes. The right angle symbol indicates the
lines intersect to form a right angle.
(b) ∠AEB and ∠CEB form a linear pair.
A
Yes. The noncommon sides form a pair of
opposite rays.
(c) EA and EB are opposite rays.
No. Point E does not lie on the same line
as A and B, so the rays are not opposite.
B
C
E
D
Homework
Pg 82-84
#1, 3-29, 39