Conditional Statements

1
Conditional Statements

• Geometry
• Chapter 2, Section 1

2
Notes

• Conditional Statement is a logical statement
  with two parts, a hypothesis and a conclusion.
• Hypothesis are the conditions that were
  considering
• Conclusion is what follows as a result of the
  conditions in the hypothesis.
• If-then form a style of stating a conditional
  statement where the hypothesis comes immediately
  after the word if and the conclusion comes
  immediately after the word then.
• That is, if the if part is satisfied then the
  then part must follow.

3
Notes

• Example
• If-then form of a conditional statement
• If it is raining outside, then the ground is
  wet.
• Hypothesis it is raining outside
• Conclusion the ground is wet.

4
Notes

• On your own
• Identify the hypothesis and the conclusion, then
  write the following conditional statement in
  if-then form.
• A number divisible by 9 is divisible by 3
• h a number is divisible by 9
• c it is divisible by 3
• If a number is divisible by 9, then it is
  divisible by 3
• The 49ers will play in the Super Bowl XLII, if
  they win their next game.
• h they win their next game
• c 49ers will play in the Super Bowl XLII
• If they win their next game then the 49ers will
  play in the Super Bowl XLII

5
Notes

• For a conditional statement to be true, it must
  be proven true for all cases that satisfy the
  conditions of the hypothesis
• A single counterexample is enough to prove a
  conditional statement false
• On Your Own
• Write a counterexample to show that the following
  statement is false.
• If x2 16, then x 4
• Counterexample if x -4 then x2 16, i.e. the
  hypothesis is satisfied, but x does not equal 4
• This proves the statement false.

6
Notes

• Related Conditionals other statements formed by
  changing the original statement.
• Converse of a statement is formed by switching
  the conclusion and the hypothesis. The converse
  of a statement is not always true!
• Example
• Original If its raining outside, then the
  ground is wet.
• Converse If the ground is wet, then it is
  raining outside.
• Q is the converse true or false?
• False

7
Notes

• On Your Own Write the converse of the following
  statement
• Original If a number is divisible by 9 then it
  is divisible by 3
• Converse If it is divisible by 3, then a number
  is divisible by 9
• Q is the converse true or false?
• False
• Original If two segments are congruent, then
  they have the same length.
• Converse If two segments have the same length,
  then they are congruent.
• Q is the converse true or false?
• True

8
Notes

• Inverse formed by negating the hypothesis and
  conclusion of the statement
• Example
• Statement If its raining outside, then the
  ground is wet.
• Inverse If its not raining outside, then the
  ground is not wet.
• On Your Own write the inverse of the following
  statement
• Statement If two segments are congruent, then
  they have the same length.
• Inverse If two segments are not congruent, then
  they do not have the same length.

9
Notes

• Contrapositive (Combination of converse and
  inverse) formed by switching the hypothesis and
  the conclusion and negating them.
• Example
• Statement If its raining outside, then the
  ground is wet.
• Contrapositive if the ground is not wet, then it
  is not raining outside
• On Your Own
• Statement If two segments are congruent, then
  they have the same length.
• Contrapositive If two segments do not have the
  same length, then they are not congruent.

10
Notes

• Logically Equivalent Statements Statements that
  have the same truth value (i.e. when one is true,
  so is the other)
• A statement and its contrapositive are equivalent
  statements
• Original If its raining outside, the ground is
  wet.
• Contrapositive If the ground isnt wet, does
  that mean it isnt raining?
• Yes
• Lets think about this, are these two saying the
  same thing?
• The converse and inverse are also logically
  equivalent.

11
Conditional Statement Activity

• Come up with your own conditional statement in
  if-then form
• Write the converse, inverse, and contrapositive.
• Judge the validity of all four statements.
• Do the equivalent statements match up as they
  should and make sense?
• Write counterexamples for the statements you
  think are false.
• Be sure to
• Label the four statements,
• Indicate whether each is true or false, and
• Show which statements are equivalent to each
  other.

12
Point, Line, and Plane Postulates

• Pg 73