Unit 2- LOGICAL REASONING & PROOFS 2-1: Conditional Statements G.3A To determine the validity of a conditional statement and its converse. G.3C To find counterexamples to disprove statements that are false.
ANSWERS: 1. y = 1 2. x = 2, 2 3. z = 2 4. x = 5. n = 0 6. m = 1
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Vocabulary an “if–then“ statement. A conditional is _______________________________________________________________ The hypothesis is ______________________________________________________________ the part that follows if in an if-then statement. The conclusion is _______________________________________________________________ the part that follows then in an if-then statement.
“true” or “false” according to the truth or falsity of the statement. The truth value of a statement is ___________________________________________________ is the conditonal “if q, then p“ . The converse of the conditional “if p, then q ” __________________________________________
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Identify the hypothesis and conclusion: If two lines are parallel, then the lines are coplanar. In a conditional statement, the clause after if is the hypothesis and the clause after then is the conclusion.
Two lines are parallel. Hypothesis: ______________________________________________________________ The lines are coplanar. Conclusion: ______________________________________________________________
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Find a counterexample to show that this conditional is false: If x2 0, then x 0. A counterexample is a case in which the hypothesis is true and the conclusion is false . This counterexample must be an example in which x2 0 (hypothesis true) and x < 0 or x 0 (conclusion false). Because any negative number has a positive square, one possible counterexample is 1. Because (-1)2 = 1, which is greater than 0, the hypothesis is Because -1 < 0, the conclusion is false
.
The counterexample shows that the conditional is false
.
true
.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Write the converse of the following conditional: If x 9, then x 3 12.
The converse of a conditional exchanges the hypothesis and the conclusion.
Conditional
Converse
Hypothesis
Conclusion
Hypothesis
Conclusion
x9
x 3 12
x 3 12
x9
If x 3 12, then x 9. So the converse is: ____________________________________
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
y 3 5 y8
New Mexico, because it is a state that includes the word “NEW” and it does not border an ocean. So the hypothesis is true but the conclusion is false.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Write the converse of the conditional. Then determine the truth value of each. If a2 25, then a 5. If a2 25, then a 5. hypothesis
If a 5, then a2 25. false
true
conclusion
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
3. Write the converse of the following conditional: If two lines are not parallel and do not intersect, then they are skew.
If two lines are skew, then they are not parallel and do not intersect. 4. Write the converse of each conditional. Determine the truth value of the conditional and its converse. (Hint: One of these conditionals is not true.) a. If two lines do not intersect, then they are parallel. Converse:
If two lines are parallel, then they do not intersect.
The conditional is
false and the converse is
true
.
b. If x 2, then ∣x∣ 2. Converse:
If ∣x∣ 2, then x 2.
The conditional is
true
and the converse is
false
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
More examples… Textbook page 81
Dogs cocker spaniel
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Textbook page 81
Converse: If two lines are skew, then they are not parallel and do not intersect.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Write the converse of each statement. Then determine the truth value of the converse. true A. If two planes have no points in common, then they are parallel. false B. If a figure has four sides, then it is a square. counterexamples: rectangle, rhombus, kite, trapezoid
true
C. If a point lies on the y-axis, then it has x-coordinate 0.
true
D. If two angles have they same measure, then they are congruent.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
More examples… Textbook page 83
Hypothesis: You send in the proof-of-purchase label.
Conclusion: They send you a get-well card.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Textbook page 83
H: You want to be fit. C: You want to get plenty of exercise. H: x + 20 = 32 C: x = 12 H: You can see the magic in a fairy tale. C: You can face the future. H: Somebody throws a brick at me. C: I can catch it and throw it back.
H: You can accept defeat and open your pay envelope without feeling guilty. C: You are stealing.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Textbook page 83 H: My fans think that I can do everything I say I can do. C: They are crazier than I am.
H: I could paint that flower in a Huge scale. C: You could not ignore its beauty.
2-1
Conditional Statements Objectives: 1) To recognize conditional statements. 2) To write converses of conditional statements.
Textbook page 83
End of section notes for 2-1 Conditional Statement.
HOMEWORK: Sept 16, 2013 (Monday) Complete Practice 2-1 of workbook, DNG page 275, due on Sept 17, 2013 before the class starts. If you feel you need homework help, a) go to Room G-8, Ms. Z. Pasion on Mondays and Wednesdays 4:05-5:00 p.m., b) at the library each morning, 7:15-7:45, Mondays through Fridays. Ask Ms. A. Rivera for more information. c) You may also click on the Geometry textbook link and right-click on video lessons for this particular section.
