Deductive Reasoning

“The proof is in the pudding.”

“Indubitably.”Je solve le crime. Pompt de pompt pompt."

Le pompt de pompt le solve de crime!"

2-1 Written Ex. P 35.

2-1 Written Ex. P 34.Underline the hypothesis and box the conclusion.

1. If 3x – 7 = 32, then x = 13.

2. I can’t sleep if I am tired.

3. I’ll try if you will.

Underline the hypothesis and box the conclusion.

4. If , then is a right angle.

5. a + b = a implies b = 0.

6. x = - 5 only if x2 = 25.

1 90m 1

Rewrite each pair of conditionals as a biconditional.

7. If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

8. If , then is a straight angle. If is a straight angle, then .

180m AOC AOC180m AOC AOC

B is between A and C if and only if AB + BC = AC

if and only if is a straight angle.AOC180m AOC

Write each biconditional as a pair of conditionals.

9. Points are collinear if and only if they all lie in one line.

10. Points lie in on plane if and only if they are coplanar.

If points are collinear, then they all lie in one line.

If all points lie in one line, then they are collinear.

If points lie in on plane, then they are coplanar.

If points are coplanar, then they lie in on plane.

Provide a counter example to show that each statement is false.

11. If ab < 0 , then a < 0.

12. If n2 = 5n , then n = 5.

13. If point G is on , then G is on .AB��������������

BA��������������

When trying these algebraic us negative numbers , 1, 0, 1, 10, and fractions between 0 and 1 as test values.

Let a = 1 and b = -1(-1)(1) < 0 and a > 0

n = 0n2 = 5n but n is not = 5

A B G

Provide a counter example to show that each statement is false.

14. If xy > 5y , then x > 5.

15. If a four-sided figure has 4 right angles, then it has four congruent sides.

15. If a four-sided figure has four congruent sides, then it has 4 right angles.

Let x = - 1 and y = -1(-1)(-1) > 5(-1) 1 > -5But – 1 is not > 5

A rectangle.

Diamond or rhombus.

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

17. If x = - 6, then . 6x

If , then x = - 6.6x

True

False x could = + 6

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

18. If , then x = - 2 .2 4x

False x could = + 2Quadratic equations have 2 roots.

18. If x = - 2 , then .2 4x

True

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

19. If b > 4 , then 5b > 20. True

If 5b > 20, then b > 4.

True

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

20. If , then is not obtuse.40m T T

True

If is not obtuse, then .40m T T

False T could = 90, 80, ..

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

21. If Pam lives in Chicago, then she lives in Illinois.

True

If Pam lives in Illinois, then she lives in Chicago.

False, there are a lot of other cities in Illinois.

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

22. If , then .A B m A m B

If , then .A B m A m B

True

True: all definitions are reversible.

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

23. If , then .

2 9a 3a

2 9a 3a

23. if .

True

False a could be -10.

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

24. x = 1 only if .2x x

If then x = 1 .2x x

Note “only if” is not the same as “if”.Only if means “then”

False x could be 0.

True

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

25. n > 5 only if n > 7 .

Note “only if” is not the same as “if”.Only if means “then”

If n > 7 , then n > 5.

False n could be 6.

True

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

26. ab = 0 implies that a = 0 or b = 0.

a = 0 or b = 0 implies that ab = 0.

True

True

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

27. If points D, E, and F are collinear, then DE + EF = DF.

D F E

If DE + EF = DF, then points D, E, and F are collinear.

D

E

F

FalseCounterexample

CounterexampleFalse

Tell whether each statement is true or false. Then write the converse and tell whether the converse is true of false.

28. P is the midpoint of implies that GH = 2 PG.

GH

True

GH = 2 PG. implies that P is the midpoint of .GH

FalseCounterexample

P

HG

1

2

29. Write a definition of congruent angles as a biconditional.

30. Write a definition of a right angle as a biconditional.

.A Bif and only if m A m B

A is a right angle if and only if090m A

C’est fini.

Good day and good luck.