“The proof is in the pudding.”
“Indubitably.”
Le pompt de pompt le solve de crime!"
Je solve le crime. Pompt de pompt pompt."
Deductive Reasoning
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 m 1  90 , then  1 is a right angle.
5.
a + b = a implies b = 0.
6.
x = - 5 only if x2 = 25.
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.
B is between A and C if and only if AB + BC = AC
8. If m  AOC 180 , then  AOC is a straight angle.
If  AOC is a straight angle, then m  AOC 180 .
m  AOC 180 if and only if  AOC is a straight angle.
Write each biconditional as a pair of conditionals.
9. Points are collinear if and only if they all lie in one line.
If points are collinear, then they all lie in one line.
If all points lie in one line, then they are collinear.
10. Points lie in on plane if and only if they are coplanar.
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.
When trying these algebraic us negative numbers , 1, 0, 1,
10, and fractions between 0 and 1 as test values.
11. If ab < 0 , then a < 0.
12. If
n2
= 5n , then n = 5.
13. If point G is on
Let a = 1 and b = -1
(-1)(1) < 0 and a > 0
n=0
n2 = 5n but n is not = 5
AB , then G is on BA
A
B G
.
Provide a counter example to show that each
statement is false.
14. If xy > 5y , then x > 5.
Let x = - 1 and y = -1
(-1)(-1) > 5(-1)
1 > -5
But – 1 is not > 5
15. If a four-sided figure has 4 right angles,
then it has four congruent sides.
A rectangle.
15. If a four-sided figure has four congruent sides,
then it has 4 right angles.
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 x  6 .
If
x 6
True
, then x = - 6.
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
x  4 , then
2
x=-2.
False x could = + 2
Quadratic equations have 2 roots.
18. If x = - 2 , then
True
x 4 .
2
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 m  T  40 , then
T
is not obtuse.
True
If
T
is not obtuse, then m  T  40 .
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
 A   B , then m  A  m  B .
True
If m  A  m  B , then
 A  B .
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. a  9 if
2
a 3
.
True
23. If
a 9
2
, then
a 3
.
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
x  x.
2
True
Note “only if” is not
the same as “if”.
Only if means
“then”
If x  x then x = 1 .
2
False x could be 0.
Tell whether each statement is true or false.
Then write the converse and tell whether the
converse is true of false.
Note “only if” is not
25. n > 5 only if n > 7 .
the same as “if”.
Only if means
“then”
False n could be 6.
If n > 7 , then n > 5.
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.
True
a = 0 or b = 0 implies that ab = 0.
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.
False
Counterexample
D
F
E
E
If DE + EF = DF, then points D, E,
and F are collinear.
False
Counterexample
D
F
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 GH implies
that GH = 2 PG.
True
GH = 2 PG. implies that P is the
midpoint of GH .
False
P
1
Counterexample
G
2
H
29. Write a definition of congruent
angles as a biconditional.
 A   Bif and only if m  A  m  B.
30. Write a definition of a right angle
as a biconditional.
 A is a right angle if and only if m  A  90
0
C’est fini.
Good day and good luck.