Preparing for the Regents Examination Geometry, AK

13 Converse: If I am not in the starting lineup,
then I will not go to practice.
Inverse: If I to go to practice, I will be in the
starting lineup.
Contrapositive: If I am in the starting lineup,
then I will go to practice.
14 Converse: If I am unhappy, then I did not
make honor roll.
Inverse: If I make honor roll, then I am happy.
Contrapositive: If I am happy, then I made
honor roll.
15 Converse: If I do not have money for the
movies, then I will go shopping.
Inverse: If I do not go shopping, then I will
have money for the movies.
Contrapositive: If I have money for the
movies, then I will not go shopping.
16 Converse: If I do not take a school bus, then I
live close to school.
Inverse: If I do not live close to school, then I
take a school bus.
Contrapositive: If I take a school bus, then I do
not live close to school.
17 Converse: If I don’t study Latin, then I will
study French.
Inverse: If I don’t study French, then I will
study Latin.
Contrapositive: If I study Latin, then I will not
study French.
18 a If the segments are congruent, then their
measures are equal. (True)
b If the measures of the segments are equal,
then the segments are congruent. (True)
2-5 Biconditionals
(pages 31–32)
1 (1) true for any value of x
2 (2) false for any value of x
3 (2) false
4 (1) If I am sleepy, then I did not get eight
hours of sleep and if I did not get eight hours
of sleep, then I am sleepy.
5 (3) Paul does not buy chicken or he does not
roast it.
6 (4) the conjunction of a conditional and its
converse
7 If I have money, then I earned it.
8 If tomorrow is Friday, then today is Thursday.
(True)
9 Converse: If a number is rational, then it is a
repeating decimal.
Biconditional: A number is rational if and only
if it is a repeating decimal.
The biconditional is false for integers and
terminating decimals.
10 If the sum of the measures of two angles is
not 45, then the two angles are not complementary.
Any pair of complementary angles makes
this statement and original false.
Any pair of angles with a sum of measures
not equal to 45 or 90 makes this statement
and the original true.
2-6 Laws of Logic
(page 36)
1 Margaret is a doctor. (Disjunctive Inference)
2 I will have straight teeth. (Detachment)
3 I did not save money. (Modus Tollens)
4 I helped my friend. (Modus Tollens)
5 It will not rain. (Detachment)
6 Joe is not a teacher. (Disjunctive Inference)
7 I will not eat dinner with Michael. (Modus
Tollens)
8 I will not play on Saturday. (Detachment)
9 If I practice the violin, my friend will be
jealous. (Chain Rule)
10 If I don’t help my brother, he cannot play
football. (Chain Rule)
2-7 Proof in Logic
(page 38)
1 1. p ∨ q
2. r → ~q
3. r
4. ~q Law of Detachment (2, 3)
5. p Law of Disjunctive Inference (1, 4)
2 1. a → b
2. c ∨ a
3. ~c
4. a Law of Disjunctive Inference (2, 3)
5. b Law of Detachment (1, 4)
2-7 Proof in Logic 5