Preparing for the Regents Examination Geometry, AK

3 1. e ∨ ~f
2. ~f → g
3. ~e
4. ~f Law of Disjunctive Inference (1, 3)
5. g Law of Detachment (2, 4)
4 1. a ∨ b
2. b → c
3. c → d
4. b → d Chain Rule (2, 3)
5. ~d
6. ~b Law of Modus Tollens (4, 5)
7. a Law of Disjunctive Inference (1, 6)
5 1. s ∨ t
2. s → r
3. r → q
4. s → q Chain Rule (2, 3)
5. ~q
6. ~s Law of Modus Tollens (4, 5)
7. t Law of Disjunctive Inference (1, 6)
8. t → v
9. v Law of Detachment (8, 7)
6 1. d → e
2. d ∨ f
3. h → ~e
4. h
5. ~e Law of Detachment (3, 6)
6. ~d Law of Modus Tollens (1, 5)
7. f Law of Disjunctive Inference (2, 6)
7 1. ~f → g
2. ~f ∨ j
3. g → ~h
4. j → k
5. h
6. ~g Law of Modus Tollens (3, 5)
7. f Law of Modus Tollens (1, 6)
8. j Law of Disjunctive Inference (2, 7)
9. k Law of Detachment (4, 8)
8 1. x → z
2. x → y
3. ~z
4. ~y Law of Modus Tollens (1, 3)
5. ~x Law of Modus Tollens (2, 4)
6. x ∨ t
7. t Law of Disjunctive Inference (5, 6)
Chapter Review (pages 38–40)
1 (1) true
2 (1) true
3 (2) false
6 Chapter 2: Logic
4 (2) false
5 (1) If I am late for school, I did not set my
alarm.
6 (2) If Marie is not bowling today, then today
is not Monday.
7 (1) The statement is always true but its converse
cannot be determined.
8 Hypothesis: The triangle has a 90 angle.
Conclusion: The square of the longest side of
a triangle is equal to the sum of the squares
of the other sides.
9 Hypothesis: A and B are alternate interior
angles.
Conclusion: They are congruent.
10 False
11 True
12 True
13 False
14 False
15 Inverse: If the altitude does not bisect the
base, the triangle is not isosceles.
Converse: If the triangle is isosceles, the altitude
bisects the base.
Contrapositive: If the triangle is not isosceles,
the altitude does not bisect the base.
Biconditional: The altitude bisects the base if
and only if it is isosceles.
16 True
17 False
18 p is true; q is false.
19 True
20 Converse: If a number is rational, then it is an
integer. (True)
Conditional statement is true.
Biconditional: A number is an integer if and
only if it is rational.
21 Converse: If a number is not prime, then a
number is a perfect square. (True)
Conditional statement is true.
Biconditional: A number is a perfect square if
and only if it is not prime.
22 Converse: If a parabola is tangent to the
x-axis, then its roots are equal. (True)
Conditional statement is true.
Biconditional: The roots of a quadratic equation
are equal if and only if the parabola is
tangent to the x-axis.