Geometry / Mr. Hansen
9/21/2001

Name: _________________________

§1.7

1.

1. undefined terms
2. postulates (a.k.a. axioms)
3. definitions
4. previously proved theorems, lemmas, corollaries

5.

i. B Þ A; original statement and converse could be both true, both false, or one of each (insufficient information to determine)
ii. wet Þ rain; both the original statement and its converse are false (you could use an umbrella, and other things besides rain can cause wetness)
iii. acute Ð Þ 45°; false
iv. If a point divides a segment into two congruent segments, then the point is the midpoint of the original segment. Both the original statement and its converse are true (by definition). [However, note that in a definition, both sides of the Û could be false. The Û means equivalence. Either both sides are true, or both sides are false.]

7.

Fallacy in reasoning: In reality, coin flips are independent. The coin has no memory of what has gone before. The odds are always 50-50 unless the coin was manufactured irregularly.

8.

Correct. [Let H represent that a person is a student with homeroom 303, F that a person is a freshman at Niles High. Since H is true for Joe Jacobs, and since the implication H Þ F is true for everyone, F must be true for Joe. If the hypothesis of a true implication is true, the conclusion must also be true.]

10.

Incorrect. Let B represent that a vehicle is a school bus, R that a vehicle stops at railroad crossings. We know B Þ R is true, and we see a vehicle that satisfies R. To conclude B would require reasoning from the converse of the known implication. That is an error. If you see a vehicle that satisfies R, you can conclude nothing.

14.

[To be done in class.]

1.

a. If a person is 18 years old, then s/he may vote in federal elections.

2a.

Original statement (TRUE): all sides of D are 10 Þ perimeter = 30
Converse (FALSE): perimeter = 30 Þ all sides of D are 10
Inverse (FALSE): at least one side of D ¹ 10 Þ perimeter ¹ 30
Contrapositive (TRUE): perimeter ¹ 30 Þ at least one side of D ¹ 10

4.

[See Venn diagram in your notes: Chicago circle should be fully inside the Illinois circle.]
a. TRUE
b. FALSE
c. FALSE
d. FALSE

5.

a. d Þ f [also acceptable: ~f Þ ~d]
b. s Þ ~p [also acceptable: p Þ ~s]

9.

p Þ ~g [also acceptable: g Þ ~p]

10.

The statements are inconsistent. In other words, they cannot all be true. [If you assume they are all true, you can prove that if the line is long, then the line is not long—a contradictory situation.]

1.
2.
3.
4.
5.
6.
7.
8.
12.
15.

3/5
1/5
1/5
0
1/3
3/10 [must list all the cases, or show work: ₃C₂ / ₅C₂]
2/5 [must list all the cases, or show work: 1 · 4 / ₅C₂]
1/10 [must list all the cases, or show work: 1 · 1 / ₅C₂]
problem equivalent to #8
a. 9/45 = 1/5
b. 1 – 9/45 = 4/5

1.

a. line AR, line AD, line RA, line RD, line DA, line DR (write « symbol above each)
b. ray BA, ray BC (write ® symbol above each)
c. ray DF
d. ray CB
e. about 60°; about 50°; about 120°
f. No.
g. No angle can be called ÐB, since there are 3 (yes, three) angles at vertex B.
h. line AC
i. segment EF
j. Ð1
k. A
l. segment EF

3.

a. 69°4'35"
b. 50°59'43"

4.

a. 46°52'30"
b. (132 1/10)°, or you could say 132.1°

5.

a. segment BC @ segment RT
b. ÐA @ ÐS

21.

a. 30° (no work needed)
b. 140° (must show diagram)
c. 127°30' (must show diagram and work)

22.

Converse (FALSE): If the angle formed by the hands of a clock is acute, then the time is 2:00.
Inverse (FALSE): If the time is not 2:00, then the angle formed by the hands of a clock is not acute.
Contrapositive (TRUE): If the angle formed by the hands of a clock is not acute, then the time is not 2:00.
[A statement and its contrapositive are always equivalent. That means they are either both true or both false. The same is true of converse and inverse: either both true or both false. However, you cannot assume that the converse is false just because the original statement is true, nor can you assume that the converse is true if the original statement is false. The converse gives you no information about the truth or falsehood of the original statement. If we list the original statement/converse/inverse/contrapositive, the possibilities are TTTT xor TFFT xor FTTF xor FFFF. Just because we happened to have TFFT in this problem does not mean you can assume that will happen in general.]

26.

30

31.

a. 44.5°
b. 44°33'

33.

7 < PR < 31

39.

The trick is to put two of the labeled points on the same side of E for one of the lines.

41.

(hard) 3:32 plus 43 7/11 seconds, or approximately 3:32:44

1.
2.
3.
4.
5.
6.

FALSE
TRUE
TRUE
segment AC
cannot be simplified
Æ

7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

segment AC
10
11/15 was intended, but 1 (or 100%) must also be accepted here
Def. trisect.
Def. acute Ð
Def. @
4 < y < 26
145 [note: since question asked for measure, 145° is ever so slightly incorrect]
ÐSQT = 13°50'54" [note: question erroneously asked for measure]
44 < x < 74
Since FH + HG = FG (not > FG), the triangle inequality fails. Therefore, F, G, and H are collinear. Since the longest distance is 18 = FG, H must be between F and G.

18.

4

19.

37°37'30" [must show work]

20.

(18 4/9)º [must show work]

21.

If the problem had said, "If a point at random is chosen from rectangle PART, what is the probability that it is in rectangle BEST?" then the answer would be 24/96 = 1/4. However, that is not what the problem said! Congratulations to Miles and Paul for finding the mistake. As worded, the question must be answered as 1 (or, if you prefer a percentage, 100%). Of course, the other wording makes a much better problem, and you should understand how the 24/96 was computed.

22.
23.
24.

If chickens did not lay eggs, then cows would have wings.
If chickens laid eggs, then cows would not have wings.
p Þ r [also acceptable: ~r Þ ~p]
[Note: "syllogism" means the same as forward chaining.]

25.

x = 40.5
y = –22.5
mÐEBD = 18

26.

By computation, mÐ1 + mÐ2 = 90, so by def. ÐPQR is a right Ð. Since it is given that ÐRST (a.k.a. ÐS) is also a right Ð, and since all right Ðs are @, ÐPQR @ ÐS. (Q.E.D.)

VI. 1.
2.
3.
4.
5.

contrapositive
converse
original statement
converse
inverse

VII.

1. ray SP bisects ÐEST
2. ÐTSP @ ÐESP
3. ÐESP @ ÐABC
4. ÐTSP @ ÐABC

1. Given
2. Def. bisect.
3. Given
4. Transitive prop. @