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Thus in Fig. 13-7, if OE
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Selecting inequality symbols Determine which inequality symbol, (a) 5 3 (b) 6 9
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Applying inequality axioms Complete each of the following statements: (a) If a (b) If x (c) If c (d) If x (e) If x (f) If e (g) If (h) If y 4x b and b y and y 20 and d y and y
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1 2 y.
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(i) If Paul and Jack have equal amounts of money and Paul spends more than Jack, then Paul will have than Jack. (j) If Anne is now older than Helen, then 10 years ago, Anne was
Solutions
(a) (b) (c) (d) , (e) (f) (g) (h) (i) less (j) older
than Helen.
Applying triangle inequality theorems (Fig. 13-8) (a) Determine the integer values that the length of side a of the triangle can have if the other two sides have lengths 3 and 7.
Inequalities and Indirect Reasoning
(b) Determine which is the longest side of the triangle if two angles have measures 59 and 60 . (c) Determine which is the longest side of parallelogram ABCD if E is the midpoint of the diagonals and m/AEB m/AED.
Fig. 13-8
Solutions
(a) Since a must be less than 3 7 10 and greater than 7 3 4, a can have the integer values of 5, 6, 7, 8, 9. 60 ) 61 . Then the longest side is opposite AD or AB ( DC) is the longest
(b) Since m/B 59 and m/C 60 , m/A 180 the largest angle, /A, so the longest side is BC. (c) In ABCD, AE CE and DE side (Principle 6).
EB. Since m /AEB
m/AED, AB
Applying circle inequality theorems In Fig. 13-9, compare (a) OD and OF if /C is the largest angle of ^ABC (b) AC and BC if mAC (c) mBC and mAC if OF mBC OE mBC
(d) m/AOB and m/BOC if mAB
Fig. 13-9
Solutions
(a) Since /C is the largest angle of the triangle, AB is the longest side, or AB OD OF by Principle 12. (b) Since mAC (c) Since OF (d) Since mAB mBC, AC OE, BC BC, the greater arc having the greater chord. mAC by Principle 10. BC; hence,
AC by Principle 13; hence, mBC
mBC, m/AOB
m/BOC, the greater arc having the greater central angle.
Inequalities and Indirect Reasoning
Proving an inequality problem Prove that in ^ABC, if M is the midpoint of AC and BM
Given: ^ABC, M is midpoint of AC.
AM, then m/A
m/B.
To Prove: m/A m/C m/B Plan: Prove m/A m/1 and m/C and then add unequals.
PROOF:
Statements 1. 2. 3. 4. M is the midpoint of AC . AM > MC BM AM BM MC 1. 2. 3. 4.
Reasons Given A midpoint divides a line into two congruent parts. Given A quantity may be substituted for its equal in any inequality. Definition of congruent segments. 5. In a triangle, the larger angle lies opposite the longer side. 6. If unequals are added to unequals, the sums are unequal in the same order.
5. In ^AMB, m/A /1. In ^BMC, m/C /2. 6. m/A m/C m/B
13.2 Indirect Reasoning
We often arrive at a correct conclusion by indirect reasoning. In this form of reasoning, the correct conclusion is reached by eliminating all possible conclusions except one. The remaining possibility must be the correct one. Suppose we are given the years 1492, 1809, and 1960 and are assured that one of these years is the year in which a president of the United States was born. By eliminating 1492 and 1960 as impossibilities, we know by indirect reasoning that 1809 is the correct answer. (Had we known that 1809 was the year in which Lincoln was born, the reasoning would have been direct.) In proving a theorem by indirect reasoning, a possible conclusion may be eliminated if we assume it is true and that assumption results in a contradiction of some given or known fact.
SOLVED PROBLEMS
Applying indirect reasoning in life situations Explain how indirect reasoning is used in each of the following situations: (a) A detective determines the murderer of a slain person. (b) A librarian determines which volume of a set of books is in use.
Solutions
(a) The detective, using a list of all those who could have been a murderer in the case, eliminates all except one. He or she concludes that the remaining one is the murderer. (b) The librarian finds all the books of the set except one by looking on the shelf and checking the records. He or she concludes that the missing one is the one in use.
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