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Inequalities and Indirect Reasoning
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Proving an inequality theorem by the indirect method Prove that in the same or equal circles, unequal chords are unequally distant from the center.
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Given: Circle O, AB CD OE ' AB, OF ' CD To Prove: OE OF Plan: Assume the other possible conclusion, OE OF, and arrive at a contradiction.
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Statements 1. Either OE OF or OE OF. 2. Assume OE OF. 3. If OE OF, then AB CD. 4. But AB CD. 5. The assumption OE 6. Hence, OE OF.
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Reasons 1. Two quantities are either equal or unequal. 2. This is one of the possible conclusions. 3. In the same or equal circles, chords equally distant from the center are equal. 4. Given 5. It leads to a contradiction. 6. This is the only remaining possibility.
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OF is not valid.
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13.1. Determine which inequality symbol, (a) If y (b) If x (c) If x 15, then 15 2, then 3x 2 and y y. or , makes each of the following true: (d) If a (e) If a (f) If b 4 and b
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(13.1) 15.
then a/b
1 4. 3, then xy 5.
5, then a2 4a. then b2 b. (13.2)
13.2. Complete each of the following statements: (a) If y (b) If a x and x b z, then y z. d, then a d c. (c) If a (d) If z b and b y, y 15, then a 15. 10, then z 10.
c and b
x, and x
13.3. Complete each of the following statements about Fig. 13-10: (a) BC BD (b) m/BAD m/BAC (c) ^ADC ^ABC (d) If m/A m/C, then AB BD.
(13.2)
Fig. 13-10
13.4. Complete each of the following statements:
(13.2)
(a) If Mary and Ann earn the same weekly wage and Mary is to receive a larger increase than Ann, then Mary will earn Ann earns. (b) If Bernice, who is the same weight as Helen, loses more weight than Helen, then Bernice will weigh Helen weighs. 13.5. Complete each of the following statements: (a) If a (b) If x (c) If 3x 3, then 4a 12. 3 15, then x 18. 18, then x 6. (d) If f (e) If x (f) If g 8, then f y, then x h, then g 7 15. 6. 9. (13.2)
5 y 10 h
Inequalities and Indirect Reasoning
13.6. Which of the following sets of numbers can be the lengths of the sides of a triangle (a) 3, 4, 8 (b) 5, 7, 12 (c) 3, 4, 6 (d) 2, 7, 8 (e) 50, 50, 5
(13.3)
13.7. What integer values can the length of the third side of a triangle have if the two sides have lengths (a) 2 and 6; (b) 3 and 8; (c) 4 and 7; (d) 4 and 6; (e) 4 and 5; (f) 7 and 7 (13.3) 13.8. In Fig. 13-11, arrange, in descending order of size, (a) the angles of ^ABC; (b) the sides of ^DEF; (c) the angles 1, 2, and 3. (13.3)
Fig. 13-11
13.9. (a) In quadrilateral ABCD of Fig. 13-12, compare m/BAC and m/ACD if AB
CD and BC
(b) In ^ABC of Fig. 13-13, compare AB and BC if BM is the median to AC and m/AMB 13.10. Arrange, in descending order of magnitude, (a) The sides of ^ABC in Fig. 13-14 (b) The central angles AOB, BOC, and AOC in Fig. 13-14 (c) The sides of trapezoid ABCD in Fig. 13-15 (d) The distances of the sides of ^DEF from the center in Fig. 13-16
m/BMC.
Fig. 13-12
Fig. 13-13
Fig. 13-14
Fig. 13-15
Fig. 13-16
Inequalities and Indirect Reasoning
(13.5)
13.11. Provide the proofs requested in Fig. 13-17.
Fig. 13-17
13.12. Explain how indirect reasoning is used in each of the following situations: (a) A person determines which of his ties has been borrowed by his roommate.
(13.6)
(b) A girl determines that the electric motor in her train set is not defective even though her toy trains do not run. (c) A teacher finds which of his students did not do their assigned homework. (d) A mechanic finds the reason why the battery in a car does not work. (e) A person accused of a crime proves her innocence by means of an alibi. 13.13. Prove each of the following: (a) The base angles of an isosceles triangle cannot be right angles. (b) A scalene triangle cannot have two congruent angles. (c) The median to the base of a scalene triangle cannot be perpendicular to the base. (d) If the diagonals of a parallelogram are not congruent, then it is not a rectangle. (e) If a diagonal of a parallelogram does not bisect a vertex angle, then the parallelogram is not a rhombus. (f) If two angles of a triangle are unequal, the sides opposite are unequal, the longer side being opposite the larger angle. (13.7)