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PRINCIPLE
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If a statement is false and its converse is true, then the conditions in the hypothesis are necessary but not sufficient for its conclusion.
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The statement if angles are supplementary, then they are right angles is false, and its converse, if angles are right angles, then they are supplementary, is true. Hence, angles need to be supplementary to be right angles, but being supplementary is not sufficient for angles to be right angles.
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PRINCIPLE 4:
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If a statement and its converse are both false, then the conditions in the hypothesis are neither necessary nor sufficient for its conclusion.
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Thus the statement if angles are supplementary, then they are congruent is false, and its converse, if angles are congruent, then they are supplementary, is false. Hence, being supplementary is neither necessary nor sufficient for the angles to be congruent.
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These principles are summarized in the table that follows.
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When the Conditions in the Hypothesis of a Statement are Necessary or Sufficient to Justify its Conclusion Principle 1 2 3 4 Statement True True False False Converse True False True False Sufficient Yes Yes No No Necessary Yes No Yes No
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CHAPTER 14 Improvement of Reasoning
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SOLVED PROBLEMS
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Determining necessary and sufficient conditions For each of the following statements, determine whether the conditions in the hypothesis are necessary or sufficient to justify the conclusion. (a) A regular polygon is equilateral and equiangular. (b) An equiangular polygon is regular. (c) A regular polygon is equilateral. (d) An equilateral polygon is equiangular.
Solutions
(a) Since the statement and its converse are both true, the conditions are necessary and sufficient. (b) Since the statement is false and its converse is true, the conditions are necessary but not sufficient. (c) Since the statement is true and its converse is false, the conditions are sufficient but not necessary. (d) Since both the statement and its converse are false, the conditions are neither necessary nor sufficient.
SUPPLEMENTARY PROBLEMS
14.1. State the order in which the terms in each of the following sets should be defined: (a) Jewelry, wedding ring, ornament, ring (b) Automobile, vehicle, commercial automobile, taxi (c) Quadrilateral, rhombus, polygon, parallelogram (d) Obtuse triangle, obtuse angle, angle, isosceles obtuse triangle 14.2. Correct each of the following definitions: (a) A regular polygon is an equilateral polygon. (b) An isosceles triangle is a triangle having at least two congruent sides and angles. (c) A pentagon is a geometric figure having five sides. (d) A rectangle is a parallelogram whose angles are right angles. (e) An inscribed angle is an angle formed by two chords. (f) A parallelogram is a quadrilateral whose opposite sides are congruent and parallel. (g) An obtuse angle is an angle larger than a right angle. 14.3. State the negative of each of the following statements: (a) x (b) 3y 2 15 4 (d) His mark was more than 65. (e) Joe is heavier than Dick. (f) a b c (14.4) (14.2) (14.1)
(c) She loves you.
CHAPTER 14 Improvement of Reasoning
(14.5)
14.4. State the inverse of each of the following statements, and indicate whether or not it is true. (a) A square has congruent diagonals. (b) An equiangular triangle is equilateral. (c) A bachelor is an unmarried person. (d) Zero is not a positive number.
14.5. State the converse, inverse, and contrapositive of each of the following statements. Indicate the truth or falsity of each, and check the logical equivalence of the statement and its contrapositive, and of the converse and inverse. (14.6) (a) If two sides of a triangle are congruent, the angles opposite these sides are congruent. (b) Congruent triangles are similar triangles. (c) If two lines intersect, then they are not parallel. (d) A senator of the United States is a member of its Congress. 14.6. Form partial converses and partial inverses of the theorems given in Fig. 14-5. (14.7)
Fig. 14-5
14.7. For each of the following statements, determine whether the conditions in the hypothesis are necessary or sufficient to justify the conclusion. (14.8) (a) Senators of the United States are elected members of Congress, two from each state. (b) Elected members of Congress are senators of the United States. (c) Elected persons are government officials. (d) If a woman lives in New York City, then she lives in New York State. (e) A bachelor is an unmarried man. (f) A bachelor is an unmarried person. (g) A quadrilateral having two pairs of congruent sides is a parallelogram.
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