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CHAPTER 14 Improvement of Reasoning
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(a) A person who is not born in the United States is not a citizen of the United States. (False, since there are naturalized citizens) (b) One who is not a sculptor is not a talented person. (False, since one may be a fine musician, etc.) (c) A figure that is not a triangle is not a polygon. (False, since the figure may be a quadrilateral, etc.)
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14.6 Forming the converse, inverse, and contrapositive State the converse, inverse, and contrapositive of the statement a square is a rectangle. Determine the truth or falsity of each, and check the logical equivalence of the statement and its contrapositive, and of the converse and inverse.
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Statement: A square is a rectangle. (True) Converse: A rectangle is a square. (False) Inverse: A figure that is not a square is not a rectangle. (False) Contrapositive: A figure that is not a rectangle is not a square. (True) Thus, the statement and its contrapositive are true and the converse and inverse are false.
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14.4 Partial Converse and Partial Inverse of a Theorem
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A partial converse of a theorem is formed by interchanging any one condition in the hypothesis with one consequence in the conclusion. A partial inverse of a theorem is formed by denying one condition in the hypothesis and one consequence in the conclusion. Thus from the theorem if a line bisects the vertex angle of an isosceles triangle, then it is an altitude to the base, we can form a partial inverse or partial converse as shown in Fig. 14-4. In forming a partial converse or inverse, the basic figure, such as the triangle in Fig. 14-4, is kept and not interchanged or denied.
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Fig. 14-4
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In Fig. 14-4(b), the partial converse is formed by interchanging statements (1) and (3). Stated in words, the partial converse is: If the bisector of an angle of a triangle is an altitude, then the triangle is isosceles. Another partial converse may be formed by interchanging (2) and (3). In Fig. 14-4(c), the partial inverse is formed by replacing statements (1) and (3) with their negatives, (1 ) and (3 ). Stated in words, the partial inverse is: If two sides of a triangle are not congruent, the line segment that bisects their included angle is not an altitude to the third side. Another partial inverse may be formed by negating (2) and (3).
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CHAPTER 14 Improvement of Reasoning
SOLVED PROBLEMS
14.7 Forming partial converses with partial inverses of a theorem Form (a) partial converses and (b) partial inverses of the statement congruent supplementary angles are right angles.
Solutions
(a) Partial converses: (b) Partial inverses: (1) Congruent right angles are supplementary. (2) Supplementary right angles are congruent. (1) Congruent angles that are not supplementary are not right angles. (2) Supplementary angles that are not congruent are not right angles.
14.5 Necessary and Sufficient Conditions
In logic and in geometry, it is often important to determine whether the conditions in the hypothesis of a statement are necessary or sufficient to justify its conclusion. This is done by ascertaining the truth or falsity of the statement and its converse, and then applying the following principles.
PRINCIPLE
If a statement and its converse are both true, then the conditions in the hypothesis of the statement are necessary and sufficient for its conclusion.
For example, the statement if angles are right angles, then they are congruent and supplementary is true, and its converse, if angles are congruent and supplementary, then they are right angles is also true. Hence, being right angles is necessary and sufficient for the angles to be congruent and supplementary.
PRINCIPLE
If a statement is true and its converse is false, then the conditions in the hypothesis of the statement are sufficient but not necessary for its conclusion.
The statement if angles are right angles, then they are congruent is true, and its converse, if angles are congruent, then they are right angles, is false. Hence, being right angles is sufficient for the angles to be congruent. However, the angles need not be right angles to be congruent.
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