Improvement of Reasoning
14.1 Definitions
“Was Lincoln an educated man?” is a question that cannot be properly answered unless we agree upon the meaning of “an educated man.” Understanding cannot exist and progress cannot be made in any discussion or problem unless the terms involved are properly defined or, by agreement, are to be undefined.
14.1A Requirements of a Good Definition
PRINCIPLE 1: All terms in a definition must have been previously defined (or be those that, by agreement, are left undefined).
Thus, if we are to define a regular polygon as an equilateral and equiangular polygon, it is necessary that equilateral, equiangular, and polygon be previously defined.
PRINCIPLE 2: The term being defined should be placed in the next larger set or class to which it belongs.
Thus, the terms polygon, quadrilateral, parallelogram, and rectangle should be defined in that order. Once the term polygon has been defined, the term quadrilateral is then defined as a kind of polygon. Then the term parallelogram is defined as a kind of quadrilateral and, lastly, the term rectangle is defined as a kind of parallelogram.
Proper sequence in definition can be understood by using a circle to represent a set of objects. In Fig. 14-1, the set of rectangles is in the next larger set of parallelograms. In turn, the set of parallelograms is in the next larger set of quadrilaterals, and, finally, the set of quadrilaterals is in the next larger set of polygons.
PRINCIPLE 3: The term being defined should be distinguished from all other members of its class.
Thus, the definition of a triangle as a polygon with three sides is a good one since it shows how the triangle differs from all other polygons.
Fig. 14-1
PRINCIPLE 4: The distinguishing characteristics of a defined term should be as few as possible.
Thus, a right triangle should be defined as a triangle having a right angle, and not as a triangle having a right angle and two acute angles.
SOLVED PROBLEMS
14.1 Observing proper sequence in definition
In which order should the terms in each of the following sets be defined: (a) Englishman, European, Londoner; (b) quadrilateral, square, rectangle, parallelogram.
(a) European, Englishman, Londoner
(b) Quadrilateral, parallelogram, rectangle, square
14.2 Correcting faulty definitions
Correct the following definition: A trapezoid is a quadrilateral having two parallel sides.
Solution
The given definition is incomplete. The correct definition is “a trapezoid is a quadrilateral having only two parallel sides.” This definition distinguishes a trapezoid from a parallelogram.
14.2 Deductive Reasoning in Geometry
The kinds of terms and statements discussed in this section comprise the deductive structure of geometry, which can be visualized as in Fig. 14-2.
Fig. 14-2
14.2A Undefined and Defined Terms
Point, line, and surface are the terms in geometry which are, by agreement, not defined. These undefined terms begin the process of definition in geometry and underlie the definitions of all other geometric terms.
Thus, we can define a triangle in terms of a polygon, a polygon in terms of a geometric figure, and a geometric figure as a figure composed of line segments, or parts of lines. However, the process of definition cannot be continued further because the term “line” is undefined.
14.2B Assumptions
Postulates and axioms are the statements which are not proved in geometry. They are called assumptions because we willingly accept them as true. These assumptions enable us to begin the process of proof in the same way that undefined terms enable us to begin the process of definition.
Thus, when we draw a line segment between two points, we justify this by using as a reason the postulate “two points determine one and only one straight line.” This reason is an assumption since we assume it to be true without requiring further justification.
14.2C Theorems
Theorems are the statements which are proved in geometry. By using definitions and assumptions as reasons, we deduce or prove the basic theorems. As we use each new theorem to prove still more theorems, the process of deduction grows. However, if a new theorem is used to prove a previous one, the logical sequence is violated.
For example, the theorem “the sum of the measures of the angles of a triangle equals 1808” is used to prove that “the sum of the measures of the angles of a pentagon is 5408.” This, in turn, enables us to prove that “each angle of a regular pentagon measures 1088.” However, it would be violating logical sequence if we tried to use the last theorem to prove either of the first two.
14.3 Converse, Inverse, and Contrapositive of a Statement
DEFINITION 1: The converse of a statement is the statement that is formed by interchanging the hypothesis and conclusion.
Thus, the converse of the statement “lions are wild animals” is “wild animals are lions.” Note that the converse is not necessarily true.
DEFINITION 2: The negative of a statement is the denial of the statement.
Thus, the negative of the statement “a burglar is a criminal” is “a burglar is not a criminal.”
DEFINITION 3: The inverse of a statement is formed by denying both the hypothesis and the conclusion.
Thus, the inverse of the statement “a burglar is a criminal” is “a person who is not a burglar is not a criminal.” Note that the inverse is not necessarily true.
DEFINITION 4: The contrapositive of a statement is formed by interchanging the negative of the hypothesis with the negative of the conclusion. Hence, the contrapositive is the converse of the inverse and the inverse of the converse.
Thus, the contrapositive of the statement “if you live in New York City, then you will live in New York State” is “if you do not live in New York State, then you do not live in New York City.” Note that both statements are true.
14.3A Converse, Inverse, and Contrapositive Principles
PRINCIPLE 1: A statement is considered false if one false instance of the statement exists.
PRINCIPLE 2: The converse of a definition is true.
Thus, the definition “a quadrilateral is a four-sided polygon” and its converse “a four-sided polygon is a quadrilateral” are both true.
PRINCIPLE 3: The converse of a true statement other than a definition is not necessarily true.
The statement “vertical angles are congruent angles” is true, but its converse, “congruent angles are vertical angles” is not necessarily true.
PRINCIPLE 4: The inverse of a true statement is not necessarily true.
The statement “a square is a quadrilateral” is true, but its inverse, “a non-square is not a quadrilateral,” is not necessarily true.
PRINCIPLE 5: The contrapositive of a true statement is true, and the contrapositive of a false statement is false.
The statement “a triangle is a square” is false, and its contrapositive, “a non-square is not a triangle,” is also false.
The statement “right angles are congruent angles” is true, and its contrapositive, “angles that are not congruent are not right angles,” is also true.
14.3B Logically Equivalent Statements
Logically equivalent statements are pairs of related statements that are either both true or both false. Thus according to Principle 5, a statement and its contrapositive are logically equivalent statements. Also, the converse and inverse of a statement are logically equivalent, since each is the contrapositive of the other.
The relationships among a statement and its inverse, converse, and contrapositive are summed up in the rectangle of logical equivalency in Fig. 14-3:
Fig. 14-3
1. Logically equivalent statements are at diagonally opposite vertices. Thus, the logically equivalent pairs of statements are (a) a statement and its contrapositive, and (b) the inverse and converse of the same statement.
2. Statements that are not logically equivalent are at adjacent vertices. Thus, pairs of statements that are not logically equivalent are (a) a statement and its inverse, (b) a statement and its converse, (c) the converse and contrapositive of the same statement, and (d) the inverse and contrapositive of the same statement.
SOLVED PROBLEMS
14.3 Converse of a statement
State the converse of each of the following statements, and indicate whether or not it is true.
(a) Supplementary angles are two angles the sum of whose measures is 1808.
(b) A square is a parallelogram with a right angle.
(c) A regular polygon is an equilateral and equiangular polygon.
Solutions
(a) Two angles the sum of whose measures is 1808 are supplementary. (True)
(b) A parallelogram with a right angle is a square. (False)
(c) An equilateral and equiangular polygon is a regular polygon. (True)
14.4 Negative of a statement
State the negative of (a) 
is the complement of 
; (d) “the point does not lie on the line.”
Solutions
(a) a ≠ b
(b) 
(c) 
is not the complement of .
(d) The point lies on the line.
14.5 Inverse of a statement
State the inverse of each of the following statements, and indicate whether or not it is true.
(a) A person born in the United States is a citizen of the United States.
(b) A sculptor is a talented person.
(c) A triangle is a polygon.
(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.)
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.
Solutions
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.
14.4 Partial Converse and Partial Inverse of a Theorem
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.
Fig. 14-4
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, (19) and (39). 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).
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
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 1: 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 2: 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.
PRINCIPLE 3: If a statement is false and its converse is true, then the conditions in the hypothesis are necessary but not sufficient for its conclusion.
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.
PRINCIPLE 4: If a statement and its converse are both false, then the conditions in the hypothesis are neither necessary nor sufficient for its conclusion.
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.
These principles are summarized in the table that follows.
When the Conditions in the Hypothesis of a Statement are Necessary or Sufficient to Justify its Conclusion
SOLVED PROBLEMS
14.8 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:
(14.1)
(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:
(14.2)
(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:
(14.4)
(a) x + 2 = 4
(b) 3y ≠ 15
(c) She loves you.
(d) His mark was more than 65.
(e) Joe is heavier than Dick.
(f) a + b ≠ c
14.4. State the inverse of each of the following statements, and indicate whether or not it is true.
(14.5)
(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.