Deductive reasoning enables us to derive true or acceptably true conclusions from statements which are true or accepted as true. It consists of three steps as follows:
1. Making a general statement referring to a whole set or class of things, such as the class of dogs: All dogs are quadrupeds (have four feet).
2. Making a particular statement about one or some of the members of the set or class referred to in the general statement: All greyhounds are dogs.
3. Making a deduction that follows logically when the general statement is applied to the particular statement: All greyhounds are quadrupeds.
Deductive reasoning is called syllogistic reasoning because the three statements together constitute a syllogism. In a syllogism the general statement is called the major premise, the particular statement is the minor premise, and the deduction is the conclusion. Thus, in the above syllogism:
1. The major premise is: All dogs are quadrupeds.
2. The minor premise is: All greyhounds are dogs.
3. The conclusion is: All greyhounds are quadrupeds.
Using a circle, as in Fig. 2-1, to represent each set or class will help you understand the relationships involved in deductive reasoning.
Fig. 2-1
1. Since the major premise or general statement states that all dogs are quadrupeds, the circle representing dogs must be inside that for quadrupeds.
2. Since the minor premise or particular statement states that all greyhounds are dogs, the circle representing greyhounds must be inside that for dogs.
3. The conclusion is obvious. Since the circle of greyhounds must be inside the circle of quadrupeds, the only possible conclusion is that greyhounds are quadrupeds.
Observation cannot serve as proof. Eyesight, as in the case of a color-blind person, may be defective. Appearances may be misleading. Thus, in each part of Fig. 2-2, AB does not seem to equal CD although it actually does.
Fig. 2-2
Measurement cannot serve as proof. Measurement applies only to the limited number of cases involved. The conclusion it provides is not exact but approximate, depending on the precision of the measuring instrument and the care of the observer. In measurement, allowance should be made for possible error equal to half the smallest unit of measurement used. Thus if an angle is measured to the nearest degree, an allowance of half a degree of error should be made.
Experiment cannot serve as proof. Its conclusions are only probable ones. The degree of probability depends on the particular situations or instances examined in the process of experimentation. Thus, it is probable that a pair of dice are loaded if ten successive 7s are rolled with the pair, and the probability is much greater if twenty successive 7s are rolled; however, neither probability is a certainty.
2.1 Using circles to determine group relationships
In (a) to (e) each letter, such as A, B, and R, represents a set or group. Complete each statement. Show how circles may be used to represent the sets or groups.
(a) If A is B and B is C, then 
.
(b) If A is B and B is E and E is R, then .
(c) If X is Y and , then X is M.
(d) If C is D and E is C, then .
(e) If squares (S) are rectangles (R) and rectangles are parallelograms (P), then .
Solutions
(a) A is C
(b) A is R
(c) Y is M
(d) E is D
(e) Squares are parallelograms
Write the statement needed to complete each syllogism:
Solutions
(a) Fluffy is a domestic animal.
(b) Jan is a person.
(c) ∠ c ≐ ∠ d.
(d) A rectangle has congruent diagonals.
(e) ΔABC is an obtuse triangle.
The entire structure of proof in geometry rests upon, or begins with, some unproved general statements called postulates. These are statements which we must willingly assume or accept as true so as to be able to deduce other statements.
POSTULATE 1: Things equal to the same or equal things are equal to each other; if a = b and c = b, then a = c. (Transitive Postulate)
Thus the total value of a dime is equal to the value of two nickels because each is equal to the value of ten pennies.
POSTULATE 2: A quantity may be substituted for its equal in any expression or equation. (Substitution Postulate)
Thus if x = 5 and y = x + 3, we may substitute 5 for x and find y = 5 + 3 = 8.
POSTULATE 3: The whole equals the sum of its parts. (Partition Postulate)
Thus the total value of a dime, a nickel, and a penny is 16 cents.
POSTULATE 4: Any quantity equals itself. (Reflexive Postulate or Identity Postulate)
Thus x = x, m ∠A = m ∠A, and AB = AB.
POSTULATE 5: If equals are added to equals, the sums are equal; if a = b and c = d, then a + c = b + d. (Addition Postulate)
POSTULATE 6: If equals are subtracted from equals, the differences are equal; if a = b and c = d, then a–c = b–d. (Subtraction Postulate)
POSTULATE 7: If equals are multiplied by equals, the products are equal; if a = b and c = d, then ac = bd. (Multiplication Postulate)
Thus if the price of one book is $2, the price of three books is $6.
Special multiplication axiom: Doubles of equals are equal.
POSTULATE 8: If equals are divided by equals, the quotients are equal; if a = b and c = d, then a/c = b/d, where c, d ≠ 0. (Division Postulate)
Thus if the price of 1 lb of butter is 80 cents then, at the same rate, the price of 
lb is 20 cents.
POSTULATE 9: Like powers of equals are equal; if a = b, then an = bn. (Powers Postulate)
Thus if x = 5, then x2 = 52 or x2 = 25.
POSTULATE 10: Like roots of equals are equal; if a = b then 
Thus if y3 = 27, then 
POSTULATE 11: One and only one straight line can be drawn through any two points.
Thus, 
is the only line that can be drawn between A and B in Fig. 2-3.
Fig. 2-3
POSTULATE 12: Two lines can intersect in one and only one point.
Thus, only P is the point of intersection of and 
in Fig. 2-4.
Fig. 2-4
POSTULATE 13: The length of a segment is the shortest distance between two points.
Thus, 
is shorter than the curved or broken line segment between A and B in Fig. 2-5.
Fig. 2-5
POSTULATE 14: One and only one circle can be drawn with any given point as center and a given line segment as a radius.
Thus, only circle A in Fig. 2-6 can be drawn with A as center and as a radius.
Fig. 2-6
POSTULATE 15: Any geometric figure can be moved without change in size or shape.
Thus, ΔI in Fig. 2-7 can be moved to a new position without changing its size or shape.
Fig. 2-7
POSTULATE 16: A segment has one and only one midpoint.
Thus, only M is the midpoint of in Fig. 2-8.
Fig. 2-8
POSTULATE 17: An angle has one and only one bisector.
Thus, only 
is the bisector of ∠A in Fig. 2-9.
Fig. 2-9
POSTULATE 18: Through any point on a line, one and only one perpendicular can be drawn to the line.
Thus, only 
⊥ at point P on in Fig. 2-10.
Fig. 2-10
POSTULATE 19: Through any point outside a line, one and only one perpendicular can be drawn to the given line.
Thus, only 
can be drawn ± from point P outside in Fig. 2-11.
Fig. 2-11
2.3 Applying postulate 1
In each part, what conclusion follows when Postulate 1 is applied to the given data from Figs. 2-12 and 2-13?
Fig. 2-12
(a) Given: a = 10, b = 10, c = 10
(b) Given: a = 25, a = c
(c) Given: a = b, c = b
(d) Given: m∠ 1 = 40°, m∠ 2 = 40°, m∠ 3 = 40°
(e) Given: m∠ 1 = m∠ 2, m∠ 3 = m∠ 1
(f) Given: m∠ 3 = m∠, m∠ 2 = m∠ 3
Fig. 2-13
Solutions
(a) Since a, b, and c each equal 10, a = b = c.
(b) Since c and 25 each equal a, c = 25.
(c) Since a and c each equal b, a = c.
(d) Since ∠ 1, ∠ 2, and ∠ 3 each measures 40°, ∠ 1 ≐ ∠ 2 ≐ ∠ 3.
(e) Since ∠ 2 and ∠ 3 each ≐ ∠ 1, ∠ 2 ≐ ∠ 3.
(f) Since ∠ 1 and ∠ 2 each ≐ ∠ 3, ∠ 1 ≐ ∠ 2.
2.4 Applying postulate 2
In each part, what conclusion follows when Postulate 2 is applied to the given data?
(a) Evaluate 2a + 2b when a = 4 and b = 8.
(b) Find x if 3x + 4y = 35 and y = 5.
(c) Given: m∠ 1 + m∠B + m∠ 2 = 180°, ∠ 1 ≐ ∠A, and ∠ 2 ≐ ∠ C in Fig. 2-14.
Fig. 2-14
Solutions
2.5 Applying postulate 3
State the conclusions that follow when Postulate 3 is applied to the data in (a) Fig. 2.15(a) and (b) Fig. 2-15(b).
Fig. 2-15
Solutions
(a) AC = 3 + 5 = 8
BD = 5 + 6 = 11
AD = 3 + 5 + 6 = 14
(b) m∠AEC = 60° + 40° = 100°
m∠BED = 40° + 30° = 70°
m∠AED = 60° + 40° + 30° = 130°
2.6 Applying postulates 4, 5, and 6
In each part, state a conclusion that follows when Postulates 4, 5, and 6 are applied to the given data.
(a) Given: a = e (Fig. 2-16)
(b) Given: a = c, b = d (Fig. 2-16)
(c) Given: m∠BAC = m∠DAF (Fig. 2-17)
(d) Given: m∠BAC = m∠BCA, m∠ 1 = m∠ 3 (Fig. 2-17)
Fig. 2-16
Fig. 2-17
Solutions
2.7 Applying postulates 7 and 8
State the conclusions that follow when the multiplication and division axioms are applied to the data in (a) Fig. 2-18 and (b) Fig. 2-19.
Fig. 2-18
Fig. 2-19
Solutions
(a) If a = b, then 2a = 2b since doubles of equals are equal. Hence, AF = DB = AG = EC. Also, 3a = 3b, using the Multiplication Postulate. Hence, AB = AC.
(b) If m∠A = m∠ C, then 
m∠ C = m∠ C since halves of equals are equal. Hence, m∠ 1 = m∠ 2.
2.8 Applying postulates to statements
Complete each sentence and state the postulate that applies.
(a) If Harry and Alice are the same age today, then in 10 years .
(b) Since 32°F and 0°C both name the temperature at which water freezes, we know that .
(c) If Henry and John are the same weight now and each loses 20 lb, then .
(d) If two stocks of equal value both triple in value, then .
(e) If two ribbons of equal size are cut into five equal parts, then .
(f) If Joan and Agnes are the same height as Anne, then .
(g) If two air conditioners of the same price are each discounted 10 percent, then .
Solutions
(a) They will be the same age. (Add. Post.)
(b) 32°F = 0°C. (Trans. Post.)
(c) They will be the same weight. (Subt. Post.)
(d) They will have the same value. (Mult. Post.)
(e) Their parts will be of the same size. (Div. Post.)
(f) Joan and Agnes are of the same height. (Trans. Post.)
(g) They will have the same price. (Subt. Post.)
2.9 Applying geometric postulates
State the postulate needed to correct each diagram and accompanying statement in Fig. 2-20.
Fig. 2-20
Solutions
(a) Postulate 17.
(b) Postulate 18.
(c) Postulate 14.
(d) Postulate 13. (AC is less than the sum of AB and BC.)
A theorem is a statement, which, when proved, can be used to prove other statements or derive other results. Each of the following basic theorems requires the use of definitions and postulates for its proof.
Note: We shall use the term principle to include important geometric statements such as theorems, postulates, and definitions.
PRINCIPLE 1: All right angles are congruent.
Thus, ∠A ≐ ∠B in Fig. 2-21.
Fig. 2-21
PRINCIPLE 2: All straight angles are congruent.
Thus, ∠ C ≐ ∠ D in Fig. 2-22.
Fig. 2-22
PRINCIPLE 3: Complements of the same or of congruent angles are congruent.
This is a combination of the following two principles:
1. Complements of the same angle are congruent. Thus, ∠ a ≐ ∠B in Fig. 2.23 and each is the complement of ∠ x.
2. Complements of congruent angles are congruent. Thus, ∠ c ≐ ∠ d in Fig. 2-24 and their complements are the congruent 
x and y.
Fig. 2-23
Fig. 2-24
PRINCIPLE 4: Supplements of the same or of congruent angles are congruent.
This is a combination of the following two principles:
1. Supplements of the same angle are congruent. Thus, ∠ a ≐ ∠B in Fig. 2-25 and each is the supplement of ∠ x.
Fig. 2-25
2. Supplements of congruent angles are congruent. Thus, ∠ c ≐ ∠ d in Fig. 2-26 and their supplements are the congruent angles x and y.
Fig. 2-26
PRINCIPLE 5: Vertical angles are congruent.
Thus, in Fig. 2-27, ∠A ≐ ∠B; this follows from Principle 4, since ∠A and ∠B are supplements of the same angle, ∠ c.
Fig. 2-27
2.10 Applying basic theorems: principles 1 to 5
State the basic angle theorem needed to prove ∠A ≐ ∠B in each part of Fig. 2-28.
Fig. 2-28
Solutions
(a) Since and are straight lines, ∠A and ∠B are straight . Hence, ∠A ≐ ∠B. Ans. All straight angles are congruent.
(b) Since 
⊥ 
and 
⊥ 
, ∠A and ∠B are rt. . Hence, ∠A ≐ ∠B. Ans. All right angles are congruent.
(c) Since ⊥ , ∠B is a rt. ∠, making ∠B the complement of ∠ 1. Since ∠A is the complement of ∠ 1, ∠A ≐ ∠B. Ans. Complements of the same angle are congruent.
The statements “A heated metal expands” and “If a metal is heated, then it expands” are two forms of the same idea. The following table shows how each form may be divided into its two important parts, the hypothesis, which tells what is given, and the conclusion, which tells what is to be proved. Note that in the if-then form, the word then may be omitted.
The converse of a statement is formed by interchanging the hypothesis and conclusion. Hence to form the converse of an if-then statement, interchange the if and then clauses. In the case of the subject-predicate form, interchange the subject and the predicate.
Thus, the converse of “triangles are polygons” is “polygons are triangles.” Also, the converse of “if a metal is heated, then it expands” is “if a metal expands, then it is heated.” Note in each of these cases that the statement is true but its converse need not necessarily be true.
PRINCIPLE 1: The converse of a true statement is not necessarily true.
Thus, the statement “triangles are polygons” is true. Its converse need not be true.
PRINCIPLE 2: The converse of a definition is always true.
Thus, the converse of the definition “a triangle is a polygon of three sides” is “a polygon of three sides is a triangle.” Both the definition and its converse are true.
2.11 Determining the hypothesis and conclusion in subject-predicate form
Determine the hypothesis and conclusion of each statement.
2.12 Determining the hypothesis and conclusion in if-then form
Determine the hypothesis and conclusion of each statement.
2.13 Forming converses and determining their truth
State whether the given statement is true. Then form its converse and state whether this is necessarily true.
(a) A quadrilateral is a polygon.
(b) An obtuse angle has greater measure than a right angle.
(c) Florida is a state of the United States.
(d) If you are my pupil, then I am your teacher.
(e) An equilateral triangle is a triangle that has all congruent sides.
Solutions
(a) Statement is true. Its converse, “a polygon is a quadrilateral,” is not necessarily true; it might be a triangle.
(b) Statement is true. Its converse, “an angle with greater measure than a right angle is an obtuse angle,” is not necessarily true; it might be a straight angle.
(c) Statement is true. Its converse, “a state of the United States is Florida,” is not necessarily true; it might be any one of the other 49 states.
(d) Statement is true. Its converse, “if I am your teacher, then you are my pupil,” is also true.
(e) The statement, a definition, is true. Its converse, “a triangle that has all congruent sides is an equilateral triangle,” is also true.
Theorems should be proved using the following step-by-step procedure. The form of the proof is shown in the example that follows the procedure. Note that accepted symbols and abbreviations may be used.
1. Divide the theorem into its hypothesis (what is given) and its conclusion (what is to be proved). Underline the hypothesis with a single line, and the conclusion with a double line.
2. On one side, make a marked diagram. Markings on the diagram should include such helpful symbols as square corners for right angles, cross marks for equal parts, and question marks for parts to be proved equal.
3. On the other side, next to the diagram, state what is given and what is to be proved. The “Given” and “To Prove” must refer to the parts of the diagram.
4. Present a plan. Although not essential, a plan is very advisable. It should state the major methods of proof to be used.
5. On the left, present statements in successively numbered steps. The last statement must be the one to be proved. All the statements must refer to parts of the diagram.
6. On the right, next to the statements, provide a reason for each statement. Acceptable reasons in the proof of a theorem are given facts, definitions, postulates, assumed theorems, and previously proven theorems.
2.14 Proving a theorem
Use the proof procedure to prove that supplements of angles of equal measure have equal measure.
and 6:
2.1. Complete each statement. In (a) to (e), each letter, such as C, D, or R, represents a set or group.
(2.2)
(a) If A is B and B is H, then .
(b) If C is D and P is C, then .
(c) If and B is R, then B is S.
(d) If E is F, F is G, and G is K, then .
(e) If G is H, H is R, and , then A is R.
(f) If triangles are polygons and polygons are geometric figures, then .
(g) If a rectangle is a parallelogram and a parallelogram is a quadrilateral, then .
2.2. State the conclusions which follow when Postulate 1 is applied to the given data, which refer to Fig. 2-29.
(2.3)
Fig. 2-29
(a) a = 7, c = 7, f = 7
(b) b = 15, b = g
(c) f = h, h = a
(d) a = c; c = f, f = h
(e) b = d, d = g, g = e
2.3. State the conclusions which follow when Postulate 2 is applied in each case.
(2.4)
(a) Evaluate a2 + 3a when a = 10.
(b) Evaluate x2–4y when x = 4 and y = 3.
(c) Does b2–8 = 17 when b = 5?
(d) Find x if x + y = 20 and y = x + 3.
(e) Find y if x + y = 20 and y = 3x.
(f) Find x if 5x–2y = 24 and y = 3.
(g) Find x if x2 + 3y = 45 and y = 3.
2.4. State the conclusions that follow when Postulate 3 is applied to the data in Fig. 2-30(a) and (b).
(2.5)
Fig. 2-30
2.5. State a conclusion involving two new equals that follows when Postulate 4, 5, or 6 is applied to the given data. (2.6)
(a) Given: b = e (Fig. 2-31).
(b) Given: b = c, a = d (Fig. 2-31).
(c) Given: ∠ 4 ≐ ∠ 5 (Fig. 2-32).
(d) Given: ∠ 1 ≐ ∠ 3, ∠ 2 ≐ ∠ 4 (Fig. 2-32).
Fig. 2-31
Fig. 2-32
2.6. In Fig. 2-33 and are trisected (divided into 3 equal parts).
(2.7)
Fig. 2-33
(a) If ≐ , why is 
≐ 
?
(b) If 
≐ 
, why is 
≐ 
?
(c) If 
≐ 
, why is ≐ ?
(d) If 
≐ 
, why is ≐ ?
2.7. In Fig. 2-34 ∠BCD and ∠ADC are trisected.
(a) If m∠BCD = m∠ADC, why does m∠ FCD = m∠ FDC?
(b) If m∠ 1 = m∠ 2, why does m∠BCD = m∠ADC?
(c) If m∠ 1 = m∠ 2, why does m∠ADF = m∠BCF?
(d) If m∠ EDC = m∠ ECD, why does m∠ 1 = m∠ 2?
Fig. 2-34
2.8. Complete each statement, and name the postulate that applies.
(2.8)
(a) If Bill and Helen earn the same amount of money each hour and their rate of pay is increased by the same amount, then .
(b) In the past year, those stocks have tripled in value. If they had the same value last year, then .
(c) A week ago, there were two classes that had the same register. If the same number of pupils were dropped in each, then .
(d) Since 100°C and 212°F are the boiling temperatures of water, then .
(e) If two boards have the same length and each is cut into four equal parts, then .
(f) Since he has $2000 in Bank A, $3000 in Bank B and $5000 in Bank C, then .
(g) If three quarters and four nickels are compared with three quarters and two dimes, .
2.9. Answer each of the following by stating the basic angle theorem needed. The questions refer to Fig. 2-35.
(2.10)
(a) Why does m∠ 1 = m∠ 2?
(b) Why does m∠ DBC = m∠ ECB?
(c) If m∠ 3 = m∠ 4, why does m∠ 5 = m∠ 6?
(d) If 
⊥ 
and 
⊥ , why does m∠ 7 = m∠ 8?
(e) If ⊥ , ⊥ , and m∠ 11 = m∠ 12, why does m∠ 9 = m∠ 10?
Fig. 2-35
2.10. Determine the hypothesis and conclusion of each statement.
(2.11 and 2.12)
(a) Stars twinkle.
(b) Jet planes are the speediest.
(c) Water boils at 212°F.
(d) If it is the American flag, its colors are red, white, and blue.
(e) You cannot learn geometry if you fail to do homework in the subject.
(f) A batter goes to first base if the umpire calls a fourth ball.
(g) If A is B’s brother and C is B’s son, then A is C’s uncle.
(h) An angle bisector divides the angle into two equal parts.
(i) A segment is trisected if it is divided into three congruent parts.
(j) A pentagon has five sides and five angles.
(k) Some rectangles are squares.
(l) Angles do not become larger if their sides are made longer.
(m) Angles, if they are congruent and supplementary, are right angles.
(n) The figure cannot be a polygon if one of its sides is not a straight line segment.
2.11. State the converse of each of the following true statements. State whether the converse is necessarily true.
(2.13)
(a) Half a right angle is an acute angle.
(b) An obtuse triangle is a triangle having one obtuse angle.
(c) If the umpire called a third strike, then the batter is out.
(d) If I am taller than you, then you are shorter than I.
(e) If I am heavier than you, then our weights are unequal.
2.12. Prove each of the following.
(2.14)
(a) Straight angles are congruent.
(b) Complements of congruent angles are congruent.
(c) Vertical angles are congruent.
