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(a) Fluffy is a domestic animal. (d) A rectangle has congruent diagonals. (e) ^ ABC is an obtuse triangle.
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2.2 Postulates (Assumptions)
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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.
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2.2A Algebraic Postulates
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Things equal to the same or equal things are equal to each other; if a a c. (Transitive Postulate)
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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)
5 and y x 3, we may substitute 5 for x and find y 5 3 8.
Thus if x
POSTULATE
3: 4: 5:
The whole equals the sum of its parts. (Partition Postulate) Any quantity equals itself. (Reflexive Postulate or Identity Postulate)
m j A, and AB AB.
Thus the total value of a dime, a nickel, and a penny is 16 cents.
POSTULATE
Thus x
x, m j A
POSTULATE
If equals are added to equals, the sums are equal; if a d. (Addition Postulate)
70 cents 20 cents 90 cents If and then x y x y 2x 12 8 20
b and c
d, then a
If and then
7 dimes 2 dimes 9 dimes
CHAPTER 2 Methods of Proof
POSTULATE
If equals are subtracted from equals, the differences are equal; if a a c b d. (Subtraction Postulate)
70 cents 20 cents 50 cents If and then x x y y 2y 12 8 4
b and c
d, then
If and then
7 dimes 2 dimes 5 dimes
POSTULATE
7: If equals are multiplied by equals, the products are equal; if a ac bd. (Multiplication Postulate)
b and c
d, then
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 where c, d 2 0. (Division Postulate)
b and c
d, then a/c
b/d,
Thus if the price of 1 lb of butter is 80 cents then, at the same rate, the price of 4 lb is 20 cents.
POSTULATE 9:
Like powers of equals are equal; if a
52 or x2 25.
b, then an
bn. (Powers Postulate)
Thus if x
5, then x2
POSTULATE 10:
Like roots of equals are equal; if a
3 227
b then 2a
Thus if y3
27, then y
2.2B Geometric Postulates
POSTULATE 11:
One and only one straight line can be drawn through any two points.
Thus, AB is the only line that can be drawn between A and B in Fig. 2-3.
Fig. 2-3
POSTULATE 12:
Fig. 2-4
Two lines can intersect in one and only one point.
Thus, only P is the point of intersection of AB and CD in Fig. 2-4.
POSTULATE 13:
The length of a segment is the shortest distance between two points.
Thus, AB is shorter than the curved or broken line segment between A and B in Fig. 2-5.
Fig. 2-5
POSTULATE 14:
Fig. 2-6
One and only one circle can be drawn with any given point as center and a given line segment as a radius. Any geometric figure can be moved without change in size or shape.
Thus, only circle A in Fig. 2-6 can be drawn with A as center and AB as a radius.
POSTULATE 15:
Thus, ^ I in Fig. 2-7 can be moved to a new position without changing its size or shape.
CHAPTER 2 Methods of Proof
Fig. 2-7
POSTULATE 16:
Fig. 2-8
A segment has one and only one midpoint. An angle has one and only one bisector.
Thus, only M is the midpoint of AB in Fig. 2-8.
POSTULATE 17:
Thus, only AD is the bisector of j 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 PC ' AB at point P on AB 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.
AB from point P outside AB in Fig. 2-11.
Thus, only PC can be drawn
Fig. 2-11
SOLVED PROBLEMS
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
Fig. 2-13
(a) Given: a (b) Given: a (c) Given: a
Solutions
10, b 25, a b, c b
10, c c
(d) Given: mj 1 (e) Given: mj 1 (f) Given: mj 3
40 , mj 2 mj 2, mj 3 mj 1, mj 2
40 , mj 3 mj 1 mj 3
(a) Since a, b, and c each equal 10, a (b) Since c and 25 each equal a, c 25.
CHAPTER 2 Methods of Proof
(c) Since a and c each equal b, a c.
(d) Since j 1, j 2, and j 3 each measures 40 , j 1 > j 2 > j 3. (e) Since j 2 and j 3 each > j 1, j 2 > j 3. (f) Since j 1 and j 2 each > j 3, j 1 > j 2.
Applying postulate 2 In each part, what conclusion follows when Postulate 2 is applied to the given data (a) Evaluate 2a (b) Find x if 3x (c) Given: mj 1 2b when a 4y 4 and b 5. 180 , j 1 > j A, and j 2 > j C in Fig. 2-14. 8.
35 and y mj 2
mj B
Fig. 2-14
Solutions
(a) Substitute 4 for a and 8 for b: 2a 2b 2(4) 2(8) 8 16 24 Ans. (b) Substitute 5 for y: 3x 4y 35 3x 4(5) 35 3x 20 35 3x 15, x 5 Ans. (c) Substitute j A for j 1 and j C for j 2: mj 1 mj B mj 2 180 mj A mj B mj C 180 Ans.
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
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