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(a) AC BD AD 3 5 3 5 6 5 8 11 6 14 (b) mj AEC mj BED mj AED 60 40 60 40 30 40 100 70 30 130
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CHAPTER 2 Methods of Proof
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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 (b) Given: a e (Fig. 2-16) c, b d (Fig. 2-16) mj DAE (Fig. 2-17) mj BCA, mj 1 mj 3 (Fig. 2-17)
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(c) Given: mj BAC (d) Given: mj BAC
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Fig. 2-16
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Fig. 2-17
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(a) a b a b CD a b a b CD e b b e EF c d c d AB Given Identity Add. Post . Subst. Given Given Add. Post. Subst. m/DAE m/1 m/DAE m/ 5 m/BCA m/3 m/BCA m/4 Given Given Subt. Post. Subst. Given Given Subt. Post. Subst.
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m/BAC m/ 1 m/BAC m/1 m/2 m/BAC m/ 1 m/BAC m/1 m/2
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m/ 3
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 using the Multiplication Postulate. Hence, AB AC. (b) If mj A mj C, then 1m/A 2
1 2 m/ C
EC. Also, 3a mj 2.
since halves of equals are equal. Hence, mj 1
CHAPTER 2 Methods of Proof
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.)
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.)
2.3 Basic Angle Theorems
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.
CHAPTER 2 Methods of Proof
All right angles are congruent.
PRINCIPLE 1:
Thus, j A > j B in Fig. 2-21.
Fig. 2-21
PRINCIPLE 2:
All straight angles are congruent.
Thus, j C > j 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, j a > j b in Fig. 2.23 and each is the complement of j x. 2. Complements of congruent angles are congruent. Thus, j c > j d in Fig. 2-24 and their complements are the cons gruent j x and y.
Fig. 2-23
PRINCIPLE 4:
Fig. 2-24
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, j a > j b in Fig. 2-25 and each is the supplement of j x. 2. Supplements of congruent angles are congruent. Thus, j c > j d in Fig. 2-26 and their supplements are the congruent angles x and y.
Fig. 2-25
PRINCIPLE 5:
Fig. 2-26
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