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2.3. State the conclusions which follow when Postulate 2 is applied in each case. (a) Evaluate a
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CHAPTER 2 Methods of Proof
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 (b) Given: b e (Fig. 2-31). c, a d (Fig. 2-31).
(c) Given: j 4 > j 5 (Fig. 2-32). (d) Given: j 1 > j 3, j 2 > j 4 (Fig. 2-32).
Fig. 2-31
Fig. 2-32
2.6. In Fig. 2-33 AD and BC are trisected (divided into 3 equal parts).
(2.7)
Fig. 2-33
(a) If AD > BC, why is AE > BF (b) If EG > FH, why is AG > BH (c) If GD > HC, why is AD > BC (d) If ED > FC, why is EG > FH 2.7. In Fig. 2-34 j BCD and j ADC are trisected. (a) If mj BCD (b) If mj 1 mj ADC, why does mj FCD mj FDC
mj 2, why does mj BCD
mj ADC
CHAPTER 2 Methods of Proof
(c) If mj 1
mj 2, why does mj ADF
mj BCF mj 2
(d) If mj EDC
mj ECD, why does mj 1
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. (a) Why does mj 1 mj 2 mj ECB mj 6 mj 8 mj 10
(2.10)
(b) Why does mj DBC (c) If mj 3
S S S
mj 4, why does mj 5
(d) If AF ' DE and GC ' DE, why does mj 7 (e) If AF ' DE, GC ' DE, and mj 11
mj 12, why does mj 9
Fig. 2-35
2.10. Determine the hypothesis and conclusion of each statement. (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.
(2.11 and 2.12)
CHAPTER 2 Methods of Proof
(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. (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. (a) Straight angles are congruent. (b) Complements of congruent angles are congruent. (c) Vertical angles are congruent.
(2.13)
(2.14)
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