Chapter 2. Reasoning and Proof
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1 Chapter 2 Reasoning and Proof
2 2.1 Use Inductive Reasoning Objective: Describe patterns and use deductive reasoning. Essential Question: How do you use inductive reasoning in mathematics? Common Core: CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO.11
3 Patterns and Inductive Reasoning If you were to see dark, towering clouds approaching, you might want to take cover. Your past experience tells you that a thunderstorm is likely to happen. When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning. 6/20/2011 LSowatsky 3
4 Patterns and Inductive Reasoning You can use inductive reasoning to find the next terms in a sequence. Find the next three terms of the sequence: 3, 6, 12, 24, 48, 96, X 2 X 2 X 2 X 2 X 2 Find the next three terms of the sequence: 7, 8, 11, 16, 23, /20/2011 LSowatsky 4
5 Patterns and Inductive Reasoning Draw the next figure in the pattern. 6/20/2011 LSowatsky 5
6 Patterns and Inductive Reasoning A conjecture is a conclusion that you reach based on inductive reasoning. In the following activity, you will make a conjecture about rectangles. 1) Draw several rectangles on your grid paper. 2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle. 3) Record your data in a table Diagonal 1 Diagonal 2 d 1 = 7.5 in. Rectangle inches 7.5 inches Make a conjecture about the diagonals of a rectangle d 2 = 7.5 in. 6/20/2011 LSowatsky 6
7 Example: Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left.
8 Patterns and Inductive Reasoning A is an educated guess. Sometimes it may be true, and other times it may be false. How do you know whether a conjecture is true or false? Try different examples to test the conjecture. If you find one example that does not follow the conjecture, then the conjecture is false. Such a false example is called a. Conjecture: The sum of two numbers is always greater than either number. Is the conjecture TRUE or FALSE? Counterexample: 6/20/2011 LSowatsky 8
9 Example: Find a counterexample to show the conjecture is false. Conjecture: The value of x 2 is always greater than x.
10 Homework: Regular: Exercises2.1 #1-30, 32, 34 Honors: Exercises2.1 #1-34, 37
11 2.2 Analyze Conditional Statements Objective: Write definitions as conditional statements. Essential Question: How do you rewrite a biconditional statement? Common Core: CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO.11
12 In mathematics, you will come across many. For Example: If a number is even, then it is divisible by two. If then statements join two statements based on a condition: A number is divisible by two only if the number is even. Therefore, if then statements are also called. 7/7/2011 LSowatsky 12
13 Conditional Statements Conditional statements have two parts. The part following if is the. The part following then is the. the number is divisible by two. If a number is even, then the number is divisible a number is even by two. Hypothesis: Conclusion: 7/7/2011 LSowatsky 13
14 Conditional Statements How do you determine whether a conditional statement is true or false? Conditional Statement True or False Why? If it is the 4 th of July (in the U.S.), then it is a holiday. If an animal lives in the water, then it is a fish. The statement is true because the conclusion follows from the hypothesis. You can show that the statement is false by giving one counterexample. Whales live in water, But whales are mammals, 7/7/2011 LSowatsky 14 not fish.
15 Conditional Statements There are different ways to express a conditional statement. The following statements all have the same meaning. If you are a member of Congress, then you are a U.S. citizen. All members of Congress are U.S. citizens. You are a U.S. citizen if you are a member of Congress. You write two other forms of this statement: If two lines are parallel, then they never intersect. Possible answers: All parallel lines never intersect. Lines never intersect if they are parallel. 7/7/2011 LSowatsky 15
16 Conditional Statements The converse of a conditional statement is formed by exchanging the hypothesis and the conclusion. Conditional: If a figure is a triangle, then it has three angles. Converse: If, then. NOTE: You often have to change the wording slightly so that the converse reads smoothly. Converse: If the figure has three angles, then it is a triangle. 7/7/2011 LSowatsky 16
17 Conditional Statements Write the converse of the following statements. State whether the converse is TRUE or FALSE. If FALSE, give a counterexample: If you are at least 16 years old, then you can get a driver s license. If, then. If today is Saturday, then there is no school. If, then. We don t have school on New Years day which may fall on a Monday. 7/7/2011 LSowatsky 17
18 Vocabulary: Negation Has opposite truth value of a given statement. Inverse Negates both hypothesis and conclusion of a conditional Contrapositive The inverse of the converse of a conditional. Equivalent Statements - Statements with the same truth value.
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20 Writing the negation of a statement: ABC is obtuse. Write the negation of the statement. (~p) Add not to the statement. ABC is not obtuse. A figure is not a square. (~p) A figure is a square.
21 Example: Write the negation of ABCD is not a convex polygon. The negation of a statement has the opposite truth value. The negation of is not in the original statement removes the word not. The negation of ABCD is not a convex polygon is ABCD is a convex polygon.
22 Writing the inverse of a conditional statement: Negate both the hypothesis and the conclusion. (~p ~q) Conditional: If a figure is a triangle, then it has exactly 180 in it. Inverse: If a figure is not a triangle, then it does not have a 180 in it. True or False
23 Writing the of a conditional statement: Switch the hypothesis and conclusion & negate both of them. (~q ~p) Conditional: If a figure is a triangle, then it has exactly 180 in it. Contrapositve: If a figure does not have a 180 in it, then it is not a triangle. True or False
24 If the conditional is true, then the contrapositive is also true. If the conditional is false, then the contrapositive is also false. They are known as.
25 Example 1: Conditional: If an is a straight, then its measure is 180. (T or F) p q Converse: If an has a measure of 180, then it is a straight. (T or F) Inverse: If an is not a straight, then its measure is not 180. (T or F) Contrapositive: q p ~p ~q If an does not have a measure of 180, then it is not a straight. (T or F) ~q ~p
26 Example: Identify the 2 statements that contradict each other. 1) ΔABC is Acute. 2) ΔABC is scalene. 3) ΔABC is equiangular.
27 When a conditional and its converse are true, you can combine them as a true. Use the phrase if and only if (iff) 7/7/2011 LSowatsky 27
28 Writing a Biconditional: Consider this true conditional statement: If x = 5, then x + 15 = 20. Write the converse: If converse is true, write as a biconditional: 7/7/2011 LSowatsky 28
29 Separating a biconditional into Parts: Consider the statement: Lines are skew iff they are noncoplanar. Write the two statements that form this biconditional. 7/7/2011 LSowatsky 29
30 Good Definitions: A good definition is a statement that can help identify or classify an object. Use clearly understood terms commonly understood, already defined Precise don t be vague Reversible can write as a true biconditional 7/7/2011 LSowatsky 30
31 Show that this definition of triangle is reversible: Definition: A triangle is a polygon with exactly three sides. Conditional: If a polygon is a triangle, it has exactly three sides. Converse: If a polygon has exactly three sides, it is a triangle. Biconditional: A polygon is a triangle iff it has exactly three sides. 7/7/2011 LSowatsky 31
32 Question: Is this a good definition? Explain. An apple is a fruit that contains seeds. 7/7/2011 LSowatsky 32
33 Homework: Regular: Exercises 2.2 #1-36 Honors: Exercises 2.2 #1-39
34 Common Core: CC.9-12.G.CO.1 CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO Apply Deductive Reasoning Objective: Use deductive reasoning to form a logical argument. Essential Question: How do you construct a logical argument?
35 (aka logical reasoning) is the process of reasoning logically from given statement to form a conclusion. Examples: Physican diagnosing illness Mechanic finding why a car won t start Troubleshooting a computer 7/7/2011 LSowatsky 35
36 Law of Detachment 7/7/2011 LSowatsky 36
37 Example: A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? 7/7/2011 LSowatsky 37
38 Example: For the given statements, what can you conclude? Given: If A is acute, m A < 90. A is acute. 7/7/2011 LSowatsky 38
39 Law of Syllogism 7/7/2011 LSowatsky 39
40 Example: Use the Law of Syllogism to draw a conclusion from the following true statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. 7/7/2011 LSowatsky 40
41 Question: Use the Law of Syllogism to draw a conclusion: If it rains, then Nestle stays inside. If Nestle stays inside, then she does not get wet. Conclusion: 7/7/2011 LSowatsky 41
42 Homework: Regular: Exercises 2.3 #1 13, 16-18, Honors: Exercises 2.3 #1 29
43 2.4 Use Postulates and Diagrams Objective: Use postulates for points, lines, and planes Essential Question: How can you identify postulates illustrated by a diagram? Common Core: CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO.11
44 A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to evey line in the plane that intersects it at that point.
45 Chapter 1 Postulates: (1) Ruler Postulate (2) Segment Addition Postulate, if A B and C are points on a segment and B is between A and C, then AB + BC = AC. (3) Protractor Postulate. (4) Angle Addition Postulate: If D is a point on the interior of ABC, then m ABD + m DBC = m ABC. We can now introduce seven more which we can illustrate with sketches.
46 Postulate 5 Postulate 6 Postulate 7 Postulate 8 Postulate 9 Through any two points there exists exactly one line. A line contains at least two points. If two lines intersect, there intersection is exactly one point. Through any three noncollinear points there exists exactly one plan. A plane contains at least three collinear points. Postulate 10 Postulate 11 If two points lie in a plane, the line containing them lies in the plane. If two planes intersect, there intersection is a line. These are in section 2.4 of your textbook!
47 Example: Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. a. b. SOLUTION a. Postulate 7: If two lines intersect, then their intersection is exactly one point. b. Postulate 11: If two planes intersect, then their intersection is a line.
48 Example 2: Identify postulates from a diagram Use the diagram to write examples of Postulates 9 and 10. Postulate 9: Plane P contains at least three noncollinear points, A, B, and C. Postulate 10: Point A and point B lie in plane P, so line n containing A and B also lies in plane P.
49 1. GUIDED PRACTICE Extension for Example 1 and 2 Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane P and plane Q is a line? ANSWER Postulate 11
50 Extension for Example 1 and 2 Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. ANSWER Postulate 5 : Postulate 6 : Postulate 7 : Line n passes through points A and B Line n contains A and B Line m and n intersect at point A
51 Summary
52 Use given information to sketch a diagram Example 3: Sketch a diagram showing TV intersecting PQ at point W, so that TW WV. SOLUTION STEP 1 Draw TV and label points T and V. STEP 2 Draw point W at the midpoint of TV Mark the congruent segments. STEP 3 Draw PQ through W.
53 Interpret a diagram in three dimensions Ex. 4: Which of the following statements cannot be assumed from the diagram? A, B, and F are collinear. E, B, and D are collinear. AB CD plane S plane T AF intersects BC at point B. SOLUTION No drawn line connects E, B, and D, so you cannot assume they are collinear. With no right angle marked, you cannot assume CD plane T.
54 Extension for Examples 3 and 4 In Questions 3 and 4, refer back to Example If the given information stated PW and QW are congruent, how would you indicate that in the diagram? ANSWER
55 xtension for Examples 3 and 4 4. Name a pair of supplementary angles in the diagram. Explain. ANSWER In the diagram the angle TWP and VWP form a linear pair So, angle TWP, VWP are a pair of supplementary angles.
56 Extension for Examples 3 and 4 5. In the diagram for Example 4, can you assume plane S intersects plane T at BC? ANSWER Yes
57 Extension for Examples 3 and 4 6. Explain how you know that AB BC in Example 4. ANSWER It is given that line AB is perpendicular to plane S, therefore line AB is perpendicular to every line in the plane that intersect it at point B. So, AB BC
58 Homework: Regular: Exercises 2.4 #1-34 Honors: Exercises 2.4 #1 38, 45
59 2.5 Reason Using Properties from Algebra Objective: Use algebraic properties in logical arguments Essential Question: How do you solve an equation? Common Core: CC.9-12.A.REI.1 CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO.11
60 Addition Property Subtraction Property Multiplication Property Division Property Reflexive Property Symmetric Property Properties of Equality If a = b, then a + c = b + c If a = b, then a c = b - c If a = b, then a c = b c a = a If a = b, then b = a Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then b can replace a in any expression. 7/7/2011 LSowatsky 60
61 Distributive Property a(b + c) = ab + ac 7/7/2011 LSowatsky 61
62 Example: Fill in each missing reason. Given: LM bisects KLN K (2x+40) o L 4x o M N LM bisects KLN Given m MLN m KLM Def. of angle bisector 4x = 2x x = 40 x = 20 Substitution Prop. Subtraction Prop of Eq Division Prop of Eq 7/7/2011 LSowatsky 62
63 Example: Suppose points A, B, and C are collinear with point B between points A and C. Solve for x if AB = 4 + 2x, BC = 15 x, and AC = 21. Justify each step. 7/7/2011 LSowatsky 63
64 Properties of Congruence Reflexive Property AB A AB A Symmetric Property If If AB A CD, then CD B, then B AB A Transitive Property If AB If A CD and CD B and B EF, then AB EF C, then A C 7/7/2011 LSowatsky 64
65 Example: Name the property that justifies each statement. A) If x = y and y + 4 = 3x, then x + 4 = 3x B) If P Q, Q R,and R S, then P S 7/7/2011 LSowatsky 65
66 Reflexive Property of Equality Real numbers For any real number a, a a. Segment Length For any segment AB, AB AB. Angle Measure For any angle A, m A m A. 66
67 Symmetric Property of Equality Real numbers For any real numbers a and b, if a b, then b a. Segment Length For any segments AB and CD, if AB CD, then CD AB. Angle Measure For any angles A and B, if m A m B, then m B m A. 67
68 Transitive Property of Equality Real numbers For any real numbers a, b, and c, if a b and b c, then a c. Segment Length For any segments AB, CD, and EF, if AB CD and CD EF, then AB EF. Angle Measure For any angles A, B, and C, if m A m B and m B m C, then m A m C. 68
69 Show that CF AD. Equation AB BC EF DE AC AB BC Reason Given Given Segment Addition Postulate DF DE EF Segment Addition Postulate DF BC AB Substitution Property of Equality 69
70 DF AC DF CD AC CD Tran sitive Property of Equality Addition Property of Equality CF AD Substitution Property of Equality 70
71 Homework: Regular: Exercises 2.5 #1 29, Honors: Exercises 2.5 #1 33
72 2.6 Prove Statements about Segments and Angles Objective: Write proofs proving geometric theorems Essential Question: How do you write a geometric proof? Common Core: CC.9-12.G.CO.9 CC.9-12.G.CO.10 CC.9-12.G.CO.11
73 Vocabulary: Proof: Two-column proof: A two-column proof has numbered statements and corresponding reasons that show an argument in logical order. Theorem:
74 Example 1:Write a two-column proof Use the diagram to prove m 1 m 4. Given m 2 m 3, m AXD m AXC Prove m 1 m 4 Note: Writing a two-column proof is a formal way of organizing your reasons to show a statement is true.
75 Example 1:Write a two-column proof Statements Reasons 1. m AXC m AXD 1. Given 2. m AXD m 1 m 2 2. Angle Addition Postulate 3. m AXC m 3 m 4 3. Angle Addition Postult ae 4. m 1 m 2 m 3 m 4 4. Substitution Property of Equality
76 Example 1:Write a two-column proof Statements Reasons 5. m 2 m 3 5. Given 6. m 1 m 3 m 3 m 4 6. Substitution Property of Equality 7. m 1 m 4 7. Subtraction Property of Equality
77 Theorem 2.1:Congruence of Segments Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment AB, AB AB. Symmetric If AB CD, then CD AB. Transitive If AB CD and CD EF, then AB EF.
78 Theorem 2.2:Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. Reflexive For any angle A, A A. Symmetric If A B, then B A. Transitive If A B and B C, then A C.
79 Example 2:Name the property shown Name the property illustrated by the statement. If 5 3, then 3 5.
80 Checkpoint Three steps of a proof are shown. Give the reasons for the last two steps. Given BC AB Prove AC AB AB Statements Reasons 1. BC AB 1. Given 2. AC AB BC 2. Segment Addition Postulate 3. AC AB AB 3. Substitution Property of Equality
81 Example 3:Use properties of equality If you know that BD bisects ABC, prove that m ABC is two times m 1. Given BD bisects ABC Prove m ABC 2 m 1 Note: Before writing a proof, organize your reasoning by copyin or drawing a diagram for the situation described. Then Identify the GIVEN and PROVE statements.
82 Example 3:Use properties of equality Statements Reasons 1. BD bisects ABC 1. Given Definition of angle bisector 3. m 1 m 2 3. Definition of congruent angles 4. m 1 m 2 m ABC 4. Angle Addition Postulate
83 Example 3:Use properties of equality Statements Reasons 5. m 1 m 1 m ABC 5. Substitution Property of Equality 6. 2 m 1 m ABC 6. Distributive Property
84 Concept Summary: Writing a Two-column Proof Proof of the Symmetric Property of Segment Congruence Given AB CD Prove CD AB Note: The statements section is based on facts that you know or conclusions you determine from deductive reasoning.
85 Concept Summary:Writing a Twocolumn Proof Statements Reasons 1. AB CD 1. Given 2. AB CD 2. Definition of congruent segments 3. CD AB 3. Symmetric Property of Equality 4. CD AB 4. Definition of congruent segments The number of statements will vary Remember to give a reason for the last statement Definitions, postulates, or proven theorems that allow you to state the corresponding statement
86 Selfeducation is, I firmly believe, the only kind of education there is. Isaac Asimov
87 Homework: Regular: Exercises 2.6 #1 18, Honors: Exercises 2.6 #1-29
88 2.7 Prove Angle Pair Relationships Objective: Use properties of special pairs of angles Essential Question: What is the relationship between vertical angles, between two angles that are supplementary to the same angle, and between two angles that are complementary to the same angle? Common Core: CC.9-12.G.CO.9
89 Congruence Recall that congruent segments have the same. also have the same measure. 7/7/2011 LSowatsky 89
90 Definition of Congruent Angles Two angles are congruent iff, they have the same. 50 B V 50 B V iff m B = m V 7/7/2011 LSowatsky 90
91 To show that 1 is congruent to 2, we use. arcs 1 2 To show that there is a second set of congruent angles, X and Z, we use double arcs. X This arc notation states that: Z X Z m X = m Z 7/7/2011 LSowatsky 91
92 Right Angles Congruence Theorem: All right angles are.
93 Example 1:Use right angle congruence Write a proof. Given JK KL, ML KL Prove K L Note:The given information in Example 1 is about perpendicular lines. You must then use deductive reasoning to show that the angles are right angles.
94 Example 1:Use right angle congruence Statements Reasons 1. JK KL, ML KL 1. Given 2. K and L are 2. Definition of righ t angles perpendicular lines 3. K L 3. Right Angles Congruence Theorem
95 When two lines intersect, angles are formed. There are two pair of nonadjacent angles. These pairs are called /7/2011 LSowatsky 95
96 Definition of Vertical Angles Two angles are vertical iff there are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: and 3 2 and 4 7/7/2011 LSowatsky 96
97 Definition of Adjacent Angles R Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common 2 1 J 1 and 2 are adjacent with the same vertex R and common side RM N 7/7/2011 LSowatsky 97
98 Two angles are complementary if and only if (iff) the sum of their degree measure is 90. Definition of Complementary Angles B A 30 C D 60 E F m ABC + m DEF = = 90 7/7/2011 LSowatsky 98
99 If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. Definition of Supplementary C D Angles A 50 B E 130 F m ABC + m DEF = = 180 7/7/2011 LSowatsky 99
100 Definition of Linear Pairs Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays A D B and 2 are a linear pair. BA and BD form AD /7/2011 LSowatsky 100
101 1) If m 1 = 4x + 3 and the m 3 = 2x + 11, then find the m 3 1 x = 4; 3 = 19 2) If m 2 = x + 9 and the m 3 = 2x + 3, then find the m 4 x = 56; 4 = 65 3) If m 2 = 6x - 1 and the m 4 = 4x + 17, then find the m 3 x = 9; 3 = 127 4) If m 1 = 9x - 7 and the m 3 = 6x + 23, then find the m 4 x = 10; 4 = 97 7/7/
102 Vertical angles are congruent. Vertical Angle m n 1 3 Theorem /7/2011 LSowatsky 102
103 Find the value of x in the figure: The angles are vertical angles. 130 x So, the value of x is /7/2011 LSowatsky 103
104 Find the value of x in the figure: The angles are vertical angles. (x 10) 125 (x 10) = 125. x 10 = 125. x = /7/2011 LSowatsky 104
105 Suppose two angles are congruent. What do you think is true about their complements? x = y = 90 x is the complement of 1 x = 90-1 y = 90-2 Because 1 2, a substitution is made. x = 90-1 y = 90-1 x = y y is the complement of 2 If two angles are congruent, their complements are congruent. x 7/7/2011 LSowatsky 105 y
106 If two angles are congruent, then their complements are. congruent Theorem The measure of angles complementary to A and B is 30. A B A B 7/7/2011 LSowatsky 106
107 If two angles are complementary to the same angle, then they are. Theorem Theorem 3 is complementary to 4 5 is complementary to If two angles are supplementary to the same angle, then they are. 1 is supplementary to 2 3 is supplementary to /7/2011 LSowatsky 107 2
108 Suppose A B and m A = 52. Find the measure of an angle that is supplementary to B. B A 52 1 m m m m m B + 1 = = 180 m B 1 = = 128 7/7/2011 LSowatsky 108
109 If 1 is complementary to 3, 2 is complementary to 3, and m 3 = 25, What are m 1 and m 2? m 1 + m 3 = 90 m 1 = 90 - m 3 Definition of complementary angles. Subtract m 3 from both sides. m 1 = Substitute 25 in for m 3. m 1 = 65 You solve for m 2 Simplify the right side. m 2 + m 3 = 90 Definition of complementary angles. m 2 = 90 - m 3 Subtract m 3 from both sides. m 2 = Substitute 25 in for m 3. m 2 = 65 Simplify the right side. 7/7/2011 LSowatsky 109
110 1) If m 1 = 2x + 3 and the m 3 = 3x - 14, then find the m 3 x = 17; 3 = ) If m ABD = 4x + 5 and the m DBC = 2x + 1, then find the E m EBC x = 29; EBC = 121 3) If m 1 = 4x - 13 and the m 3 = 2x + 19, then find the m 4 x = 16; 4 = 39 4) If m EBG = 7x + 11 and the m EBH = 2x + 7, then find the m 1 x = 18; 1 = 43 G D A B C 7/7/2011 LSowatsky H 2
111 Suppose you draw two angles that are congruent and supplementary. What is true about the angles? 7/7/2011 LSowatsky 111
112 Theorem 2-5 If two angles are congruent and supplementary then each is a. 1 is supplementary to 2 1 and 2 = All right angles are. Theorem 2-4 C A A B C 7/7/2011 LSowatsky 112 B
113 If 1 is supplementary to 4, 3 is supplementary to 4, and m 1 = 64, what are m 3 and m 4? 1 3 They are vertical angles. m 1 = m 3 A 2 E B C 3 is supplementary to 4 Given m 3 + m 4 = 180 Definition of supplementary m 4 = 180 m 4 = D 7/7/2011 LSowatsky 113
114 Homework: Regular: Exercises 2.7 # 1 29, Honors: Exercises 2.7 # 1 29, 36-44
115 Chapter 2 Test
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