2 Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the properties and theorems in this chapter. D. Correctly interpret the information contained in a conditional. E. Understand and properly use algebraic properties F. Properly prove theorems relating to lines and angles.
4 Objectives A. Identify and properly use conditional statements. B. Identify the hypothesis and conclusion of conditionals. C. Identify and state the converse of a conditional. D. Provide correct counterexamples to prove statements false. E. Identify and use biconditional statements.
5 Conditional Statements We use conditional statements in our everyday language, as well as in our mathematical language. The common form of a conditional statement, or conditional, is: If hypothesis, then conclusion.
6 Hypothesis Conditional Statements According to Merriam-Webster dictionary a tentative assumption made in order to draw out and test its logical or empirical consequences More simply a set of pre-conditions from which we attempt to reach a conclusion. Or the information which must be known in order to apply the conclusion to a problem. In geometry, it is common for the hypothesis to describe a diagram, or to be part of a diagram. Conclusion Information which can be added to a problem when the criteria of the hypothesis have been met.
7 Conditional Statements Conditional statements can be TRUE or FALSE: If they are considered TRUE, they must be TRUE in ALL cases. If there is a SINGLE case where the statement is false, then the ENTIRE conditional is considered FALSE! Examples: If I do not eat, then I will eventually starve. If I live in Bexley, then I live in Ohio. If I add 3 to 4, then I will have 7. If you cheat on homework, then you won t do well in this class. If you want the freedom of an adult, then you must accept adult responsibilities.
8 Other Forms of Conditional Statements Thanks to the English language, you have several other ways of expressing a conditional statement: IF hypothesis, THEN conclusion. hypothesis IMPLIES conclusion. hypothesis ONLY IF conclusion. conclusion IF hypothesis.
9 Equivalent Conditionals: Examples If you live in Ohio, then you live in the United States. You live in Ohio implies that you live in the United States. You live in Ohio only if you live in the United States. You live in the United States if you live in Ohio. 4
10 Say it ain t so! A major outcome of your work in this class will be your ability to prove or disprove conditionals. Remember, a conditional is always true or it is false, there is no sometimes this, sometimes that. To prove a conditional or theorem to be true usually takes a number of steps. The proof MUST show that the statement to be true for ALL cases. To prove something is false we need ONLY ONE example where the hypothesis contradicts the conclusion. Such an example is known as a counterexample: A counterexample proves a conditional false by agreeing with the hypothesis but disagreeing with the conclusion.
11 Counterexamples If it is a week night, then you have geometry homework. Counterexample December 25 th may be a week night this year, but you don t have geometry homework on Christmas. This statement agrees with the hypothesis, but disagrees with the conclusion. Since we have found one counterexample for the conditional we say the conditional is false. If we find at least one counterexample that proves the conditional false, it makes no difference how many examples we can find where it is true. Because the conditional was false once, it may be false again. So the conditional has no value in predicting the future or judging the present. 4
12 Converse of a Conditional converse: The converse of a conditional is another if-then statement formed by interchanging the hypothesis and the conclusion of a given statement. Conditional statement If p then q Converse of above statement If q then p Example: Conditional If tomorrow is Saturday, then today is Friday. Converse If today is Friday, then tomorrow is Saturday.
13 Converses of Conditionals NOTE! Just because the conditional statement is true does NOT make the converse of the statement true! Likewise, just because the converse of a statement is true does not make the conditional true. Example: Conditional If I have 2 dimes and a nickel, then I have 25 cents. Converse If I have 25 cents, I have 2 dimes and a nickel. Remember, you need only ONE counterexample to prove a statement false.
14 Biconditionals For a statement to be biconditional, both the original conditional (statement) and its converse must be true. One sign that you have a biconditional statement is the key phrase if and only if to connect the parts of the statement. In a biconditional, the order of the phrases can be switched without changing the meaning.
15 Biconditional Example Conditional If I draw a right angle, then I draw a 90 degree angle. Converse If I draw a 90 degree angle, then I draw a right angle. Biconditional I draw a right angle if and only if I draw a 90 degree angle.
16 Definitions Written as Biconditionals An angle is acute if and only if it measures between 0 and 90. An angle measures between 0 and 90 if and only if the angle is acute. A ray bisects an angle if and only if it divides the angle into two congruent adjacent angles. A ray divides the angle into two congruent adjacent angles if and only if the ray bisects an angle. Points are collinear if and only if the points lie on one line. Points lie on one line if and only if the points are collinear. 3
17 Recommendations When it is possible, I HIGHLY recommend that you reword any definitions, postulates, and theorems in their biconditional form. In all other cases, I HIGHLY recommend you reword any definitions, postulates, and theorems in their IF/THEN form.
18 Section 2-1 Sample Problems Write the hypothesis and the conclusion of each conditional. 1. If 3x - 7 = 32, then x = 13 If 3x - 7 = 32, then x = I ll try if you will. I ll try if you will. 5. a + b = a implies b = 0. a + b = a implies b = 0.
19 Section 2-1 Sample Problems Rewrite each pair of conditionals as a biconditional. 7. If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. B is between A and C if and only if AB + BC = AC. AB + BC = AC if and only if B is between A and C. Write each biconditional as two conditionals that are converses of each other. 9. Points are collinear if and only if they all lie on one line. If points are collinear, then they all lie on one line. If points all lie on one line, then they are collinear.
20 Sample Problems Section 2-1 Provide a counterexample to show that each statement is false. You may use words or diagrams. 11. If ab < 0, then a < 0. Let a = 2 and b = -3. Therefore, ab = -6. In this case, ab < 0 (agrees with hypothesis), but a > 0 (disagrees with conclusion). 13. If point G is on AB, then G is on BA. 15. If a four sided figure has four right angles, then it has four congruent sides.
21 Sample Problems Section 2-1 Tell whether each statement is true or false. Then write the converse and tell whether it is true or false. 17. If x = - 6, then x = 6. True If x = 6, then x = -6. False, x = If b > 4, then 5b > If Pam lives in Chicago, then she lives in Illinois. 23. a 2 > 9 if a > 3.
22 25. n > 5 only if n > 7. Sample Problems Section If points D, E and F are collinear, then DE + EF = DF. 29. Write a definition of congruent angles as a biconditional.
23 Section 2-2 Properties from Algebra Homework Pages 41-42: 1-14
24 Objectives A. Properly use and describe algebraic properties. B. Relate the algebraic properties to geometric properties. C. Properly apply geometric properties.
25 Algebraic Properties of Equality: Transformations Addition: You may add the same value to both sides of an equation. If a = b, then a + c = b + c. Subtraction (add a negative): You may subtract the same value from both sides of an equation. If a = b, then a - c = b - c. Multiplication: You may multiply the same value to both side of an equation. If a = b, then a * c = b * c.
26 Algebraic Properties of Equality: Transformations Division (multiply by a reciprocal): You may divide the same value into both sides of an equation. If a = b, then a / c = b / c. HOWEVER: c cannot be zero! Distribution: You may multiply a factor next to a grouping symbol to every term inside the grouping symbol. If a (b + c + d) = e, then ab + ac + ad = e. Substitution: Left and right sides of an equation are interchangeable. Either statement may be used in another equation. If a + b = c AND d e = c, then: a + b = d e
27 Algebraic Properties of Equality: Transformations Reflexive: A value must equal itself. a = a Symmetric: The left and right sides of an equation can be switched. If a = b, then b = a. Transitive: Any two values in a chain of equality are equal. If a = b AND If b = c. then a = c.
28 But what about GEOMETRIC properties? Remember, we cannot talk about two FIGURES being EQUAL! Two geometric figures can be CONGRUENT. So certain properties, such as the Addition Property of Equality, cannot be applied to figures. However, SOME algebraic properties can be applied to figures. Lengths of line segments and measures of angles are real numbers. Therefore, we can apply algebraic properties of equalities (such as the Addition Property) to these real numbers.
29 Properties of Congruence Reflexive: Any object must be congruent (same size and shape) to itself. DE DE D D Symmetric: The objects on the left and right sides of a congruence statement may be switched. If If DE D FG E then FG DE. then E D. Transitive: Any two objects in a chain of congruence statements are congruent (same size and shape). If DE FG and FG JK then DE JK. If D E and E F then D F.
30 So, what do we do with these properties? We use these algebraic and geometric properties to prove statements. For example: If 3X + 5 = 17, then X = 4. 3X + 5 (- 5) = 17 (- 5) 3X = 12 3X 12X X X = 4 Given. Subtraction Property of Equality. Division Property of Equality. Symmetric Property of Equality.
31 Sample Problems Section 2-2 Justify each step. (Give the reason for each step.) 1. 4x 5 = -2 Given. 4x = 3 Addition Property of Equality x 3 4 Division Property of Equality
32 Sample Problems Section 2-2 Justify each step. (Give the reason for each step.) 3. z 7 11 Given. 3 z 7 33 Multiplication Property of Equality z = -40 Addition Property of Equality
33 Sample Problems Section 2-2 Justify each step. (Give the reason for each step.) 5. 2 b 8 2b 3 2b 3(8 2b) 2b 8b b 24 6b 3 24
34 Sample Problems Section 2-2 Given: AOD as shown Prove: m AOD = m 1 + m 2 + m 3 A B O 1 C 2 3 D Statements 0. AOD as shown 1. m AOD = m AOC + m 3 2. m AOC = m 1 + m 2 3. m AOD = m 1 + m 2 + m Reasons Given. Angle Addition Postulate Angle Addition Postulate Substitution Property of Equality
35 Sample Problems Section 2-2 Given: DW = ON Prove: DO = WN D O W N Statements 1. DW = ON 2. DW = DO + OW ON = OW = OW Reasons 3. Substitution 4. 5.
36 1. m 1 m 2; m 3 m 4 2. m SRT m 1 m 3; m STR m 2 m 4 3. m SRT m 2 m 3 Sample Problems Section Given: m 1 = m 2; m 3 = m 4 Prove: m SRT = m STR 4. m SRT m 2 m 4 5. m SRT m STR R 1. Given P S Z Q Angle Addition Postulate 3. Substitution Property of = 4. Substitution Property of = 5. Substitution Property of = T
37 Sample Problems Section Given: RQ = TP ZQ = ZP Prove: RZ = TZ S P Z Q R T
39 Objectives A. Use the Midpoint Theorem and the Angle Bisector Theorem correctly. B. Understand the valid reasons used in proofs. C. Apply valid reasons to prove theorems and conditionals.
40 Also known as direct proof Deductive Reasoning Deductive reasoning is one logical process used to prove conditionals true by building an argument based upon valid reasoning.
41 Valid Reasons Used in Proofs The items you may use in a proof are: Information given in the hypothesis, Information given in diagrams, Accepted postulates, Algebraic and geometric properties, Definitions, Previously proven or accepted theorems, and Previously proven or accepted corollaries.
42 Theorem 2-1 (Midpoint Theorem) If M is the midpoint of segment AB, A M B then AM = ½AB and MB = ½AB.
43 Proof Of Midpoint Theorem If M is the midpoint of segment AB, then AM = ½AB and MB = ½AB. A M B Now that I know M is the midpoint, what else can I say? Statements Reasons M is the midpoint of AM MB AM MB AM + MB = AB AM + AM = AB 2AM = AB AM = ½ AB AB Given Definition of Midpoint. Definition of Congruence. Segment Addition Postulate. Substitution Property. Simple addition. Division Property of Equality
44 Theorem 2-2 (Angle Bisector Theorem) If ray BX is the bisector of ABC, A X B C then m ABX = ½m ABC and m XBC = ½m ABC.
45 Proof Of Angle Bisector Theorem If ray BX is the bisector of ABC, then m ABX = ½ m ABC and m XBC = ½ m ABC. A X Statements B Reasons C
46 Sample Problems Section 2-3 Name the definition, postulate or theorem that justifies the statement about the diagram. 1. If D is the midpoint of BC, then BD 3. If AD bisects BAC, then 1 2. DC A 5. If BD DC, then D is the midpoint of BC m 1 + m 2 = m BAC 1. Definition of midpoint. 3. Definition of angle bisector. 5. Definition of midpoint. 7. Angle Addition Postulate B 3 4 D C
47 Sample Problems Section 2-3 Write the number that is paired with the bisector of CDE. C E 11. D C E D
48 Sample Problems Section The coordinates of points L and X are 16 and 40, respectively. N is the midpoint of LX, and Y is the midpoint of LN. Sketch a diagram and find: a. LN b. the coordinate of N c. LY d. the coordinate of Y L Y N X LN = Length of the segment LN = = -24 = 24 N = Midpoint of the segment LX = ( ) / 2 = 28 LY = Length of the segment LY = ½ LN LN = ½ LX = ½ (24) = 12 LY = ½ LN = ½ (12) = 6 Y = Midpoint of the segment LN = ( ) / 2 = 22
49 Sample Problems Section a. Suppose M and N are the midpoints of LK and GH, respectively. What segments are congruent? b. What additional information about the picture would enable you to deduce that LM = NH. G N H LM = NH LM = GN MK = NH LK = GH
50 Sample Problems Section 2-3 What can you deduce from the given information? 17. Given: AE = DE; A CE = BE B E D C AE = BE? DE = EC? AC = DB?
51 Section 2-4 Special Pairs of Angles Homework Pages 52-54: 1-33
52 Objectives A. Use the terms complementary, supplementary, and vertical angles correctly. B. Apply these terms to proofs. C. Use the Vertical Angle Theorem correctly.
53 Definitions complementary angles: A pair of coplanar angles, called complements, whose measurements add up to be 90. supplementary angles: A pair of coplanar angles whose measurements add up to be 180. A supplement of an angle is another angle that when added to the first makes 180. vertical angles: A pair of coplanar angles such that the sides of one angle are opposite rays to the sides of the other angle.
54 Complementary Angles 3
55 Supplementary Angles 6
56 Vertical Angles 6
57 Vertical angles are congruent. Theorem 2-3
58 Proof Of Theorem 2-3 (Vertical Angles Theorem) Given: Diagram Prove: Statements Reasons 1. Diagram 1. Given 2. Angles 1 & 2 are vertical angles 2. Definition of vertical angles 3. m 1 m m 2 m m 1 m 3 m 2 m 3 5. m 3 m 3 6. m 1 m Definition of straight angles (Angle Addition Postulate) 4. Substitution Property of Equality 5. Reflexive Property of Equality 6. Subtraction Property of Equality 7. Definition of congruence
59 Sample Problems Section 2-4 Find the measures of a complement and a supplement of K. 1. m K = m K = x 5. Two complementary angles are congruent. Find their measures. 1. Complimentary angles add to 90. Supplementary angles add to 180. Compliment + m K = 90 Compliment + 20 = 90 Compliment = Compliment + m K = 90 Compliment + x = 90 Compliment = 90 - x 5. x + x = 90 2x = 90 x = 45 Supplement + m K = 180 Supplement + 20 = 180 Supplement = 160 Supplement + m K = 180 Supplement + x = 180 Supplement = x
60 Sample Problems Section 2-4 In the diagram, AFB is a right angle. Name the figures described. 7. Another right angle. 9. Two congruent supplementary angles. 11. Two acute vertical angles. 7. AFD 9. AFD and AFB A E 11. EFD and BFC B F D How can you PROVE these angles are acute and vertical? C
61 In the diagram, Sample Problems Section 2-4 OTbisects SOU, m UOV = 35, and m YOW = 120. Find the measure of each angle. 13. m ZOY 17. ½ (120 ) = m VOW 17. m TOU T U 35 V S = 25 O W Z Y X
62 Find the value of x. Sample Problems Section x 19. (3x 5) = 70 3x = 75 x = = 100 WHY? 4x = 100 WHY? x = 25 WHY?
64 Sample Problems Section 2-4 If A and B are supplementary, find the value of x, m A and m B. 25. m A = x + 16, m B = 2x m A + m B = 180 REASON? x x - 16 = 180 REASON? 3x = 180 REASON? x = 60 REASON? m A = 76 m B = 104 REASON?
65 Sample Problems Section 2-4 If C and D are complementary, find the value of y, m C and m D. 27. m C = y - 8, m D = 3y + 2 Use the information given to write an equation and solve the problem. 29. Find the measure of an angle that is half as large as its complement. 31. A supplement of an angle is six times as large as a complement of the angle. Find the measures of the angle, its supplement and its complement.
66 Sample Problems Section 2-4 Find the values of x and y for each diagram. 33. x 50 (3x - y)
68 Objectives A. Use the term perpendicular lines correctly. B. Apply the term perpendicular lines to theorems. C. Use the theorems associated with perpendicular lines (Theorems 2-4, 2-5, and 2-6) correctly.
69 Definition of Perpendicular Lines Perpendicular lines are two lines that intersect to form right angles. Notice how we are using previous terms (defined and undefined) to build new terms.
70 Theorem 2-4 If two lines are perpendicular, then they form congruent adjacent angles.
71 Theorem 2-5 If two lines form congruent adjacent angles, then the lines are perpendicular.
72 Comparing Theorems 2-4 and 2-5 How is Theorem 2-4 (If two lines are perpendicular, then they form congruent adjacent angles) related to Theorem 2-5 (If two lines form congruent adjacent angles, then the lines are perpendicular)? Are they the same? Could they be written as a single statement? What type of statement?
73 Theorem 2-6 If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. A D B C m ABD + m DBC = 90
74 Proof Of Theorem 2-6: If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given : OA OC Prove : AOB and BOC are complimentary. A O B C Statements 1. OA OC 1. Given 2. AOC is a right angle. 3. m AOC 90. Reasons 2. Definition of lines. 3. Definition of right angle. 4. m AOB m BOC m AOC 4. Angle addition postulate. 5. AOB m BOC 90 m 5. Substitution property of = m 6. Def. of complimentary s. 6. AOB and m BOC complimentary s
75 Sample Problems Section In the diagram, UL MJ and m JUK = x. Express in terms of x the measures of the angles: a. LUK b. MUK L K x M U J a. m LUK 90 x b. m MUK 180 x
76 Sample Problems Section 2-5 Name the definition or state the theorem that justifies the statement about the diagram. 3. If EBC is a right angle, then BE AC. 5. If BE AC, then ABD and DBE are complementary. 7. If BE AC, then m ABE = 90. D E F A B C 3. Definition of perpendicular lines. 5. If the exterior sides of 2 adjacent acute angles are perpendicular, then the angles are complementary. 7. Definition of perpendicular lines and right angles.
77 Sample Problems Section 2-5 In the diagram, BE AC and BD BF. Find the value of x. 9. m ABD = 2x - 15, m DBE = x 11. m ABD = 3x - 12, m DBE = 2x + 2, m EBF = 2x + 8 E D F A B C 9. m ABD + m DBE = 90 2x x = 90 3x - 15 = 90 3x = 105 x = m ABD + m DBE = 90 3x x + 2 = 90 5x - 10 = 90 5x = 100 x = 20
78 Sample Problems Section 2-5 In the figure BF AE, m BOC = x, and m GOH = y. Express the measures of the angles in terms of x and y. 15. COH 17. DOE 15. m COH = y 17. m DOE = 90 - (x + y ) B x C D A H y O E G F
79 Sample Problems Section 2-5 Can you conclude from the information given for each exercise that XY XZ? and 3 are complementary 21. m 1 = m m 1 = m 2 and m 3 = m and No, m 1 = 40 m 2 = 10 m 3 = No, what if m 1 = 70? 23. Yes: m 1 + m 2 + m 3 + m 4 = 180 2m 2 + 2m 3 = 180 m 2 + m 3 = No, what if m 1 = 70? X Y Z
80 Sample Problems Section 2-5 What can you conclude from the information given. 27. Given: AD AC; CE AC; m 1 = m 4 D B E A C
81 Section 2-6 Planning a Proof Homework Pages 63-64: 1-22
82 Objectives A. Understand and apply theorems 2-7 (supplementary angles) and 2-8 (complementary angles) correctly. B. Apply the two-column deductive proof method to prove statements, theorems, and/or corollaries.
83 Theorem 2-7 If two angles are supplements of congruent angles (or the same angle), then the two angles are congruent. A B D C ABC & BCD are supplementary ADC & BCD are supplementary ABC & ADC are congruent
84 Theorem 2-8 If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. A B C O D AOB & BOC are complementary COD & BOC are complementary AOC & COD are congruent
85 Producing a Proof All proofs you will produce will be two-column proofs, regardless of the directions in the book. Therefore, if the book says produce a paragraph proof, you will instead produce a two-column proof. All two-column proof should appear as a two-column table The left column will be the statements column. The right column will be the reasons column. Statements Reasons
86 Statements: Parts of a Two Column Proof Each statement should be numbered and supported in the reason column. The first statement is a list of the given information found in the hypothesis of the conditional or from information in a diagram. The body of the proof consists of a logical series of statements and reasons flowing from the givens and from information that can be proved from the diagram. The last statement must be what you were required to prove, found in the conclusion of the conditional.
87 Reasons: Parts of a Two Column Proof Each reason should be numbered so that it matches the statement it supports. The first reason will be given provided that the first statement was a list of the given information. Only given, definitions, postulates, algebraic properties, theorems and their corollaries are acceptable reasons for the body of the proof. The final reason will depend upon the logical structure of the whole proof, but it still must come from the list above (except that it cannot be given ).
88 Each Row in the Two-column Proof Remember! For each row in the proof: The hypothesis in the reason must be shown to be true in preceding statements. The conclusion in the reason must match the statement. Writing all definitions, postulates, properties, theorems and corollaries as standard if-then conditionals will make this a lot easier! C A O B D Statements Reasons 1. Given lines AB and CD are perpendicular and intersect at point O. 1. Given 2. Angle BOC is a right angle. 2. If two lines are perpendicular then they intersect at right angles.
89 Steps for Writing a Two Column Proof Step 1: Identify the conditional you are required to prove. Step 2: Draw, label, and annotate a diagram for the proof. Step 3: List from the conditional, in terms of the figure, what is given. This is the first statement/reason in the proof. Step 4: Determine from the conditional, in terms of the diagram, what is to be proven. This will be the last statement in the proof. Step 5: List from the diagram, in terms of the figure, what can be proven and why. Step 6: Select and arrange, in a logical order, those statements from steps 3, 4 & 5 that will allow you to move from the given information to the statement to be proved. N.B. If you get stuck writing the proof from the top down, then work up from the conclusion by deciding what you had to know to make that statement. 22 N
90 Example of a Two Column Proof: #21 page 64 Step 1: If AC BC and 3 is complementary to 1, then 3 2. Step 2: A 3 C 1 2 D B Step 3: Given: AC BC 3 is complementary to 1 Step 4: Prove: 3 2 Step 5: m ACB = 90, Definition of perpendicular lines, right angles m 3 + m 1 = 90, def comp s m 2 + m 1 = m ACB, Angle Addition Postulate m 2 + m 1 = 90, substitution property of equality 2 is complementary to 1, def comp s m 3 + m 1 = m 2 + m 1, transitive m 3 = m 2, subtraction property of equality 3 2, Th. 2-8: Complementary Angles P
91 Example of a Two Column Proof: #21 page 64 Step 5: m ACB = 90, definition of right angles, definition of m 3 + m 1 = 90, def comp s m 2 + m 1 = m ACB, Angle Addition Postulate m 2 + m 1 = 90, substitution property of = 2 is complementary to 1, def comp s m 3 + m 1 = m 2 + m 1, transitive property of = m 3 = m 2, subtraction property of equality 3 2, Th. 2-8: Complimentary Angles Step 6: Statements Reasons 1. AC BC, 3 is complementary to 1 1. Given 2. m ACB = def 3. m 2 + m 1 = m ACB 3. Angle Add. Post. 4. m 2 + m 1 = Substitution prop = 5. 2 is complementary to 1 5. def comp s Th. 2-8 P
92 Sample Problems Section 2-6 Write the name or statement of the definition, postulate, property, or theorem that justifies the statement about the diagram. 1. AD + DB = AB If D is on the line segment AB, then AD + DB = AB. 3. If vertical angles exist, then the angles are congruent. A H 6 C D 5 E G F B
93 Sample Problems Section 2-6 Write the name or statement of the definition, postulate, property, or theorem that justifies the statement about the diagram. 5. If DF bisects CDB, then If CD AB, then m CDB = If two lines are perpendicular, then they meet to form right angles. If an angle is a right angle, then it measures If a ray bisects an angle, then it cuts the angle into 2 congruent angles. A H 6 C D 5 E G F B
94 Sample Problems Section 2-6 Write the name or statement of the definition, postulate, property, or theorem that justifies the statement about the diagram. 9. If m 3 + m 4 = 90, then 3 & 4 are complementary. 11. If AB CE, then ADC ADE. 13. If FDG is a right angle, then DF DG. 9. If the sum of the measure of two angles is 90, then the angles are complementary. 11. If two lines are perpendicular, then they form congruent adjacent angles. A H 6 C D 5 E G F B
95 Sample Problems Section Given: 1 & 5 are supplementary 3 & 5 are supplementary Prove: & 5 are supplementary 1. 3 & 5 are supplementary 2. m 1 + m 5 = m 3 + m 5 = m 1 + m 5 = m 3 + m m 5 = m m 1 = m 3 or
96 Sample Problems Section Given: PQ QR, PS SR, 1 4 Prove: 2 5 Q P R 4 5 S 6 1. PQ QR, PS SR, is comp. to is comp to
97 Sample Problems Section Given: 2 3 Prove: Given: AC BC 3 is comp. to 1 Prove: A C 1 2 D B
98 Chapter Two Deductive Reasoning Review Homework Pages 68-69: 1-12
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Lesson 7 Inductive ing 1. I CAN understand what inductive reasoning is and its importance in geometry 3. I CAN show that a conditional statement is false by finding a counterexample Can you find the next
Name Period GP GOTRI PROOFS 1) I can define, identify and illustrate the following terms onjecture Inductive eductive onclusion Proof Postulate Theorem Prove Given ates, assignments, and quizzes subject
Geometry Topic 5: Conditional statements and converses page 1 Student Activity Sheet 5.1; use with Overview 1. REVIEW Complete this geometric proof by writing a reason to justify each statement. Given:
Chapter 1 Line and Angle Relationships SECTION 1.1: Sets, Statements, and Reasoning 1. a. Not a statement. b. Statement; true c. Statement; true d. Statement; false. a. Statement; true b. Not a statement.
Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
Testing for Congruent Triangles Examples 1. Why is congruency important? In 1913, Henry Ford began producing automobiles using an assembly line. When products are mass-produced, each piece must be interchangeable,
UNIT 5 LINES AND ANGLES (A) Main Concepts and Results An angle is formed when two lines or rays or line segments meet or intersect. When the sum of the measures of two angles is 90, the angles are called
Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Formal Geometry S1 (#2215)
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
1. State the assumption you would make to start an indirect proof of each statement. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. 2. is a scalene triangle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
1. Determine whether each quadrilateral is a Justify your answer. 3. KITES Charmaine is building the kite shown below. She wants to be sure that the string around her frame forms a parallelogram before
Geometry Week 7 Sec 4.2 to 4.5 section 4.2 **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle. Protractor Postulate:
Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 10 Year 10 Term 1 Week 10 Homework 1 10.1 Deductive geometry.................................... 1 10.1.1 Basic
Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
Parallelogram Bisectors Geometry Final Part B Problem 7 By: Douglas A. Ruby Date: 11/10/2002 Class: Geometry Grades: 11/12 Problem 7: When the bisectors of two consecutive angles of a parallelogram intersect
Triangles Triangle A triangle is a closed figure in a plane consisting of three segments called sides. Any two sides intersect in exactly one point called a vertex. A triangle is named using the capital
Geometry Sample Problems Sample Proofs Below are examples of some typical proofs covered in Jesuit Geometry classes. Shown first are blank proofs that can be used as sample problems, with the solutions
Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015 Mathematics Without Numbers The
Geometry: 2.1-2.3 Notes NAME 2.1 Be able to write all types of conditional statements. Date: Define Vocabulary: conditional statement if-then form hypothesis conclusion negation converse inverse contrapositive
Name: Class: Date: ID: A Chapter 4 Study guide Numeric Response 1. An isosceles triangle has a perimeter of 50 in. The congruent sides measure (2x + 3) cm. The length of the third side is 4x cm. What is
5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
Geometry Chapter 2 Study Guide Short Answer ( 2 Points Each) 1. (1 point) Name the Property of Equality that justifies the statement: If g = h, then. 2. (1 point) Name the Property of Congruence that justifies
ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram
73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Holt 5-1 Perpendicular and Angle Bisectors Warm Up Construct each of the following. 1.
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 3. PROOF Write a two-column proof to prove that if ABCD is a rhombus with diagonal. 1. If, find. A rhombus is a parallelogram with all
/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the
Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A
HPTR 9 HPTR TL OF ONTNTS 9-1 Proving Lines Parallel 9-2 Properties of Parallel Lines 9-3 Parallel Lines in the oordinate Plane 9-4 The Sum of the Measures of the ngles of a Triangle 9-5 Proving Triangles
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries, 4th Ed by Marvin Jay Greenberg (Revised: 18 Feb 2011) Logic Rule 0 No unstated assumptions may be
1 Geometry Proofs Reference Sheet Here are some of the properties that we might use in our proofs today: #1. Definition of Isosceles Triangle says that If a triangle is isosceles then TWO or more sides