Common Core Math 3. Proofs. Can you find the error in this proof "#$%&!!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""!

Common Core Math 3 Proofs Can you find the error in this proof "$%& a = b'()$&2 = 1 *+,+$-$%+.. /$,0)%. " a = b $%&'( ) a 2 = ab = a 2 - b 2 = ab - b 2? (a + b)(a - b) = b(a - b) @ (a + b) = b B a + a = a D 2a = a E 2 = 1 *+,-%.,%/0-%1(.21.'2-314'5+0,%-3 67+,-%.,381-9:%;':83"< >+8-20/-%1(.21.'2-314'5+0,%-3 6:+8-20/- ) 421781-9:%;':< >+8:-%-+-%1(.21.'2-314'5+0,%-3 6:+8:-%-+-'40/-12';4127< A%&%:%1(.21.'2-314'5+0,%-3 6;%&%;'81-9:%;':836"$<< >+8:-%-+-%1(.21.'2-314'5+0,%-3 6:+8:-%-+-'"412C:%(/'"%< >+8:-%-+-%1(.21.'2-314'5+0,%-3 6"&"%'"( A%&%:%1(.21.'2-314'5+0,%-36;%&%;'81-9 :%;':830< "$%&"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" A P E X HIGH SCHOOL 1501 LAURA D UNCAN R OAD A P E X, NC 27502

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Getting Blood From a Stone The rules of writing a mathematical proof are very similar to the rules of "getting blood from a stone". The old saying "You can't get blood from a stone means that nobody can give you anything that they, themselves, do not have. The "rules of the game", and how these rules relate to mathematical proof: 1) You must start with the word STONE and end up with the word BLOOD by a series of logical steps (much like the steps in a mathematical proof). 2) You must change one letter in each step. Each step must follow directly from the previous step. (In a proof, you must address one concept at a time. Each step in a proof must follow directly from the step before it - including from the 'Given', and lead to the next.) 3) The word in each step must be a real word, and in the dictionary. (In a proof, each step must be true: a definition, postulate, or theorem). If you use an unusual word, you must write the definition. 4) In this game, some people can do this in 10 steps; others may take 15, and both can be correct. There is more than one route, and no one way is necessarily better than another. One may be longer, but the best one is the one is the one that the student "sees" when trying to do the problem (This is very true of proof; there are dozens of proofs of the Pythagorean Theorem, for example, and even in classroom assignments there is often more than one method.) 5) In this game, some people like to work backwards, or even from both ends toward the middle. (This works very well with proofs, too) Example: Can you get a DOG from a CAT? (Can you go from the word CAT to the word DOG?) Here are some possible solutions to this problem: Solution 1: CAT - BAT - BAG - BOG - DOG Solution 2: CAT - COT - DOT - DOG 3

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Algebraic PROPERTIES used in Geometry Property Example(s) Distributive 3(n + 5) = 3n + 15 -(x 8) = -x + 8 Reflexive -3 + x = -3 + x ONE EQUATION AB @ AB a quantity is equal (congruent) to itself Symmetric if y = 7 then 7 = y TWO EQUATIONS can switch sides of the equation If AB @ CD then CD @ AB (or congruence) Transitive If 2+3=5 and 5=10-5, then 2+3=10-5 THREE EQUATIONS if a = b and b = c then a = c If AB @ BC and BC @ CD then AB @ CD (also applies to congruence) Properties of Equality Substitution if 5x - 2x + 12 = 18, then 3x + 12 = 18 If x = y, then x may be Property of = replaced by y in any If 7x + y = 9 and y = -3, then 7x 3 = 9 equation or expression. Addition IF 3x 8 = 10, then 3x = 18 add an equal amount to Property of = both sides of an equation If AB = DE then AB + BC = DE + BC x Multiplication prop. if = -4, then x = -12 multiply both sides of an equation Property of = 3 by an equal amount Subtraction if x + 5 = -6, then x = -11 subtract an equal amount from Property of = both sides of an equation If AB + BC = CD + BC, then AB = CD Division if 3x = 12, then x = 4 divide both sides of an equation Property of = by an equal amount If 3AB = 6EF, then AB = 2EF 6

Algebraic Properties Worksheet I. Name the property that justifies each statement. 1. If m A = m B, then m B = m A. 2. If x + 3 = 17, then x = 14 3. xy = xy 4. If 7x = 42, then x = 6 5. If XY YZ = XM, then XY = XM + YZ 6. 2(x + 4) = 2x + 8 7. If m A + m B = 90, and m A = 30, 8. If x = y + 3 and y + 3 = 10, then x = 10. then 30 + m B = 90. II. Complete the reasons in each algebraic proof. 9. Prove that if 2(x 3) = 8, then x = 7. Given: 2(x 3) = 8 Prove: x = 7 Statements Reasons a) 2 (x 3) = 8 a) b) 2x 6 = 8 b) c) 2x = 14 c) d) x = 7 d) 10. Prove that if 3x 4 = x + 6, then x = 4. Given: 3x 4 = x + 6 Prove: x = 4 Statements Reasons a) 3x 4 = x + 6 a) b) 2(3x 4) = 2( x + 6) b) c) 6x 8 = x + 12 c) d) 5x 8 = 12 d) e) 5x = 20 e) f) x = 4 f) 7

Name the property that justifies each statement. 1. If 3x = 120, then x = 40. 2. If 12 = AB, then AB = 12. 3. If AB = BC and BC = CD, then AB = CD. 4. If y = 75 and y = m A, then m A = 75. 5. If 5 = 3x 4, then 3x 4 = 5. Ê 6. If 3 x - 5 ˆ Ë 3 = 1, then 3x 5 = 1. 7. If m 1 = 90 and m 2 = 90, then m 1 = m 2. 8. For XY, XY = XY. 9. If EF = GH and GH = JK, then EF = JK. 10. If m 1 + 30 = 90, then m 1 = 60. Choose the number of reason in the right column that best matches each statement in the left column. Statements Reasons A. If x 7 = 12, then x = 19 B. If MK = NJ and BG = NJ, then MK = BG 1. Distributive Property 2. Addition Property of Equality C. If y = m 5 30 and m 5 = 90, then y = 90 30 3. Symmetric Property D. If ST = UV, then UV = ST. E. If x = -3(2x 4), then x = -6x + 12 4. Substitution Property 5. Transitive Property Name the property that justifies each statement. 11. If AB + BC = DE + BC, then AB = DE. 12. m ABC = m ABC 13. If XY = PQ and XY = RS, then PQ = RS. 14. If, then x = 15. 15. If 2x = 9, then x =. 8

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Writing Reasons Fill in the blank with a reason for the statement. Write your answer in if-then form if it is a definition. STATEMENTS REASONS 1. a) a) Given b) AB = CD b) 2. a) M is the midpoint of. a) Given b) b) 3. a) bisects at point X. a) Given b) X is the midpoint of. b) 4. a) and a) Given b) b) 5. a) M is between A and B, a) Given b) M is the midpoint of b) 6. a) M is between C and D a) Given b) CM + MD = CD b) 7. a) MN = RS a) Given b) b) 8. a) a) Given b) b) 9. a) AB = CD and BC = BC a) Given b) AB + BC = CD + BC b) 10

Presenting a proof: A proof is a logical argument that establishes the truth of a statement. A proof is a written account of the complete thought process that is used to reach a conclusion. Each step of the process is supported by a theorem, postulate or definition verifying why the step is possible. Proofs are developed such that each step in the argument is in proper chronological order in relation to earlier steps. No steps can be left out. There are three classic styles for presenting proofs: Method 1: The Paragraph Proof is a detailed paragraph explaining the proof process. The paragraph is lengthy and contains steps and reasons that lead to the final conclusion. Be careful when using this method -- it is easy to leave out critical steps (or supporting reasons) if you are not careful. Method 2: The Flowchart Proof is a concept map that shows the statements and reasons needed for a proof using boxes and connecting arrows. Statements, written in the logical order, are placed in the boxes. The reason for each statement is placed under that box. Method 3: The Two-Column Proof consists of two columns where the first column contains a numbered chronological list of steps ("statements") leading to the desired conclusion. The second column contains a list of "reasons" which support each step in the proof. These reasons are properties, theorems, postulates and definitions. A formal 2-column proof contains the following elements: Statement of the original problem Diagram, marked with "Given" information Re-statement of the "Given" information in the proof Complete supporting reasons for each step in the proof The "Prove" statement as the last statement 11

EXAMPLE: "$%& RT @ SU '()$&"$%& Method 1 - Paragraph Proof: "$%"&'(")*+$* RT @ SU '()*+,-.-/-0.10.234+./5+26+./5789++:48; ;+.2/75</745"%$&'(+1845+-5=+/>++."8.*%</7++26+./?**-/-0. @05/4;8/+/+;;545/7+"%$"A%'(+1845+%-5=+/>++.8.*&</7++26+./?**-/-0.@05/4;8/+/+;;545/7+&$%A%&'B+18./74554=5/-/4/+"A%,03"%< 8.*%A%&,03&-./0/7++:48;-/)"%$&/02-9+"A%$%A%&'%$%=) /7+3+,;+C-9+D30D+3/)'&5-.2/7+54=/381/-0.D30D+3/)0,+:48;-/)<>+18./745 54=/381/%,306=0/75-*+50,/7++:48;-/)"A%$%A%&/0D309+/78/"$%&' Method 2 - Flowchart Proof: RT @ SU Given Method 3 - Two-Column Proof: Statements 1. RT @ SU -./0123?./01/2@20 D.23120@03 E./2@20120@03 I.20120 K./2103 " % & "%$& If 2 segments are congruent their lengths are =. "%$"A% &$%A%& Segment Addition Postulate,'(") "$%&$' 45*678"&9")*8$%":7)&%;")*<*+") *+"'%=")&*+8$%"">;$=. 2"&9")*ABB'*'7)C78*;=$*" 2"&9")*ABB'*'7)C78*;=$*" 2;F8*'*;*'7)C%7G"%*H /"5="J'("C%7G"%*H "%$& Subtraction Property of Equality 2;F*%$:*'7)G%7G"%*H75L>;$='*H "A%$%A%& Substitution Property %$% Reflexive Property 12

Definition Proofs EXAMPLE: Given: XY = YZ Prove: Y is the midpoint of X Y Z STATEMENT REASON 1. XY = YZ 1. Given 2. 2. If 2 segments have = lengths, then they are @. 3. Y is the midpoint of 3. If a point divides a segment into 2 @ segments, then it is the midpoint 1. Given: Line m bisects Prove: XP = YP X P m STATEMENT 1. line m bisects. 2. P is the midpoint of. REASON Y 3. 4. XP = YP 2. Given: B is the midpoint of D is the midpoint of Prove: AB + CD = BC + DE STATEMENT 1. B is the midpoint of 2. D is the midpoint of AB @ BC A B C D E REASON CD @ DE 3. AB = BC CD = DE 4. AB + CD = BC + DE 13

3. Given: PQ = MN MN = QR Prove: Q is the midpoint of STATEMENT P REASON Q R M N 4. Given: line k bisects line k bisects Prove: AB + DE = BC + EF STATEMENT REASON A B D E F C k D 5. Given: C is the midpoint of Prove: AB + BC = CE A B C E STATEMENT REASON F 14

Practice with Segment Proofs Fill in the reason for each statement below. Then rewrite the proof as a FLOWCHART PROOF. 1. Given: DE = EF, AD = BF Prove: AE = BE A D E F B Statements 1. AD = BF 1. DE = EF 2. AD + DE = BF + FE 2. Reasons 3. AD + DE = AE 3. BE = BF + FE 4. AE = BE 4. FLOWCHART PROOF: 2. Given: AX = BY, XC = YC Prove: Statements 1. AX = BY 1. XC = YC 2. AX + XC = BY + YC 2. 3. AX + XC = AC 3. BY + YC = BC 4. AC = BC 4. 5. 5. FLOWCHART PROOF: A X C Y Reasons B 15

B Write 2-column proofs (from scratch) C 3. Given: AE = FD Prove: AF = ED Statements Reasons A F E D D C 4. Given: AB = BE BC = BD Prove: AC = ED Statements Reasons A B E B 5. Given: AF = EC Prove: AE = FC Statements A Reasons E F D C 16

6. Given: AC = BD BE = EC Prove: AE = DE A E B Statements Reasons D C C 7. Given: AB = DE BC = CD Prove: B D A E Statements Reasons 17

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EXAMPLE: Given: 1 and 2 are supplementary 2 @ 3 Prove: m 1 + m 3 = 180 Method 1 - Paragraph Proof: We are given that 1 and 2 are supplementary and that 2 @ 3. By definition the measures of supplementary angles add to 180 so m 1 + m 2 = 180. By definition congruent angles have equal measures so m 2 = m 3. We can use the substitution property of equality to substitute m 3 for m 2 in the equation m 1 + m 2 = 180 and prove that m 1 + m 3 = 180. Method 2 - Flowchart Proof: 1 and 2 are supplementary m 1 + m 2 = 180 Given 2 @ 3 Given Definition of supplementary angles m 2 = m 3 Definition of congruent angles m 1 + m 3 = 180 Substitution property of equality Method 3 - Two-Column Proof: Statements Reasons 1. 1 and 2 are supplementary Given 2. 2 @ 3 Given 3. m 1 + m 2 = 180 Definition of supplementary angles 4. m 2 = m 3 If 2 angles are congruent then they have equal measure. 5. m 1 + m 3 = 180 Substitution property of equality 21

Prove Angle Theorems "$%&'()*+,'-+-.)/$'()*+,0$/+($/+1'-+2&()-3+($45-&6+$/.,$/+&-+74 1. Given: A and B are right angles Prove: A B " " "$%&'()*+,'-+6+-$.2'*'()*+,0$/+($/+1'-+2&()-3+($45-&6+$/.,$/+&-+74 2. Given: Lines k and m intersect Prove: 1 3 %" $" &" '" " "" " " " " " " " " "$%&'()*+,/'6+$/+,'7+,388*+7+($0$/+($/+1'-+2&()-3+($45-&6+$/.,$/+&-+74 $%""&'()*+" ",*-" ",.)"/0112)3)*4,.5",*62)/" " """ ",*-" 7",.)"/0112)3)*4,.5",*62)/" """""8.9()+" " " 7" " " 22

ANGLES Worksheet ANGLE ADDITION POSTULATE: If point B lies in the interior of AOC, then + = If two angles form a linear pair then they are. All right angles are. All vertical angles are. If two angles are @ and supplementary, then each is a angle. If two angles are @, then their are @. If two angles are @, then their are @ If two angles have the same complement (supplement), then they are. If two angles are supplements (complements) of @ angles, then the two angles are. Fill in the blank with Always, Sometimes, or Never. 1. Vertical angles have a common vertex. 2. Two right angles are complementary. 3. Right angles are vertical angles. 4. Angles A, B, and C are supplementary. 5. Vertical angles have a common supplement. Solve for the given variables. 1. 2. 3x 5 6x 23 x + 16 2x 16 3. 4. x 2y 17 3x 8 x 2x 16 50 3x y 5. bisects KAT m 1 = x + 12 m 3 = 6x 20 K 2 L 3 1 S A T 6. If m C = 40 x, and C and D are complementary, what is the m D? 7. If m A = x + 80 and A and B form a linear pair, what is the m B? 23

Use the figure to the right to answer 8-15 8. Name all angles shown. 9. Name 1 in 2 other ways. 10. Name all angles which have side. G H 2 1 3 D E F 11. Name a point in the interior of GEF. 12. Name a point in the exterior of 2. 13. Name 2 pairs of adjacent angles. 14. What is the union of and? 15. What is the union of and? State whether the numbered angles shown in 16-23 are adjacent or not. If NOT, explain. 16. 17. 18. 19. ABC, ABD A 2 1 1 2 1 2 D C B 29. 21. 22. 23. 1 2 1 2 1 2 1 2 Answers: 16. 17. 18. 19. 20. 21. 22. 23. 24

Angle Proofs Worksheet B 1. Given: bisects NDH Prove: 1 @ 3 E 1 D 3 2 N Statements Reasons H I 2. Given: 1 @ 2 Prove: 1 @ 3 3 1 2 Statements Reasons 3. Given: 1 is a supplement of 2 Prove: 1 @ 3 1 2 3 Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 25

B 4. Given: ADC @ CEA Prove: CDB @ AEB D E Statements Reasons A C 1 2 3 4 5. Given: 1 @ 4 Prove: 2 @ 3 Statements Reasons 26

6. Given: DMH @ DHM RMH @ RHM Prove: DMR @ DHR Statements Reasons 1. 1. 2. 2. M D R H 3. 3. 4. 4. 5. 5. 6. 6. 7. Given: Prove: ABD and DBC are complementary C D B A Statements Reasons 27

8. Given: ; 1 is a complement of 3 Prove: 2 @ 3 A 1 2 Statements " is the complement of 2. m BAC = 90 1. 2. B Reasons 3 C 3. 3. 4. m 1 + m 2 = 90 4. 5. m 1 + m 3 = 90 5. 6. 6. 7. 7. 8. 8. C 9. Given: ; 2 @ 3 Prove: 1 @ 4 Statements B D 2 3 1 4 A O E Reasons 28

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y a x 4 2 1 3 8 7 5 6 9 14 12 11 10 15 13 16 18 17 19 b ""$%&&&'&((&)*&&&+,$-.%/%&/0%&10'2/* ANGLES TYPE, SUPPL., OR NONE 1. " 1 and " 14 2. " 2 and " 15 3. " 7 and " 9 4. " 9 and " 16 5. " 10 and " 17 6. " 16 and " 14 7. " 9 and " 14 8. " 18 and " 19 9. " 1 and " 16 10. " 3 and " 8 11. " 6 and " 9 12. " 12 and " 13 13. " 7 and " 11 14. " 6 and " 8 15. " 4 and " 13 33

Parallel line Proofs 1. Given: l // m ; s // t Prove: 2 @ 4 s Diagram for 1-4 t l 2 1 2. Given: l // m ; s // t Prove: 1 @ 5 m 3 5 4 3. Given: l // m; 1 @ 4 Prove: s // t 4. Given: l // m; 2 @ 5 Prove: s // t 5. Given: ; Prove: B and E are supplementary A D B P C F E 34

6. Given: ; Prove: B @ E B A P D C E F 7. Given: g // h; g // j Prove: 2 @ 3 2 1 g h 3 j 8. Given: ; Prove: B @ D A C E B D 35

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Triangle Proofs Practice If two sides of a triangle are congruent, the angles opposite those sides are congruent. (Isosceles Triangle). Prove this theorem. 1. Given: AC @ BC Prove: CAD @ CBD Statements Reasons 1. AC @ BC 2. Draw CD bisecting ACB " &" '" " %" $" 3. 1@ 2 4. CD @ CD 5. DACD @ DBCD 6. CAD @ CBD $" 2. Given: BD bisects AC. BD is perpendicular to AC. Prove: ABC is isosceles Statements Reasons 1. BD bisects AC " %" " 2. BD is perpendicular to AC 3. AD @ CD 4. BD @ BD 5. ADB and BDC are right angles 6. ADB BDC 7. "ABD CBD 8. AB @ CB 9. ABC is isosceles 40

3. Given: AB @ DE BC @ EF B @ E Prove: DABC @ DDEF 4. Given: E is the midpoint of BD AE @ EC Prove: DAEB @ DCED 5. Given: BA @ DC BD"$%&'$ AC AC"$%&'$ BD Prove: DABE @ DCDE 41

6. Given: BAC @ DAE AE @ AC A is the midpoint of BD Prove: DBEA @ DDCA Statements Reasons 7. Given: A @ E AB @ BE Prove: AD @ EC Statements Reasons 42

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Worksheet - Construct a copy of a given line segment http://www.mathopenref.com/wscopysegment.html MATH OPEN REFERENCEwww.mathopenref.com NAME: Constructing a copy of a given line segment with compass and straightedge (For assistance see www.mathopenref.com/constcopysegment.html) 1. Construct a copy of the line segment below. 2. Construct a copy of the line segment AB from the triangle. The copy should have one endpoint at P. 1 of 1 8/20/14 11:05 PM 51

Worksheet - Construct a copy of a given angle http://www.mathopenref.com/wscopyangle.html MATH OPEN REFERENCEwww.mathopenref.com NAME: Constructing a copy of a given angle with compass and straightedge (For assistance see www.mathopenref.com/constcopyangle.html) 1. Construct a copy of the angle below. 2. Using the fewest arcs and lines, construct a copy of the shape below. (The record: 5 arcs, 2 lines ) 1 of 1 8/20/14 10:37 PM 52

Worksheet - Perpendicular line bisector http://www.mathopenref.com/wsbisectline.html MATH OPEN REFERENCEwww.mathopenref.com NAME: Constructing the perpendicular bisector of a line segment with compass and straightedge (For assistance see www.mathopenref.com/constbisectline.html) 1. Draw the perpendicular bisector of both of the two lines below 2. Construct the four perpendicular bisectors of the sides of the rectangle below, using the fewest arcs and lines. (The record: 3 arcs, 2 lines) 1 of 1 8/20/14 10:13 PM 54

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