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Geometry 1 Unit 2: Reasoning and Proof 1 Geometry 1 Unit 2 2.1 Conditional Statements 2 Conditional Statements Conditional StatementA statement with two parts If-then form A way of writing a conditional statement that clearly showcases the hypothesis and conclusion Hypothesis The “if” part of a conditional Statement Conclusion The “then” part of a conditional Statement 3 Conditional Statements Examples of Conditional Statements If today is Saturday, then tomorrow is Sunday. If it’s a triangle, then it has a right angle. If x2 = 4, then x = 2. If you clean your room, then you can go to the mall. If p, then q. 4 Conditional Statements Example 1 Circle the hypothesis and underline the conclusion in each conditional statement If you are in Geometry 1, then you will learn about the building blocks of geometry If two points lie on the same line, then they are collinear If a figure is a plane, then it is defined by 3 distinct points 5 Conditional Statements Example 2 Rewrite each statement in if…then form A line contains at least two points If a figure is a line, then it contains at least two points When two planes intersect their intersection is a line If two planes intersect, then their intersection is a line. Two angles that add to 90° are complementary If two angles add to equal 90°, then they are complementary. 6 Conditional Statements Counterexample An example that proves that a given statement is false Write a counterexample If x2 = 9, then x = 3 7 Conditional Statements Example 3 Determine if the following statements are true or false. If false, give a counterexample. If x + 1 = 0, then x = -1 If a polygon has six sides, then it is a decagon. If the angles are a linear pair, then the sum of the measure of the angles is 90º. 8 Conditional Statements Negation In most cases you can form the negation of a statement by either adding or deleting the word “not”. 9 Conditional Statements Examples of Negations Statement: mA 30 Negation : mA 30 Statement: Mr. Ross is not more than 6 feet tall. Negation: Mr. Ross is more than 6 feet tall I am doing my homework. Negation: 10 Conditional Statements Example 4 Write the negation of each statement. Determine whether your new statement is true or false. Stanfield is the largest city in Arizona. All triangles have three sides. Dairy cows are not purple. Some VGHS students have brown hair. 11 Conditional Statements Converse Formed by switching the if and the then part. Original If you like green, then you will love my new shirt. Converse If you love my new shirt, then you like green. 12 Conditional Statements Inverse Formed by negating both the if and the then part. Original If you like green, then you will love my new shirt. Inverse If you do not like green, then you will not love my new shirt. 13 Conditional Statements Contrapositive Formed by switching and negating both the if and then part. Original If you like green, then you will love my new shirt. Contrapositive If you do not love my new shirt, then you do not like green. 14 Conditional Statements Write in if…then form. Write the converse, inverse and contrapositive of each statement. I will wash the dishes, if you dry them. A square with side length 2 cm has an area of 4 cm2. 15 Conditional Statements Point-line postulate: Through any two points, there exists exactly one line Point-line converse: A line contains at least two points Intersecting lines postulate: If two lines intersect, then their intersection is exactly one point 16 Conditional Statements Point-plane postulate: Through any three noncollinear points there exists one plane Point-plane converse: A plane contains at least three noncollinear points Line-plane postulate: If two points lie in a plane, then the line containing them lies in the plane Intersecting planes postulate: If two planes intersect, then their intersection is a line 17 Use the diagram to state the postulate that verifies the truth of the statement. The points E, R, and T lie in a plane (labeled A). The points E and R lie on a line (labeled y). The planes A and P intersect in a line (labeled m). The points E and R lie in a plane A. Therefore, line y lies in plane A. m P ●T E R y A 18 Geometry 1 Unit 2 2.2: Definitions and Biconditional Statements 19 Check your answers on Worksheet 2.1A 20 Do you remember….. At the bottom of Page 5 of your notes packet, make a strip containing 10 boxes. In those boxes, write any of these terms. We are about to play…. collinear perpendicular congruent vertical angles inverse supplementary midpoint adjacent angles contrapositive complementary counterexample coplanar conditional statement converse 21 Biconditional Statement Two Statements Combined into One I will pass if and only if I earn a 70% or better in this class. I am happy if and only if I smile at Mrs. Dolezal. I have good smelling breath if and only if I brush my teeth. I attract more friends if and only if I learn geometry. 22 Definitions and Biconditional Statements Can be rewritten with “if and only if” Only occurs when the statement and the converse of the statement are both true. A biconditional can be split into a conditional and its converse. 23 Definitions and Biconditional Statements Example 1 (Write as a conditional statement and its converse.) An angle is right if and only if its measure is 90º A number is even if and only if it is divisible by two. A point on a segment is the midpoint of the segment if and only if it bisects the segment. You attend school if and only if it is a weekday. You get an A if and only if you bring the teacher gifts. 24 Definitions and Biconditional Statements Perpendicular lines Two lines are perpendicular if they intersect to form a right angle To the left of “Perpendicular Lines,” draw 5 lines that intersect. Put your pen on the intersection so that it goes straight up from the intersection. A line perpendicular to a plane A line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it The symbol is read, “is perpendicular to. 25 Definitions and Biconditional Statements Example 2 Write the definition of perpendicular as a biconditional statement. _________________________ if and only if _____________________________________ Definition of perpendicular: Perpendicular objects form right angles at their intersection. 26 Definitions and Biconditional Statements Example 3 Give a counterexample that demonstrates that the converse is false. (not on paper…) If you are in Mrs. Dolezal’s class, you are having fun. If two lines are perpendicular, then they intersect. 27 Definitions and Biconditional Statements Example 4 The following statement is true. Write the converse and decide if it is true or false. If the converse is true, combine it with its original to form a biconditional. If x2 = 4, then x = 2 or x = -2 28 Definitions and Biconditional Statements Example 5 Consider the statement x2 < 49 if and only if x < 7. Is this a biconditional? Is the statement true? 29 Person Lower Limit Upper Limit Whiz Kid 90% 100% Smarty 80% 89% Average 70% 79% Below Average 0% 69% Use the information in the table to write a definition for each type of student. For example: A student who has an average of 90% to 100% is called a whiz kid. 30 WORKSHEET 2.2A ANSWERS 31 Each person selects a slogan from the basket. You and your table mate decides which slogan you use. • Write the slogan as a conditional statement (if-then format). • Write its converse, inverse, and contrapositive. SLOGAN ACTIVITY 32 Geometry 1 Unit 2 2.3 Deductive Reasoning 33 Deductive Reasoning Symbolic Logic Statements are replaced with variables, such as p, q, r. Symbols are used to connect the statements. 34 Deductive Reasoning Symbol ~ → Λ V → ↔ Meaning not if…then and or if…then if and only if 35 Deductive Reasoning Example 1 Let p be “the sum of the measure of two angles is 180º” and Let q be “two angles are supplementary”. What does p → q mean? What does q → p mean? 36 Deductive Reasoning Example 2 p: Jen cleaned her room. q: Jen is going to the mall. What does p → q mean? What does q → p mean? What does ~q mean? What does p Λ q mean? 37 Deductive Reasoning Example 3 t: Jeff has a math test today s: Jeff studied tvs s → t ~s → t What does ~t mean? 38 If Ana completes all her homework, then she will go to the movies. Ana completed all of her homework. What will Ana do now? If Joe wins the football game, he will get a new movie. Joe did not win the football game. Will John get a new movie? 39 If Derrick cleans his room, he will go to the mall. If Derrick goes to the mall, he will get new shoes. Derrick cleaned his room. Does he get new shoes? LAW OF SYLLOGISM 40 Deductive Reasoning Deductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument. 41 Deductive Reasoning Law of Detachment When you have a true conditional statement and you know the hypothesis is true, you can conclude the conclusion is true. Given: Given: Conclusion: p→q p q 42 Deductive Reasoning Example 4 Determine if the argument is valid. If Jasmyn studies then she will ace her test. Jasmyn studied. Jasmyn aced her test. 43 Deductive Reasoning Example 5 Determine if the argument is valid. If Mike goes to work, then he will get home late. Mike got home late. No Conclusion – Cannot make a valid argument. 44 Now Try These If an angle is right, then it is not acute. TOP is a right angle. Valid or Invalid: TOP is not acute. If it is finger-licking good, then it is Kentucky Fried chicken. It is Kentucky Fried chicken. Valid or Invalid: It is finger-licking good. If you use Pro-Active, then you will not have acne. You don’t have acne. Valid or Invalid: You use Pro-Active 45 And more… If you chew gum at VGHS, then you could get lunch detention. You chew gum. Valid or Invalid: You could get lunch detention. If you go to Burger King, then you have it your way. You go to Burger King. Valid or Invalid: You have it your way. If you have an LG television, then life is good. Life is good. Valid or Invalid: You have an LG television. If you want to reach out and touch someone, then use AT & T. You want to reach out and touch someone. Valid or Invalid: You use AT & T. 46 WHY ARE FIRETRUCKS RED? Cause there's eight wheels on them and four people, and four plus eight is twelve, and twelve is a foot and a foot is a ruler, and Queen Elizabeth was a ruler, and Queen Elizabeth was also a ship, and the ship sails the sea and in the sea is fish and fish have fins, and the Finns fought the Russians and the Russians were red and that's why firetrucks are red. 47 Use the Law of Syllogism to complete the statement, ”If there is a fire, then __________________.” If the robot sets off a fire alarm, then it concludes there is a fire. If the robot senses high levels of smoke and heat, then it sets off a fire alarm. If the robot locates the fire, then the robot extinguishes the fire. If there is a fire, then the robot senses high levels of smoke and heat. If the robot concludes there is a fire, then it locates the fire. 48 Use the Law of Syllogism to complete the statement, “If an Old Lady swallowed a bat, then _______________.” From There was an Old Woman Who Swallowed a Bat by Lucille Colandro. If she swallowed an owl, then she swallowed a cat. If she swallowed a ghost, then she swallowed a goblin. If an Old Lady swallowed a bat, then she swallowed an owl. If she swallowed a wizard, then she yelled, “Trick of treat!” If she swallowed a cat, then she swallowed a ghost. If she swallowed a goblin, then she swallowed some bones. If she swallowed some bones, then she swallowed a wizard. 49 A nursery rhyme ends with “For want of a nail, the _________.” If the horseshoe is lost, then the horse will be lost. If the horse is lost, then the knight will be lost. If the horseshoe nail is lost, then the horseshoe will be lost. If the battle is lost, then the kingdom will be lost. If the knight is lost, then the battle will be lost. 50 Take out a sheet of paper. Write an if-then statement on your paper. It can be about anything. For example….If I forgot my homework today, then I get a 0. Pass the paper to the person at the desk numbered one higher than yours. Using the conclusion of the last sentence as your hypothesis, write another conditional. For example…If I got a 0, then my grade will drop. Pass your paper to the next person again. AN EXERCISE IN SYLLOGISM 51 Deductive Reasoning Law of Syllogism Given two linked conditional statements you can form one conditional statement. Given: Given: Conclusion: p→q q→r p→r 52 Deductive Reasoning Example 6 Determine if the argument is valid. If today is your birthday, then Joe will bake a cake. If Joe bakes a cake, then everyone will celebrate. If today is your birthday, then everyone will celebrate. 53 Deductive Reasoning Example 7 Determine if the argument is valid. If it is a square, then it has four sides. If it has four sides, then it is a quadrilateral. If it is a square, then it is a quadrilateral. 54 Now Try These Write a conclusion using the true statements. If no conclusion is possible, write no conclusion. If Jesse is late, then he is tardy. If he is tardy, then he will get lunch detention. Jesse is tardy…. If Casey is friendly, then she will have a date. If she has a date, then she will go to up-and-coming. Casey is going to up-and-coming. If Mary goes to Vista Grande, then she is a Spartan. If she is a Spartan, then she has school pride. Mary is a Spartan. 55 More… Write a conclusion using the true statements. If no conclusion is possible, write no conclusion. If Tim misses practice, then he cannot play in the game. Tim goes to practice. If Deb does her homework, then it will be graded. If the homework is graded, then it will help her pass. Deb did her homework. If Sara attends class every day, then she will have perfect attendance. If she has perfect attendance, then she will not have to take finals. If she does not have to take finals, then she will have get out of school 2 days early. Sara attends class every day. 56 If you are tall, then you play basketball. You are tall. You play basketball. You are not tall. You do not play basketball. If you are late, then you get lunch detention. You are tardy. You have lunch detention. You are not tardy. You do not have lunch detention. ANOTHER WAY TO DETERMINE VALIDITY USING VENN DIAGRAMS 57 Geometry 1 Unit 2 2.4 Reasoning with Properties from Algebra 58 Reasoning with Properties from Algebra Addition property If Subtraction property If a = b, then a – c = b – c Multiplication property If a = b, then a + c = b + c a = b, then ac = bc Division property If a = b, then a c b c 59 Reasoning with Properties from Algebra Reflexive property For Symmetric property If a=b, then b = a Transitive Property If any real number a, a = a a = b and b = c, then a = c Substitution property If a = b, then a can be substituted for b in any equation or expression Distributive property 2(x + y) = 2x + 2y 60 USING THE PROPERTY IN LIFE Property Example Addition Property Allowance Reflexive Property I am who I am. Substitution Property Cooking Ingredients Transitive Property If I’m as good as you and you’re as good as ….., then… Symmetric Property Turning About Face Multiplication Property Wages on an Hourly Basis Subtraction Property Payments Division Property Sharing Distributive Property Passing Out Rewards 61 USING THE PROPERTY IN GEOMETRY Property Example Addition Property If mTRY = 90°, then 10 + mTRY = _____ Reflexive Property M = _____ Substitution Property If mH = 120°, then mH + mI = 120 + mI Transitive Property If GR = EA and ____ = TM, then _____________. Symmetric Property CAT + 50 = 50 + CAT Multiplication Property If mTRY = 90°, 2(mTRY) = _____ Subtraction Property If mNOT = mTRY, then mNOT - LOW = mTRY - LOW. Division Property If mLOW = mTRY, then (mLOW)/2 = (mTRY)/2 Distributive Property If 2MAD +2 TRY = 90, then 2(MAD + TRY) = 90. 62 CAN YOU IDENTIFY THE PROPERTY? 1. Cut into strips and cut properties from examples. 2. Match the property to the example. Property Example Addition Property If mK = 10°, then 4(mK) = 40° Reflexive Property If mM = mA and mA = mT, then mM = mT. Substitution Property If 8(mV) = 120°, then mV = 15° Transitive Property If AB = 10 cm, then AB + 5 cm = 15 cm. Symmetric Property CAT = CAT Multiplication Property If TH = UM, then UM = TH Subtraction Property If TRY = 90°, then MIS + TRY = MIS + 90. Division Property If LOW + TRY = HAM + TRY, then LOW = HAM Distributive Property If 2(LOW + TRY) = 90, then 2LOW +2 TRY = 90. 63 Reasoning with Properties from Algebra Example 1 Solve 6x – 5 = 2x + 3 and write a reason for each step Statement 6x – 5 = 2x + 3 Reason Given 4x – 5 = 3 4x = 8 x=2 64 Reasoning with Properties from Algebra Example 2 2(x – 3) = 6x + 6 1. 2. 3. 4. 5. Given 65 Create Your Own Step Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 66 Another Try (but not on notes) Step Reason 1. 3x - 7= 5(2 + x) + 5 1. 2. 3x - 7 = 10 + 5x + 5 2. 3. 3x - 7 = 15 + 5x 3. 4. -22 = 2x 4. 5. 5. 2x = - 22 6. x = - 11 6. 67 Still Another Try (but not on notes) Step Reason 1. x = 4 + y 3x + 7(y + 3) = 53 1. 2. 3(4 + y) + 7(y + 3) = 53 2. 3. 12 + 3y + 7y + 21 = 53 3. 4. 33 + 10y = 53 4. 5. 5. 10y = 20 6. y = 2 6. 7. x = 4 + y 7. 8. x = 4 + 2 8. 9. x = 6 9. 68 Reasoning with Properties from Algebra Determine if the equations are valid or invalid. (x + 2)(x + 2) = x2 + 4 x3x3 -(x = x6 + y) = x – y 69 More Reasoning with Properties from Algebra (not on notes) Determine if the equations are valid or invalid. (x + 2)(x - 2) = x2 - 4 (x3)3 = x6 -5(x - y) = -5x + y 70 Reasoning with Properties from Algebra G Geometric Properties of Equality Reflexive property of equality For any segment AB, AB = AB Symmetric property of equality If mA mB, then mB mA Transitive property of equality B A P If AB = CD and CD = EF, then, AB = EF 71 Reasoning with Properties from Algebra Example 3 A B C D In the diagram, AB = CD. Show that AC = BD Statement Reason 1. AB = CD 2. AB + BC = BC + CD 3. AC = AB + BC 4. BD = BC + CD 5. AC = BD 72 A In the diagram, ABC = DBF. Show that ABD = CBF C D B F Statement Reason 73 Reasoning with Properties from Algebra Example 4 A (not on notes) B C D In the diagram, AC = DB. Show that AB = CD Statement Reason 1. AC = DB 2. BC = BC 3. AB + BC = AC 4. CD + BC = DB 5. AB + BC = CD + BC 6. AB = CD 74 A In the diagram, ABD = CBF. Show that ABC = DBF C Not on notes. D B F Statement Reason 75 Geometry 1 Unit 2 2.5: Proving Statements about Segments 76 Marking Diagrams LN MP 77 Proving Statements about Segments Key Terms: 2-column proof A way of proving a statement using a numbered column of statements and a numbered column of reasons for the statements Theorem A true statement that is proven by other true statements 78 Proving Statements about Segments Properties of Segment Congruence Reflexive For any segment AB, AB AB Symmetric If AB CD, then CD AB Transitive If AB EF and AB CD ,then CD EF 79 Proving Statements about Segments Example 1 K In triangle JKL, Given: LK = 5, JK = 5, JK = JL Prove: LK = JL J L Statement 1. Reason 1. Given 2. 3. 4. 2. Given 3. Substitution 4. Transitive Property of Congruence 80 Proving Statements about Segments (not on notes) Example 2 L Given: M is the midpoint of LN Prove: LM = ½ LN and MN = ½ LN Statement Reason 1. 1. Given 2. LM = MN 2. 3. LM + MN = LN 3. 4. LM + LM = LN 4. 5. 2 * LM = LN 5. 6. LM = ½ LN 6. 7. 7. M N 81 Proving Statements about Segments (not on notes) Example 3 W X Given: Collinear Points W, X, Y and Z Prove: WZ = WX + XY + YZ Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. Y Z 82 Proving Statements about Segments (not on notes) Example 5 M A T Given: Collinear Points M, A, T, and H Prove: MH = MA + AT + TH Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. H 83 Proving Statements about Segments (not on notes) Example 6 G R M B GR EA Prove: GR BC Given: EA TM Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. C TM MN N T M MN BC 84 Proving Statements about Segments Duplicating a Segment Tools Straight edge: Ruler or piece of wood or metal used for creating straight lines Compass: Tool used to create arcs and circles Steps 1. 2. 3. A C B D 4. 5. 6. Use a straight edge to draw a segment longer than segment AB Label point C on new segment Set compass at length of segment AB Place compass point at C and strike an arc on line segment Label intersection of arc and segment point D Segment CD is now congruent to segment AB85 NOT ON NOTES A AB B C D AB – CD CD 3AB – 2CD AB + CD 86 A 2AB B C D 3CD - AB 87 Let’s Try More Constructions Bisect an Angle Bisect a Segment 88 Geometry 1 Unit 2 2.6: Proving Statements about Angles 89 Proving Statements about Angles Properties of Angle Congruence Reflexive For any angle A, A A. Symmetric If A B, thenB A. Transitive If A Band B C , thenA C. 90 Proving Statements about Angles Right Angle Congruence Theorem All right angles are congruent. 91 Proving Statements about Angles Congruent Supplements Theorem If two angles are supplementary to the same angle, then they are congruent. If m1 m2 180 and m2 m3 180 , then1 3. 1 2 3 92 Visual of Supplementary Angles (not on notes) Sketch supplementary angles where one angle measures 135°. Sketch supplementary angles where one angle measures 30°. 93 Proving Statements about Angles Congruent Complements Theorem If two angles are complementary to the same angle, then the two angles are congruent. If m4 m5 90 and m5 m6 90 , then4 6. 5 6 4 94 Visual of Complementary Angles (not on notes) Sketch complementary angles where one angle measures 60°. Sketch complementary angles where one angle measures 35°. 95 Proving Statements about Angles Linear Pair Postulate If two angles form a linear pair, then they are supplementary. m1 m2 180 1 2 96 Visual of Linear Pair (not on notes) Sketch a linear pair where one angle measures 35°. Sketch a linear pair where one angle measures 135°. 97 Proving Statements about Angles Vertical Angles Theorem Vertical angles are congruent. 2 1 3 4 1 3, 2 4 98 Visual of Vertical Angles (not on notes) Sketch vertical angles which measure 40°. Sketch vertical angles which measure 110°. 99 Proving Statements about Angles Example 1 Given:1 2, 3 4, 2 3. B Prove:1 4 1 A Statement 1. 2. 3. 4. Reason 1. 2. 3. 4. 3 2 4 C 100 Proving Statements about Angles Example 2 Given: m1 63 , 1 3, 3 4 Prove: m4 63 Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 1 3 2 4 101 Proving Statements about Angles D Example 3 A Given: DAB, ABC are right angles ABC BDC Prove: DAB BDC Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. C B 102 Proving Statements about Angles Example 4 Given: m1 = 24º, m3 = 24º 1 and 2 are complementary 3 and 4 are complementary Prove: 2 4 1 2 Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 3 4 103 Proving Statements about Angles Example 5 the diagram m1 = 60º and BFD is right. Explain how to show m4 = 30º. In C B D 1 A 2 3 F 4 E 104 Proving Statements about Angles Example 6 1 and 2 are a linear pair, 2 and 3 are a linear pair Prove: 1 3 Given: 1 2 3 Statement Reason 1. 1. 2. 2. 3. 3. 105 Write a two-column proof Given: 8 = 5 Prove: 7 = 6 6 5 8 7 106