Algebraic and Geometric Proofs

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1 UNIT 7 Module 0 lgebraic and Geometric Proofs Contents Prep for MCC9-.G.CO.9 MCC9-..REI. MCC9-.G.CO.9 MCC9-.G.CO.9 MCC9-.G.CO.9 0- iconditional and Definitions lgebraic Proof Geometric Proof Task 0- Design Plans for Proofs Flowchart and Paragraph Proofs Ready to Go On? Module Quiz MTHEMTICL The Common Core Georgia Performance Standards for Mathematical Practice PRCTICES describe varieties of expertise that all students should seek to develop. Opportunities to develop these practices are integrated throughout this program. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 6 ttend to precision. 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning. 568 Module 0 lgebraic and Geometric Proofs

2 Unpacking the Standards Multilingual Glossary Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter. MCC9-.G.CO.9 Prove theorems about lines and angles. Key Vocabulary proof (demostración) n argument that uses logic to show that a conclusion is true. theorem (teorema) statement that has been proven. line (línea) n undefined term in geometry, a line is a straight path that has no thickness and extends forever. angle (ángulo) figure formed by two rays with a common endpoint. What It Means For You With just a few definitions, properties, and postulates, you can begin to prove simple theorems about line segments, linear pairs, right angles, vertical angles, and complementary or supplementary angles. Given: and are vertical angles. Prove: and are vertical angles. Given and are formed by intersecting lines. Def. of vert. s and are a linear pair. and are a linear pair. Def. of lin. pair. and are supplementary. and are supplementary. Lin. Pair Thm. Supps. Thm. Unpacking the Standards 569

3 0- iconditional and Definitions Essential Question: How can definitions be written as biconditional statements? Objective Write and analyze biconditional statements. Vocabulary biconditional statement definition polygon triangle quadrilateral Who uses this? gardener can plan the color of the hydrangeas she plants by checking the ph of the soil. The ph of a solution is a measure of the concentration of hydronium ions in the solution. If a solution has a ph less than 7, it is an acid. lso, if a solution is an acid, it has a ph less than cidic Neutral asic The biconditional p if and only if q can also be written as p iff q or p q. When you combine a conditional statement and its converse, you create a biconditional statement. biconditional statement is a statement that can be written in the form p if and only if q. This means if p, then q and if q, then p. means and So you can define an acid with the following biconditional statement: solution is an acid if and only if it has a ph less than 7. Prep for MCC9-.G.CO.9 Online Video Tutor Identifying the Conditionals within a iconditional Statement Write the conditional statement and converse within each biconditional. Two angles are congruent if and only if their measures are equal. Let p and q represent the following. p : Two angles are congruent. q : Two angle measures are equal. The two parts of the biconditional p q are p q and q p. Conditional: If two angles are congruent, then their measures are equal. Converse: If two angle measures are equal, then the angles are congruent. solution is a base it has a ph greater than 7. Let x and y represent the following. x : solution is a base. y : solution has a ph greater than 7. The two parts of the biconditional x y are x y and y x. Conditional: If a solution is a base, then it has a ph greater than 7. Converse: If a solution has a ph greater than 7, then it is a base. Write the conditional statement and converse within each biconditional. a. n angle is acute iff its measure is greater than 0 and less than 90. b. Cho is a member if and only if he has paid the $5 dues. Steffan Hauser/botanikfoto/lamy 570 Module 0 lgebraic and Geometric Proofs

4 Prep for MCC9-.G.CO.9 Online Video Tutor Writing a iconditional Statement For each conditional, write the converse and a biconditional statement. If x + 5 =, then x =. Converse: If x =, then x + 5 =. iconditional: x + 5 = if and only if x =. If a point is a midpoint, then it divides the segment into two congruent segments. Converse: If a point divides a segment into two congruent segments, then the point is a midpoint. iconditional: point is a midpoint if and only if it divides the segment into two congruent segments. For each conditional, write the converse and a biconditional statement. a. If the date is July 4th, then it is Independence Day. b. If points lie on the same line, then they are collinear. For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. MCC.MP. Online Video Tutor nalyzing the Truth Value of a iconditional Statement Determine if each biconditional is true. If false, give a counterexample. square has a side length of 5 if and only if it has an area of 5. Conditional: If a square has a side length of 5, then it has an area of 5. The conditional is true. Converse: If a square has an area of 5, then it has a side length of 5. The converse is true. Since the conditional and its converse are true, the biconditional is true. The number n is a positive integer n is a natural number. Conditional: If n is a positive integer, then n is a natural number. The conditional is true. Converse: If n is a natural number, then n is a positive integer. The converse is false. If n =, then n =, which is not an integer. ecause the converse is false, the biconditional is false. Determine if each biconditional is true. If false, give a counterexample. a. n angle is a right angle iff its measure is 90. b. y = -5 y = 5 In geometry, biconditional statements are used to write definitions. definition is a statement that describes a mathematical object and can be written as a true biconditional. Most definitions in the glossary are not written as biconditional statements, but they can be. The if and only if is implied. 0- iconditional and Definitions 57

5 In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments. Each segment intersects exactly two other segments only at their endpoints, and no two segments with a common endpoint are collinear. Polygons Not Polygons triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon. Triangles Polygons Quadrilaterals sides 4 sides Think of definitions as being reversible. Postulates, however, are not necessarily true when reversed. good, precise definition can be used forward and backward. For example, if a figure is a quadrilateral, then it is a four-sided polygon. If a figure is a four-sided polygon, then it is a quadrilateral. To make sure a definition is precise, it helps to write it as a biconditional statement. MCC.MP.6 Online Video Tutor 4 Writing Definitions as iconditional Write each definition as a biconditional. triangle is a three-sided polygon. figure is a triangle if and only if it is a three-sided polygon. segment bisector is a ray, segment, or line that divides a segment into two congruent segments. ray, segment, or line is a segment bisector if and only if it divides a segment into two congruent segments. Write each definition as a biconditional. 4a. quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 80. THINK ND DISCUSS MCC.MP.6 MTHEMTICL PRCTICES. How do you determine if a biconditional statement is true or false?. Compare a triangle and a quadrilateral.. GET ORGNIZED Copy and complete the iconditional graphic organizer. Use the definition of a polygon to write a conditional, converse, Conditional and biconditional in the appropriate boxes. Converse 57 Module 0 lgebraic and Geometric Proofs

6 0- Exercises Homework Help GUIDED PRCTICE. Vocabulary How is a biconditional statement different from a conditional statement? SEE Write the conditional statement and converse within each biconditional.. Perry can paint the entire living room if and only if he has enough paint.. Your medicine will be ready by 5 P.M. if and only if you drop your prescription off by 8.M. SEE For each conditional, write the converse and a biconditional statement. 4. If a student is a sophomore, then the student is in the tenth grade. 5. If two segments have the same length, then they are congruent. SEE Multi-Step Determine if each biconditional is true. If false, give a counterexample. 6. xy = 0 x = 0 or y = figure is a quadrilateral if and only if it is a polygon. SEE 4 Write each definition as a biconditional. 8. Parallel lines are two coplanar lines that never intersect. 9. hummingbird is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. Independent Practice For See Exercises Example Online Extra Practice PRCTICE ND PROLEM SOLVING Write the conditional statement and converse within each biconditional. 0. Three points are coplanar if and only if they lie in the same plane.. parallelogram is a rectangle if and only if it has four right angles.. lunar eclipse occurs if and only if Earth is between the Sun and the Moon. For each conditional, write the converse and a biconditional statement.. If today is Saturday or Sunday, then it is the weekend. 4. If Greg has the fastest time, then he wins the race. 5. If a triangle contains a right angle, then it is a right triangle. Multi-Step Determine if each biconditional is true. If false, give a counterexample. 6. Felipe is a swimmer if and only if he is an athlete. 7. The number n is even if and only if n is an integer. Getty/Stone Write each definition as a biconditional. 8. circle is the set of all points in a plane that are a fixed distance from a given point. 9. catcher is a baseball player who is positioned behind home plate and who catches throws from the pitcher. 0- iconditional and Definitions 57

7 lgebra Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 0. If a = b, then a = b.. If x - =, then 4 _ 5 x + 8 =.. If y = 64, then y = 4.. If x > 0, then x > 0. Use the diagrams to write a definition for each figure iology Equilateral triangle Not an equilateral triangle Square Not squares White blood cells live less than a few weeks. drop of blood can contain anywhere from 7000 to 5,000 white blood cells. 6. iology White blood cells are cells that defend the body against invading organisms by engulfing them or by releasing chemicals called antibodies. Write the definition of a white blood cell as a biconditional statement. Explain why the given statement is not a definition. 7. n automobile is a vehicle that moves along the ground. 8. calculator is a machine that performs computations with numbers. 9. n angle is a geometric object formed by two rays. Chemistry Use the table for Exercises 0. Determine if a true biconditional statement can be written from each conditional. 0. If a solution has a ph of 4, then it is tomato juice.. If a solution is bleach, then its ph is.. If a solution has a ph greater than 7, then it is not battery acid. ph Examples attery cid cid rain, tomato juice Saliva Sea water leach, oven cleaner Drain cleaner Real-World Connections Complete each statement to form a true biconditional.. The circumference of a circle is 0π if and only if its radius is?. 4. Four points in a plane form a? if and only if no three of them are collinear. 5. Critical Thinking Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 6. Write bout It Use the definition of an angle bisector to explain what is meant by the statement good definition is reversible. 7. a. Write I say what I mean and I mean what I say as conditionals. b. Explain why the biconditional statement implied by lice is false. Then you should say what you mean, the March Hare went on. I do, lice hastily replied; at least at least I mean what I say that s the same thing, you know. (tl) Nibsc/Photo Researchers, Inc.; (bl) Victoria Smith/HMH; (br) The Granger Collection 574 Module 0 lgebraic and Geometric Proofs

8 TEST PREP 8. Which is a counterexample for the biconditional n angle measures 80 if and only if the angle is acute? m S = 60 m S = 5 m S = 90 m S = Which biconditional is equivalent to the spelling phrase I before E except after C? The letter I comes before E if and only if I follows C. The letter E comes before I if and only if E follows C. The letter E comes before I if and only if E comes before C. The letter I comes before E if and only if I comes before C. 40. Which conditional statement can be used to write a true biconditional? If a number is divisible by 4, then it is even. If a ratio compares two quantities measured in different units, the ratio is a rate. If two angles are supplementary, then they are adjacent. If an angle is right, then it is not acute. 4. Short Response Write the two conditional statements that make up the biconditional You will get a traffic ticket if and only if you are speeding. Is the biconditional true or false? Explain your answer. CHLLENGE ND EXTEND 4. Critical Thinking Describe what the Venn diagram of a true biconditional statement looks like. How does this support the idea that a definition can be written as a true biconditional? 4. Consider the conditional If an angle measures 05, then the angle is obtuse. a. Write the inverse of the conditional statement. b. Write the converse of the inverse. c. How is the converse of the inverse related to the original conditional? d. What is the truth value of the biconditional statement formed by the inverse of the original conditional and the converse of the inverse? Explain. 44. Suppose,, C, and D are coplanar, and,, and C are not collinear. What is the truth value of the biconditional formed from the true conditional If m D + m DC = m C, then D is in the interior of C? Explain. 45. Find a counterexample for n is divisible by 4 if and only if n is even. MTHEMTICL PRCTICES FOCUS ON MTHEMTICL PRCTICES 46. Error nalysis lake wrote, n angle is obtuse if and only if it is not an acute angle. What did lake overlook? Complete this statement: n angle is obtuse if and only if?. 47. Precision Is the following a good, precise definition? Explain. If a polygon has exactly three acute angles, then it is a triangle. 48. Justify conditional statement is true, and so is its inverse. Explain why the conditional statement is a valid biconditional statement. 0- iconditional and Definitions 575

9 0- lgebraic Proof Essential Question: What kinds of justifications can you use in writing algebraic and geometric proofs? Objectives Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Vocabulary proof Who uses this? Game designers and animators solve equations to simulate motion. (See Example.) proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. If you ve ever solved an equation in lgebra, then you ve already done a proof! n algebraic proof uses algebraic properties such as the properties of equality and the Distributive Property. The Distributive Property states that a (b + c) = ab + ac. Properties of Equality ddition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality If a = b, then ac = bc. If a = b and c 0, then a c = b c. a = a Symmetric Property of Equality If a = b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Substitution Property of Equality If a = b, then b can be substituted for a in any expression. s you have learned, if you start with a true statement and each logical step is valid, then your conclusion is valid. n important part of writing a proof is giving justifications to show that every step is valid. For each justification, you can use a definition, postulate, property, or a piece of information that is given. MCC9-..REI. Online Video Tutor Solving an Equation in lgebra Solve the equation -5 = n +. Write a justification for each step. -5 = n + Given equation - - Subtraction Property of Equality -6 = n Simplify. _-6 n_ Division Property of Equality = - = n Simplify. n = - Symmetric Property of Equality. Solve the equation _ t = -7. Write a justification for each step. Getty Images 576 Module 0 lgebraic and Geometric Proofs

10 MTHEMTICL PRCTICES MCC9-..REI. Online Video Tutor Make sense of problems and persevere in solving them. Problem-Solving pplication To simulate the motion of an object in a computer game, the designer uses the formula sr =.6p to find the number of pixels the object must travel during each second of animation. In the formula, s is the desired speed of the object in kilometers per hour, r is the scale of pixels per meter, and p is the number of pixels traveled per second. The graphics in a game are based on a scale of 6 pixels per meter. The designer wants to simulate a vehicle moving at 75 km/h. How many pixels must the vehicle travel each second? Solve the equation for p and justify each step. Understand the Problem The answer will be the number of pixels traveled per second. List the important information: sr =.6p s = 75 km/h p: pixels traveled per second r = 6 pixels per meter Make a Plan Substitute the given information into the formula and solve. Solve sr =.6p (75)(6) =.6p 450 =.6p _ 450 _.6 =.6p.6 5 = p p = 5 pixels Given equation Substitution Property of Equality Simplify. Division Property of Equality Simplify. Symmetric Property of Equality 4 Look ack Check your answer by substituting it back into the original formula. sr =.6p (75) (6) =.6 (5) 450 = 450 represents the length of, so you can think of as a variable representing a number.. What is the temperature in degrees Celsius C when it is 86 F? Solve the equation C = _ 5 (F - ) for C and justify each step. 9 Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. 0- lgebraic Proof 577

11 Prep for MCC9-.G.CO.9 Online Video Tutor Solving an Equation in Geometry 5x - 4 Write a justification for each step. K x + L x - M KM = KL + LM 5x - 4 = (x + ) + (x - ) 5x - 4 = x + x - 4 = x = 6 x =. Write a justification for each step. m C = m D + m DC 8x = (x + 5) + (6x - 6) 8x = 9x - -x = - x = Segment ddition Postulate Substitution Property of Equality Simplify. Subtraction Property of Equality ddition Property of Equality Division Property of Equality (x + 5) D (6x -6) C m C = 8x You have learned that segments with equal lengths are congruent and angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Properties of Congruence SYMOLS Reflexive Property of Congruence figure figure (Reflex. Prop. of ) EF EF Numbers are equal (=) and figures are congruent ( ). Symmetric Property of Congruence If figure figure, then figure figure. (Sym. Prop. of ) Transitive Property of Congruence If figure figure and figure figure C, then figure figure C. (Trans. Prop. of ) If, then. If PQ RS and RS TU, then PQ TU. Prep for MCC9-.G.CO.9 4 Identifying Properties of Equality and Congruence Identify the property that justifies each statement. m = m Reflex. Prop. of = XY VW, so VW XY. Sym. Prop. of C C C D, and. So. Reflex. Prop. of Trans. Prop. of Online Video Tutor Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4b. 94 = 94 4c. 0 = a, and a = x. So 0 = x. 4d. Y, so Y. 578 Module 0 lgebraic and Geometric Proofs

12 MCC.MP.8 THINK ND DISCUSS. Tell what property you would use to solve the equation k _ 6 =.5.. Explain when to use a congruence symbol instead of an equal sign. MTHEMTICL PRCTICES. GET ORGNIZED Copy and complete the Property Equality Congruence graphic organizer. In each box, write an Reflexive example of the property, using the Symmetric correct symbol. Transitive 0- Exercises GUIDED PRCTICE. Vocabulary Write the definition of proof in your own words. Homework Help SEE Multi-Step Solve each equation. Write a justification for each step.. y + = 5. t -. = -8. x + 4. p - 0 = -4p _ - = 8 6. _ n = _ 7. 0 = (r - ) SEE 8. Nutrition my s favorite breakfast cereal has 0 Calories per serving. The equation C = 9f + 90 relates the grams of fat f in one serving to the Calories C in one serving. How many grams of fat are in one serving of the cereal? Solve the equation for f and justify each step. 9. Movie Rentals The equation C = $ $0.89m relates the number of movie rentals m to the monthly cost C of a movie club membership. How many movies did Elias rent this month if his membership cost $.98? Solve the equation for m and justify each step. SEE Write a justification for each step. 0. 5y + 6 y +. C = C 9n 5 P n Q 5 R PQ + QR = PR 5y + 6 = y + n + 5 = 9n -5 y + 6 = 5 = 6n -5 y = 5 y = 5 0 = 6n 5 = n SEE 4 Identify the property that justifies each statement... m = m, and m = m 4. So m = m x = y, so y = x. 5. ST YZ, and YZ PR. So ST PR. 0- lgebraic Proof 579

13 Independent Practice For See Exercises Example PRCTICE ND PROLEM SOLVING Multi-Step Solve each equation. Write a justification for each step. 6. 5x - = 4 (x + ) 7..6 =.n 8. z_ - = (h + ) = y + 7 = -9. _ ( p - 6) =. Ecology The equation T = 0.0c b relates the numbers of cans c and bottles b collected in a recycling rally to the total dollars T raised. How many cans were collected if $47 was raised and 50 bottles were collected? Solve the equation for c and justify each step. Online Extra Practice Write a justification for each step.. m XYZ = m + m 4. m WYV = m + m 4n - 6 = 58 + (n - ) 5n = (n - ) n - 6 = n n = n n - 6 = 46 5n = n + 5 n = 5 n = 5 n = 6 n = 6 X V (n - ) 58 (n - ) W Y Z m WYV = 5n m XYZ = (4n - 6) Identify the property that justifies each statement. 5. KL PR, so PR KL = 4 7. If a = b and b = 0, then a = figure figure 9. Estimation Round the numbers in the equation (.x ) = 94.6 to the nearest whole number and estimate the solution. Then solve the equation, justifying each step. Compare your estimate to the exact solution. Use the indicated property to complete each statement. 0. Reflexive Property of Equality: x - =?. Transitive Property of Congruence: If X and X T, then?.. Symmetric Property of Congruence: If C NP, then?.. Recreation The north campground is midway between the Northpoint Overlook and the waterfall. Use the midpoint formula to find the values of x and y, and justify each step. 4. usiness computer repair technician charges $5 for each job plus $ per hour of labor and 0% of the cost of parts. The total charge for a -hour job was $ What was the cost of parts for this job? Write and solve an equation and justify each step in the solution. (, y) Northpoint Overlook (, 5) North campground (x, ) Waterfall 5. Finance Morgan spent a total of $,7.65 on her car last year. She spent $9.50 on registration, $79.96 on maintenance, and $98 on insurance. She spent the remaining money on gas. She drove a total of 0,80 miles. a. How much on average did the gas cost per mile? Write and solve an equation and justify each step in the solution. b. What if? Suppose Morgan s car averages miles per gallon of gas. How much on average did Morgan pay for a gallon of gas? 6. Critical Thinking Use the definition of segment congruence and the properties of equality to show that all three properties of congruence are true for segments. 580 Module 0 lgebraic and Geometric Proofs

14 Real-World Connections 7. Recall from lgebra that the Multiplication and Division Properties of Inequality tell you to reverse the inequality sign when multiplying or dividing by a negative number. a. Solve the inequality x and write a justification for each step. b. Solve the inequality -x > 6 and write a justification for each step. 8. Write bout It Compare the conclusion of a deductive proof and a conjecture based on inductive reasoning. TEST PREP 9. Which could NOT be used to justify the statement CD? Definition of congruence Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence 40. club membership costs $5 plus $ each time t the member uses the pool. Which equation represents the total cost C of the membership? 5 = C + t C + 5 = t C = 5 + t C = 5t + 4. Which statement is true by the Reflexive Property of Equality? x = 5 CD = CD RT TR CD = CD 4. Gridded Response In the triangle, m + m + m = 80. If m = m and m = m, find m in degrees. CHLLENGE ND EXTEND 4. In the gate, P = Q, Q = R, and P = 8 in. Find PR, and justify each step. 44. Critical Thinking Explain why there is no ddition Property of Congruence. 45. lgebra Justify each step in the solution of the inequality 7 - x > 9. P C N S T D Q R (tl) rand X Pictures/Getty Images; (cr) Paul. Souders/CORIS MTHEMTICL PRCTICES FOCUS ON MTHEMTICL PRCTICES 46. Reasoning Petra is writing an algebraic proof. She states that x = 9 is true because of the Symmetric Property of Equality. What was her statement just prior to that? Explain. 47. Draw Conclusions Students are asked to prove that n =.5 is a solution of n - = 5. Kevyn used the ddition Property of Equality followed by the Division Property of Equality. Carla used the Substitution Property of Equality and simplified the expression. re both proofs valid? Explain. 0- lgebraic Proof 58

15 0- Geometric Proof Essential Question: How can you organize the deductive reasoning of a geometric proof? Objectives Write two-column proofs. Prove geometric theorems by using deductive reasoning. Vocabulary theorem two-column proof Who uses this? To persuade your parents to increase your allowance, your argument must be presented logically and precisely. When writing a geometric proof, you use deductive reasoning to create a chain of logical steps that move from the hypothesis to the conclusion of the conjecture you are proving. y proving that the conclusion is true, you have proven that the original conjecture is true. Hypothesis Definitions Postulates Properties Theorems Conclusion MCC9-.G.CO.9 Online Video Tutor When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Writing Justifications Write a justification for each step, given that and are complementary and C. C. and are complementary. Given information. m + m = 90 Def. of comp.. C Given information 4. m = m C Def. of 5. m C + m = 90 Subst. Prop. of = Steps, 4 6. C and are complementary. Def. of comp.. Write a justification for each step, given that is the midpoint of C and EF.. is the midpoint of C. C. C F. EF 4. C EF E theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. Theorem 0-- THEOREM HYPOTHESIS CONCLUSION Linear Pair Theorem If two angles form a linear pair, then they are supplementary. and form a linear pair. and are supplementary. REL LIFE DVENTURES c 004 GarLanco. Reprinted with permission of UNIVERSL PRESS SYNDICTE. ll rights reserved. 58 Module 0 lgebraic and Geometric Proofs

16 Theorem THEOREM HYPOTHESIS CONCLUSION 0-- Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. and are supplementary. and are supplementary. geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. MCC9-.G.CO.9 Completing a Two-Column Proof Fill in the blanks to complete a two-column proof of the Linear Pair Theorem. Given: and form a linear pair. Prove: and are supplementary. Proof: C Online Video Tutor linear pair of angles is a pair of angles in the same plane that have a common vertex and a common side, no common interior points, and noncommon sides that are opposite rays.. and form a linear pair.. and C form a line.. m C = a.? 5. b.? 6. and are supplementary.. Given. Def. of lin. pair. Def. of straight 4. dd. Post. 5. Subst. Steps, 4 6. c.? Use the existing statements and reasons in the proof to fill in the blanks. a. m + m = m C The dd. Post. is given as the reason. b. m + m = 80 Substitute 80 for m C. c. Def. of supp. The measures of supp. add to 80 by def.. Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: and are supplementary, and and are supplementary. Prove: Proof:. a.?. m + m = 80 m + m = 80. b.? 4. m = m 5. m = m 6. d.?. Given. Def. of supp.. Subst. 4. Reflex. Prop. of = 5. c.? 6. Def. of 0- Geometric Proof 58

17 efore you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. Theorems THEOREM HYPOTHESIS CONCLUSION 0-- Right ngle Congruence Theorem ll right angles are congruent. and are right angles Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. and are complementary. and are complementary. MCC9-.G.CO.9 Writing a Two-Column Proof from a Plan Use the given plan to write a two-column proof of the Right ngle Congruence Theorem. Given: and are right angles. Prove: Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles. Proof: Online Video Tutor. and are right angles.. m = 90, m = 90. m = m 4.. Given. Def. of rt.. Trans. Prop. of = 4. Def. of. Use the given plan to write a two-column proof of one case of the Congruent Complements Theorem. Given: and are complementary, and and are complementary. Prove: Plan: The measures of complementary angles add to 90 by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that. The Proof Process. Write the conjecture to be proven.. Draw a diagram to represent the hypothesis of the conjecture.. State the given information and mark it on the diagram. 4. State the conclusion of the conjecture in terms of the diagram. 5. Plan your argument and prove the conjecture. 584 Module 0 lgebraic and Geometric Proofs

18 THINK ND DISCUSS. Which step in a proof should match the Prove statement?. Why is it important to include every logical step in a proof?. List four things you can use to justify a step in a proof. 4. GET ORGNIZED Copy and complete the graphic organizer. In each box, describe the steps of the proof process MCC.MP. MTHEMTICL PRCTICES 0- Exercises Homework Help GUIDED PRCTICE Vocabulary pply the vocabulary from this lesson to answer each question.. In a two-column proof, you list the? in the left column and the? in the right column. (statements or reasons).? is a statement you can prove. (postulate or theorem) SEE. Write a justification for each step, given that m = 60 and m = m.. m = 60, m = m. m = (60 ). m = 0 4. m + m = m + m = and are supplementary. SEE 4. Fill in the blanks to complete the two-column proof. Given: Prove: and are supplementary. Proof:.. m = m. b.? 4. m + m = m + m = d.?. Given. a.?. Lin. Pair Thm. 4. Def. of supp. 5. c.? Steps, 4 6. Def. of supp. SEE 5. Use the given plan to write a two-column proof. Given: X is the midpoint of Y, and Y is the midpoint of X. Prove: X Y Plan: y the definition of midpoint, X XY, and XY Y. Use the Transitive Property to conclude that X Y. X Y 0- Geometric Proof 585

19 Independent Practice For See Exercises Example PRCTICE ND PROLEM SOLVING 6. Write a justification for each step, given that X bisects C and m XC = 45.. X bisects C.. X XC. m X = m XC 4. m XC = m X = m X + m XC = m C = m C = m C 9. C is a right angle. X C Fill in the blanks to complete each two-column proof. Online Extra Practice 7. Given: and are supplementary, and and 4 are supplementary. Prove: 4 4 Proof:. and are supplementary. and 4 are supplementary.. a.?. m + m = m + m m = m 6. c.? Given. Def. of supp.. b.? 4. Given 5. Def. of 6. Subtr. Prop. of = Steps, 5 7. d.? 8. Given: C is a right angle. Prove: and are complementary. Proof: C. C is a right angle.. m C = 90. b.? 4. m + m = c.? 7. m + m = e.?. Given. a.?. dd. Post. 4. Subst. Steps, 5. Given 6. Def. of 7. d.? Steps 4, 6 8. Def. of comp. Use the given plan to write a two-column proof. 9. Given: E CE, DE E Prove: CD Plan: Use the definition of congruent segments to write the given information in terms of lengths. Then use the Segment ddition Postulate to show that = CD and thus CD. E D C 586 Module 0 lgebraic and Geometric Proofs

20 Use the given plan to write a two-column proof. 0. Given: and are complementary, and and 4 are complementary. 4 Prove: Plan: Since and are complementary and and 4 are complementary, both pairs of angle measures add to 90. Use substitution to show that the sums of both pairs are equal. Since 4, their measures are equal. Use the Subtraction Property of Equality and the definition of congruent angles to conclude that. 4 Engineering Find each angle measure.. m 48. m 6. m The Oresund ridge, at 7845 meters long, is the world s longest single bridge. It carries both rail and car traffic between Denmark and Sweden. 4. Engineering The Oresund ridge, which connects the countries of Denmark and Sweden, was completed in 999. If, which theorem can you use to conclude that 4? 5. Critical Thinking Explain why there are two cases to consider when proving the Congruent Supplements Theorem and the Congruent Complements Theorem. Tell whether each statement is sometimes, always, or never true. 6. n angle and its complement are congruent. 7. pair of right angles forms a linear pair. 8. n angle and its complement form a right angle. 9. linear pair of angles is complementary. 4 lgebra Find the value of each variable. 0.. (9x - (4n + 5) (8n - 6) 5) (8.5x + ). 4z (z + 6) (cl) Corbis Images; (cr) Nordicphotos/lamy; (bl) rand X Pictures/Getty Images Real-World Connections. Write bout It How are a theorem and a postulate alike? How are they different? 4. Sometimes you may be asked to write a proof without a specific statement of the Given and Prove information being provided for you. For each of the following situations, use the triangle to write a Given and Prove statement. a. The segment connecting the midpoints of two sides of a triangle is half as long as the third side. b. The acute angles of a right triangle are complementary. c. In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Y C 0- Geometric Proof 587 X

21 TEST PREP 5. Which theorem justifies the conclusion that 4? Linear Pair Theorem Congruent Supplements Theorem Congruent Complements Theorem Right ngle Congruence Theorem 4 6. What can be concluded from the statement m + m = 80? and are congruent. and are supplementary. and are complementary. and form a linear pair. 7. Given: Two angles are complementary. The measure of one angle is 0 less than the measure of the other angle. Conclusion: The measures of the angles are 85 and 95. Which statement is true? The conclusion is correct because 85 is 0 less than 95. The conclusion is verified by the first statement given. The conclusion is invalid because the angles are not congruent. The conclusion is contradicted by the first statement given. CHLLENGE ND EXTEND 8. Write a two-column proof. Given: m LN = 0, m = 5 Prove: M bisects LN. L M N Multi-Step Find the value of the variable and the measure of each angle (a +.5) (4x (a +.5) - 6) (-x + 9x) (.5a - 5) MTHEMTICL PRCTICES FOCUS ON MTHEMTICL PRCTICES. Proof The left side of a two-column proof is shown along with the supporting figure. Identify any given. Justify your choices. R W. m WST = 46. m RSW = 64. m RST = 0 S T 4. WST is an acute angle.. Communication Lynne wants to prove that if two angles form a linear pair, then they are supplementary. Can she use the Linear Pair Theorem as a reason to justify the statement? Why or why not? 588 Module 0 lgebraic and Geometric Proofs

22 0- Use with Geometric Proof ctivity MTHEMTICL PRCTICES Prove the Common ngles Theorem. Given: X CXD Prove: XC XD Design Plans for Proofs Sometimes the most challenging part of writing a proof is planning the logical steps that will take you from the Given statement to the Prove statement. Like working a jigsaw puzzle, you can start with any piece. Write down everything you know from the Given statement. If you don t see the connection right away, start with the Prove statement and work backward. Then connect the pieces into a logical order. Construct viable arguments and critique the reasoning of others. C Start by considering the difference in the Given and Prove statements. How does X compare to XC? How does CXD compare to XD? In both cases, XC is combined with the first angle to get the second angle. The situation involves combining adjacent angle measures, so list any definitions, properties, postulates, and theorems that might be helpful. Definition of congruent angles, ngle ddition Postulate, properties of equality, and Reflexive, Symmetric, and Transitive Properties of Congruence Start with what you are given and what you are trying to prove and then work toward the middle. X CXD The first reason will be Given. m X = m CXD Def. of?????? m XC = m XD??? XC XD The last statement will be the Prove statement. 4 ased on Step, XC is the missing piece in the middle of the logical flow. So write down what you know about XC. XC XC Reflex. Prop. of m XC = m XC Reflex. Prop. of = 5 Now you can see that the ngle ddition Postulate needs to be used to complete the proof. m X + m XC = m XC dd. Post. m XC + m CXD = m XD dd. Post. 6 Use the pieces to write a complete two-column proof of the Common ngles Theorem. X D MCC9-.G.CO.9 Prove theorems about lines and angles. Try This. Describe how a plan for a proof differs from the actual proof.. Write a plan and a two-column proof.. Write a plan and a two-column proof. Given: D bisects C. Given: LXN is a right angle. Prove: m = m C D Prove: and are M complementary. C L X N Geometry Task Design Plans for Proofs 589

23 0-4 Flowchart and Paragraph Proofs Essential Question: What are some formats you can use to organize geometric proofs? Objectives Write flowchart and paragraph proofs. Prove geometric theorems by using deductive reasoning. Vocabulary flowchart proof paragraph proof Why learn this? Flowcharts make it easy to see how the steps of a process are linked together. second style of proof is a flowchart proof, which uses boxes and arrows to show the structure of the proof. The steps in a flowchart proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. Theorem 0-4- Common Segments Theorem THEOREM HYPOTHESIS CONCLUSION Given collinear points,, C, and D arranged as shown, if CD, then C D. C D CD C D MCC9-.G.CO.9 Online Video Tutor Reading a Flowchart Proof Use the given flowchart proof to write a two-column proof of the Common Segments Theorem. Given: CD Prove: C D C D Flowchart proof: CD C = C Given Reflex. Prop. of = + C = C, C + CD = D Seg. dd. Post. nimated Math = CD Def. of segs. + C = C + CD dd. Prop. of = C = D Subst. C D Def. of segs. Two-column proof:. CD. = CD. C = C 4. + C = C + CD 5. + C = C, C + CD = D 6. C = D 7. C D 590 Module 0 lgebraic and Geometric Proofs. Given. Def. of segs.. Reflex. Prop. of = 4. dd. Prop. of = 5. Seg. dd. Post. 6. Subst. 7. Def. of segs. HRW Photo

24 . Use the given flowchart proof to write a two-column proof. Given: RS = UV, ST = TU Prove: RT TV Flowchart proof: R S T U V RS = UV, ST = TU Given RS + ST = RT, TU + UV = TV Seg. dd. Post. RS + ST = TU + UV dd. Prop of = RT = TV Subst. RT TV Def. of segs. MCC9-.G.CO.9 Writing a Flowchart Proof Use the given two-column proof to write a flowchart proof of the Converse of the Common Segments Theorem. Given: C D Prove: CD C D Two-column proof: Online Video Tutor. C D. C = D. + C = C, C + CD = D 4. + C = C + CD 5. C = C 6. = CD 7. CD. Given. Def. of segs.. Seg. dd. Post. 4. Subst. Steps, 5. Reflex. Prop. of = 6. Subtr. Prop. of = 7. Def. of segs. Like the converse of a conditional statement, the converse of a theorem is found by switching the hypothesis and conclusion. Flowchart proof: C D Given C = D Def. of segs. + C = C, C + CD = D Seg. dd. Post + C = C + CD C = C Reflex. Prop. of = = CD Subst. Subtr. Prop. of = CD Def. of segs.. Use the given two-column proof to write a flowchart proof. Given: 4 Prove: m = m 4 Two-column proof:. 4. and are supplementary. and 4 are supplementary.. 4. m = m. Given. Lin. Pair Thm.. Supps. Thm. 4. Def. of 0-4 Flowchart and Paragraph Proofs 59

25 paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. lthough this style of proof is less formal than a two-column proof, you still must include every step. Theorems THEOREM HYPOTHESIS CONCLUSION 0-4- Vertical ngles Theorem Vertical angles are congruent. and are vertical angles If two congruent angles are supplementary, then each angle is a right angle. ( supp. rt. ) and are supplementary. and are right angles. MCC9-.G.CO.9 Reading a Paragraph Proof Use the given paragraph proof to write a two-column proof of the Vertical ngles Theorem. Given: and are vertical angles. Prove: Online Video Tutor Paragraph proof: and are vertical angles, so they are formed by intersecting lines. Therefore and are a linear pair, and and are a linear pair. y the Linear Pair Theorem, and are supplementary, and and are supplementary. So by the Congruent Supplements Theorem,. Two-column proof: Vertical angles are angles formed by intersecting lines that share a vertex but do not have common sides.. and are vertical angles.. and are formed by intersecting lines.. and are a linear pair. and are a linear pair. 4. and are supplementary. and are supplementary. 5.. Given. Def. of vert.. Def. of lin. pair 4. Lin. Pair Thm. 5. Supps. Thm. 59 Module 0 lgebraic and Geometric Proofs. Use the given paragraph proof to write a two-column proof. Given: WXY is a right angle. Prove: and are complementary. Paragraph proof: Since WXY is a right W angle, m WXY = 90 by the definition of a right angle. y the ngle ddition Postulate, m WXY = m + m. y substitution, m + m = 90. Since, m = m by the definition of congruent angles. Using substitution, m + m = 90. Thus by the definition of complementary angles, and are complementary. X Y Z lamy Images

26 Writing a Proof Claire Jeffords Riverbend High School When I have to write a proof and I don t see how to start, I look at what I m supposed to be proving and see if it makes sense. If it does, I ask myself why. Sometimes this helps me to see what the reasons in the proof might be. If all else fails, I just start writing down everything I know based on the diagram and the given statement. y brainstorming like this, I can usually figure out the steps of the proof. You can even write each thing on a separate piece of paper and arrange the pieces of paper like a flowchart. MCC9-.G.CO.9 4 Writing a Paragraph Proof Use the given two-column proof to write a paragraph proof of Theorem Given: and are supplementary. Prove: and are right angles. Two-column proof: Online Video Tutor. and are supplementary.. m + m = 80. m = m 4. m + m = m = m = m = and are right angles.. Given. Def. of supp.. Def. of Step 4. Subst. Steps, 5. Simplification 6. Div. Prop. of = 7. Trans. Prop. of = Steps, 6 8. Def. of rt. Paragraph proof: and are supplementary, so m + m = 80 by the definition of supplementary angles. They are also congruent, so their measures are equal by the definition of congruent angles. y substitution, m + m = 80, so m = 90 by the Division Property of Equality. ecause m = m, m = 90 by the Transitive Property of Equality. So both are right angles by the definition of a right angle. 4. Use the given two-column proof to write a paragraph proof. Given: 4 Prove: Two-column proof: 4 Danilo Donadoni/ruce Coleman Inc. 4., Given. Vert. Thm.. Trans. Prop. of Steps, 4. Trans. Prop. of Steps, 0-4 Flowchart and Paragraph Proofs 59

27 THINK ND DISCUSS. Explain why there might be more than one correct way to write a proof.. Describe the steps you take when writing a proof.. GET ORGNIZED Copy and complete the graphic organizer. Two-column In each box, describe the proof style in your own words. Proof Styles Flowchart MCC.MP. Paragraph MTHEMTICL PRCTICES 0-4 Exercises Homework Help GUIDED PRCTICE Vocabulary pply the vocabulary from this lesson to answer each question.. In a? proof, the logical order is represented by arrows that connect each step. (flowchart or paragraph). The steps and reasons of a? proof are written out in sentences. (flowchart or paragraph) SEE. Use the given flowchart proof to write a two-column proof. Given: Prove: and are right angles. Flowchart proof: and are supplementary. Given and are right angles. Lin. Pair Thm. supp. rt. SEE 4. Use the given two-column proof to write a flowchart proof. Given: and 4 are supplementary. 4 Prove: m = m Two-column proof:. and 4 are supplementary.. and 4 are supplementary.. 4. m = m. Given. Lin. Pair Thm.. Supps. Thm. Steps, 4. Def. of 594 Module 0 lgebraic and Geometric Proofs

28 SEE 5. Use the given paragraph proof to write a two-column proof. Given: 4 4 Prove: Paragraph proof: y the Vertical ngles Theorem,, and 4. It is given that 4. y the Transitive Property of Congruence, 4, and thus. SEE 4 6. Use the given two-column proof to write a paragraph proof. Given: D bisects C. Prove: G bisects FH. Two-column proof: D C 4 F H G. D bisects C... 4, G bisects FH.. Given. Def. of bisector. Vert. Thm. 4. Trans. Prop. of Steps, 5. Trans. Prop. of Steps, 4 6. Def. of bisector Independent Practice For See Exercises Example PRCTICE ND PROLEM SOLVING 7. Use the given flowchart proof to write a two-column proof. Given: is the midpoint of C. D = EC Prove: D = E D E C Flowchart proof: is mdpt. of C. Given D + D =, E + EC = C Seg. dd. Post D = EC Given C = C D + D = E + EC D = E Def. of mdpt. Def. of segs. Subst. Subtr. Prop. of = Online Extra Practice 8. Use the given two-column proof to write a flowchart proof. Given: is a right angle. Prove: 4 is a right angle. 4 Two-column proof:. is a right angle.. m = 90. and 4 are supplementary. 4. m + m 4 = m 4 = m 4 = is a right angle.. Given. Def. of rt.. Lin. Pair Thm. 4. Def. of supp. 5. Subst. Steps, 4 6. Subtr. Prop. of = 7. Def. of rt. 0-4 Flowchart and Paragraph Proofs 595

29 9. Use the given paragraph proof to write a two-column proof. Given: 4 Prove: and are supplementary. Paragraph proof: 4 and form a linear pair, so they are supplementary by the Linear Pair Theorem. Therefore, m 4 + m = 80. lso, and are vertical angles, so by the Vertical ngles Theorem. It is given that 4. So by the Transitive Property of Congruence, 4, and by the definition of congruent angles, m 4 = m. y substitution, m + m = 80, so and are supplementary by the definition of supplementary angles. 0. Use the given two-column proof to write a paragraph proof. Given: and are complementary. Prove: and are complementary. Two-column proof: and are complementary.. m + m = m = m 5. m + m = and are complementary.. Given. Def. of comp.. Vert. Thm. 4. Def. of 5. Subst. Steps, 4 6. Def. of comp. Find each measure and name the theorem that justifies your answer.. cm. m. m cm cm C D 7 lgebra Find the value of each variable in. 7 in. x + 4 5x - 5. y 6. (x + 40) (5x + 6) 7. /ERROR NLYSIS / elow are two drawings for the given proof. Which is incorrect? Explain the error. Given: C Prove: C C C Real-World Connections 8. Rearrange the pieces to create a flowchart proof. m + m = 80 Def. of supp. m = 7 Subtr. Prop. of = and are supplementary. Lin. Pair Thm. m = 6 Given m + 6 = 80 Subst. rand X Pictures/Getty Images 596 Module 0 lgebraic and Geometric Proofs

30 9. Critical Thinking Two lines intersect, and one of the angles formed is a right angle. Explain why all four angles are congruent. 0. Write bout It Which style of proof do you find easiest to write? to read? TEST PREP. Which pair of angles in the diagram must be congruent? and 5 5 and 8 and 4 None of the above. What is the measure of? Which statement is NOT true if and 6 are supplementary? m + m 6 = 80 and are supplementary. and 6 are supplementary. m + m 4 = CHLLENGE ND EXTEND 4. Textiles Use the woven pattern to write a flowchart proof. Given: Prove: m 4 + m 5 = m 6 5. Write a two-column proof. Given: OC OD Prove: O COD 6. Write a paragraph proof. Given: and 5 are right angles. m + m + m = m 4 + m 5 + m 6 Prove: 4 7. Multi-Step Find the value of each variable and the measures of all four angles. O C D (x + y + ) (x + ) (6y + x - 6) Victoria Smith/HMH MTHEMTICL PRCTICES FOCUS ON MTHEMTICL PRCTICES 8. Reasoning One of the statements within a proof with the diagram shown at the right is PRS SRT, and the Transitive Property of Congruence is given as the reason. What other statement or statements may also be in the proof? Why? S P R Q 9. Communication Compare a flowchart proof to a two-column proof. In what ways are they similar? In what ways are they different? 0-4 Flowchart and Paragraph Proofs 597 T

31 MODULE 0 QUIZ Ready to Go On? ssessment and Intervention 0- iconditional and Definitions. For the conditional If two angles are supplementary, the sum of their measures is 80, write the converse and a biconditional statement.. Determine if the biconditional x = 4 if and only if x = 6 is true. If false, give a counterexample. 0- lgebraic Proof Solve each equation. Write a justification for each step.. m - 8 = 4. 4y - = x _ = Identify the property that justifies each statement. 6. m XYZ = m PQR, so m PQR = m XYZ , and. So k = 7, and m = 7. So k = m. 0- Geometric Proof 0. Fill in the blanks to complete the two-column proof. Given: m + m = 80 Prove: 4 Proof: 4. m + m = 80. b.?. and 4 are supplementary d.?. a.?. Def. of supp.. Lin. Pair Thm. 4. c.? 5. Supps. Thm.. Use the given plan to write a two-column proof of the Symmetric Property of Congruence. Given: EF E Prove: EF F Plan: Use the definition of congruent segments to write EF as a statement of equality. Then use the Symmetric Property of Equality to show that EF =. So EF by the definition of congruent segments. 598 Module 0 Ready to Go On?

32 0-4 Flowchart and Paragraph Proofs Use the given two-column proof to write the following. Given: Prove: 4 Proof: 4.., Given. Vert. Thm.. Trans. Prop. of 4. Trans. Prop. of. a flowchart proof. a paragraph proof PRCC ssessment Readiness Selected Response. Write the conditional statement and converse within the biconditional. rectangle is a square if and only if all four sides of the rectangle have equal lengths. Conditional: If all four sides of the rectangle have equal lengths, then it is a square. Converse: If a rectangle is a square, then its four sides have equal lengths. Conditional: If a rectangle is a square, then it is also a rhombus. Converse: If a rectangle is a rhombus, then it is also a square. Conditional: If all four sides have equal lengths, then all four angles are 90. Converse: If all four angles are 90, then all four sides have equal lengths. Conditional: If a rectangle is not a square, then its sides are of different lengths. Converse: If the sides are of different lengths, then the rectangle is not a square.. Give the missing justifications in the solution of 4x - 6 = 4 shown below. 4x - 6 = 4 Given equation [] 4x = 40 Simplify. _ 4x 4 = _ 40 4 [] x = 0 Simplify. Mini-Task [] Substitution Property of Equality; [] Division Property of Equality [] ddition Property of Equality; [] Division Property of Equality [] Division Property of Equality; [] Subtraction Property of Equality [] ddition Property of Equality; [] Reflexive Property of Equality. Two angles with measures (5x + 5) and (7x + 85) are supplementary. Find the value of x and the measure of each angle. 599