Unit 1 Proof, Parallel and Perpendicular Lines


 Jeremy Carpenter
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1 Unit 1 Proof, Parallel and Perpendicular Lines SpringBoard Geometry Pages before pages) (add in comma after the course and write the unit and dash Overview In this unit, students study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. They are then introduced to mathematical proof by applying formal definitions and logical reasoning to develop proofs about basic figures. Finally, students learn how to write equations of parallel and perpendicular lines. Standards: Standards in this Unit: MAFS.912.GCO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MAFS.912.GGPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MAFS.912.GCO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Embedded Assessment 1: Geometric Figures and Basic ReasoningThe Art and Math of Folding Paper (page 3738) Make use of geometric figures (Lesson 11, 12)(describe step by step process to create geometric figure) Complete two column Algebraic proofs (Lesson 21, 22)(write a prove statement when given a statement) Axiomatic system of geometry(write an if then statement, given a conditional statement write the converse, inverse, and contrapositive, write a biconditional statement)(ex: That means that from a small, basic set of agreedupon assumptions and premises, an entire structure of logic is devised.) (Lesson 31, 32, 33) MAFS.912.GGPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 1
2 Embedded Assessment 2: Distance, Midpoint, and Angle Measurement A Walk in the Park (6162) (include page ) (no bold)use Segment, Midpoint, and Angle Bisectors (include more detail about used for finding distance)(lesson 41, 42) Use Distance and Midpoint formulas on a coordinate plane (Lesson 51, 52) Embedded Assessment 3: Angles, Parallel Lines, and Perpendicular Lines Graph of Steel (99100) (include page ) (no bold)create Geometric proofs about line segments and angles (Lesson 61, 62) Using and Proving Parallel and perpendicular lines with proofs (Lesson 71, 72, 73) Writing Equations with the slops of parallel and perpendicular lines (Lesson 81, 82) (Refer to finding measure of angles, justify reasoning of equations) Vocabulary (Academic/Math): compare and contrast, justify, argument, interchange, negate, format, confirm, inductive reasoning, conjecture, deductive reasoning, proof, theorem, axiomatic system, undefined terms, twocolumn proof, conditional statement, hypothesis, conclusion, counterexample, converse, inverse, contrapositive, truth value, biconditional statement, postulates, midpoint, congruent, bisect, bisector of an angle, parallel, transversal, sameside interior angles, alternate interior angles, corresponding angles, perpendicular, perpendicular bisector(no bold) 2
3 Support for Lesson 11 Learning Targets for lesson are found on page 3. Main Ideas for success in 11 Identify, describe, and name points, lines, line segments, rays, and planes using correct notation. Identify and name angles. Vocabulary used in this lesson includes: point, line, plane, line segment, ray, and angle Standards for Mathematical Practice to be demonstrated are: Critique the reasoning of others, and Reason abstractly Practice Support for Lesson 11 This example is a model to help solve Practice problem 16. Identify all possible names of this geometric figure. Since the figure has points of J, K, M, and L the figure should be called plane JKLM or you can use the figure letter R and call it plane R. This example is a model to help solve Practice problem 17. Identify all rays in the figure. To identify a ray you have to look for the endpoint that the ray starts at and then extends through a point. You then use the endpoint as the first letter and the point it extends through as the second point. So there are three rays,, and. This example is a model to help solve Practice problem 18. Explain why in the figure above is not a correct name for. B is not correct because it describes the entire angle ABD, not the smaller portion of ABC. 1
4 Practice Support for Lesson 11 Continued: 2
5 Additional Practice: Answers to additional practice: 1. a. angle; T, STQ, QTS b. line segment; AC,CA 2. C 3
6 Support for Lesson 12 Learning Targets for lesson are found on page 7. Main Ideas for success in 12. Describe angles and angle pairs. Identify and name parts of circles. Vocabulary used in this lesson includes: compare and contrast complementary angles, supplementary angles, acute angles, right angles, obtuse angles, straight angles, chord, diameter, radius, and justify Standards for Mathematical Practice to be demonstrated are: Make sense of problems, Model with mathematics, Construct viable arguments, Reason quantitatively Practice Support for Lesson 12 This example is a model to help solve Practice problem 11. a. List all possible correct names for segment. a. Since has the endpoint on the sides of the circle we can call it a chord, but it also goes through the center of the circle so we can call it a diameter. b. all possible correct names for segment. b. Since has its endpoints at the center and on the edge of the circle we call it a radius. c. List all possible correct names for segment. c. Since has endpoints on the circle but does not go through the center we call it a chord. 4
7 Practice Support for Lesson 12 Continued: This example is a model to help solve Practice problem 12 Identify the pairs of Supplementary angles in the figure above. Since supplementary angles have to be 180 degrees we look for linear pairs which equals 180. PKO, PKM; PKL, LKN; OKL, LKM; PKM, NKM are all linear pairs. This example is a model to help solve Practice problem 13 Name the angles that appear to be right in the figure above. PKL, and NKL because they both equal 90 degrees. This example is a model to help solve Practice problem 14. Can two acute angles be Supplementary? No, for an angle to be acute it must be less than 90 degrees, so therefore two angles less than 90 cannot add to more than 178 degrees. This example is a model to help solve Practice problem 15 Is it possible to draw two nonadjacent angles that share vertex A, and are complementary? Yes it is possible. As shown below two angles can be nonadjacent (meaning not directly next to each other and touching) and still have a sum of 90 degrees. If angle 1 was 45 degrees, and angle 2 was 45 degrees then the two would add to 90 and still be nonadjacent. 5
8 Additional Practice 12: additional practice answers:
9 Learning Targets for lesson are found on page 13. Main Ideas for success in 21: Make conjectures by applying inductive reasoning. Recognize the limits of inductive reasoning. Support for Lesson 21 Vocabulary used in this lesson includes: Inductive Reasoning, conjecture, Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Critique the reasoning of others, and Make use of structure, Practice Support for Lesson 21 This example is a model to help solve Practice problems 11. Use inductive reasoning to find the next 3 items in the sequence. a. A, C, E, G,I,. Since we see A then C we skipped B, also we see C then E so we skipped D. therefore J, L, N would be next. b. 2, 4, 6, 10, 16 b. Since we see 2 then 4 followed by 6 we might think the sequence is even numbers, but we skip 8 and go to 10 followed by 16. If we look at it as a sum we see 2+4=6, and 4+6=10, and 6+10=16. Therefore 26, 36, 62 would be the next three items in the sequence. This example is a model to help solve Practice problems 12. Write the first five terms of two different sequences that have 2 as the second term. 1, 2, 3, 4, 5 (you would just need to count up from ones.) 0, 2, 2, 4, 6, 10 (you would start at 0 + 2, and add the two adjacent terms together to get the next.) This example is a model to help solve Practice problems 13. EXAMPLE Generate a sequence with 4 terms using this description: The first term in the sequence is 2, and each other term is three more than twice the previous term. (ie. 2 x + 3 if x is the previous term) 2, 7, 17, 37 7
10 Practice Support for Lesson 21 Continued: This example is a model to help solve Practice problems 14. The diagram shows the first three figures in a pattern. Each figure is made of small triangles. How many small triangles will be in the 4 th figure of the pattern? Support your answer. We notice that the first is 1 and the second is 4 then 9. From there we try to demine the pattern is the pattern the x is the number in the sequence ie: and, so from this patern we take =16. So 16 triangles will exist in the 4 th figure of this sequence. This example is a model to help solve Practice problems 15. The first two terms of a number pattern are 3 and 5. Joe conjectures that the next term will be 7. Mike conjectures that the next term will be 8. Whose conjecture is reasonable? Explain. Both students can be correct 3+5=8 and with a series of odd numbers we get 3,5,7. 8
11 Additional Practice 21: additional practice answers:
12 Support for Lesson 22 Learning Targets for lesson are found on page 18. Main Ideas for success in 22: Use deductive reasoning to prove that a conjecture is true. Develop geometric and algebraic arguments based on deductive reasoning. Vocabulary used in this lesson includes: argument, deductive reasoning, proof, and theorem Standards for Mathematical Practice to be demonstrated are: Express regularity in repeated reasoning, Model with mathematics, Reason abstractly, Construct viable arguments, Critique the reasoning of others, and Reason quantitatively This example is a model to help solve Practice problems 15. Practice Support for Lesson 22 Use expressions for even and odd integers to confirm the conjecture that the sum of an odd integer and an odd integer is an even integer. m + m+2 = 3m+2 and if m is odd all products of (3m+2) will equal an even integer. This example is a model to help solve Practice problems 16. Prove this conjecture geometrically: Any even integer can be expressed as the sum of an even integer and an even integer. A figure representing an even integer can be broken apart into a rectangle representing an even integer and a square representing an even integer. The even integer can be expressed as the sum of the even integer and the even integer 2. This example is a model to help solve Practice problems 17. EXAMPLE Use deductive reasoning to prove that the solution of the equation x 4 = 2 is x = 2. Be sure to justify each step in your proof. X 4 = 2 Given X = Addition Property of Equality X + 0 = 2 Addition X = 2 Identity Property 10
13 Practice Support for Lesson 22 Continued: This example is a model to help solve Practice problems 18. Based solely on the pattern in the table, Kevin states that the number of edges of a polyhedron is equal to its number of Vertices plus the Faces minus 2. Is Andre s statement a conjecture or a theorem? Explain. Shape Vertices Faces Edges Square Pyramid Square Prism Tetrahedron A conjecture. Kevin makes the statement based on three types of polygons. He has not proved that the statement is true for every polygon. A theorem is an item that first needs to proven to be used as a true statement. This example is a model to help solve Practice problems 19. DNA found at a crime scene is consistent with those of an escaped prisoner. Based on this evidence, an investigator concludes that the suspect was at the crime scene. Is this an example of inductive or deductive reasoning? Explain. Inductive reasoning, the evidence shows that the hair could have come from the prisoner, but it does not prove that the suspect was actually at the crime scene. It could have been planted there by a second person. So, the reasoning is inductive rather than deductive. Deductive reasoning is when your case or idea is based on a proof that is built on rules of logic 11
14 Additional Practice 22: 1. Use expressions for even integers to show that the product of two even integers is an even integer. 2. Make use of structure. Use deductive reasoning to prove that x = 5 is not in the solution set of the inequality 2x+1 7. Be sure to justify each step in your proof. 3. During the first month of school, students recorded each day on which they had a quiz in math class. A student stated that there is a math quiz every Tuesday morning. Is the student s statement a conjecture or a theorem? Explain. 4. Answers to additional practice: 1. Use 2p and 2q to represent two even integers. Then (2p)(2q)=2(2pq) We know that the expression 2pq represents an integer because when you find the product of two or more integers, the result is also an integer. So the expression 2(2pq) is an even integer because it is 2 times an integer. 2. Sample answer: 2x+1 7 2x 6 Subtraction Property of Inequality x 3 Division Property of Inequality 5 is not less than or equal to 3 because 5 is to the right of 3 on the number line. So x = 5 is not in the solution set of x The student s statement is a conjecture because it is a generalization based on a pattern of data. The statement is not a theorem because it has not been proved using deductive reasoning. 4. a. Deductive reasoning. The student s conclusion is based on a proof, so the reasoning is deductive rather than inductive. 12
15 Learning Targets for lesson are found on page 25 Main Ideas for success in: Distinguish between undefined and defined terms Support for Lesson 31 Use properties to complete and write algebraic twocolumn proofs Use undefined terms to create definitions of defined terms Vocabulary used in this lesson includes: axiomatic system, undefined terms, ray, collinear points, coplanar points, angle, vertex, complementary angles, supplementary angles, twocolumn proofs. Standards for Mathematical Practice to be demonstrated are: express regularity in repeated reasoning, construct viable arguments, look for and make use of structure. This example is a model to help solve Practice problem 8. Practice Support for Lesson 31 Identify the property that justifies the statement: If, then. The property that justifies the statement would be the Subtraction Property of Equality because 10 has been subtracted to both sides of the equation This example is a model to help solve Practice problems 9 and 11. EXAMPLE Complete the prove statement and write a twocolumn proof for the equation: Given: Prove: Given: Prove: Statements Reasons Complete algebraic solutions: Given equation Distributive Property Subtraction Property of Equality Addition Property of Equality Division Property of Equality 1
16 Practice Support for Lesson 31 Continued: This example is a model to help solve Practice problem 10. Explain why ray is considered a defined term in geometry. Ray is a defined term because it can be defined in terms of the undefined terms line and point. A ray is a part of a line bounded by one endpoint and extending infinitely in one direction. This example is a model to help solve Practice problem 12. Suppose you are given that and. What can you prove by using these statements and the Substitution Property? 1. The Substitution Property says that if, then can be substituted for in any equation or inequality. 2. Since we know we can use the Substitution Property to substitute 10 for in the first equation. 3. This would give the equation: 4. Using the Subtraction Property of Equality we can subtract 2 from both sides to give the equation: 5. Using the Division Property of Equality we can divide both sides by 4 to find. 6. We are able to prove using the given statements and the Substitution Property. 2
17 Additional Practice 31: Answers to additional practice: 1. a. Given b. Multiplication Property of Equality c. Addition Property of Equality d. Division Property 2. C 3. C of Equality 3
18 Learning Targets for lesson are found on page 29. Main Ideas for success in: Support for Lesson 32 Identify the hypothesis and conclusion of a conditional statement. Give counterexamples for false conditional statements. Restate conditional statements in ifthen form. Vocabulary used in this lesson includes: conditional statement, hypothesis, conclusion, and counterexample. Standards for Mathematical Practice to be demonstrated are: make use of structure, reason abstractly, critique the reasoning of others, and construct viable arguments. This example is a model to help solve Practice problem 8. Practice Support for Lesson 32 Write the statement in ifthen form: All 90 o angles are right angles. 1. First, identify the hypothesis and the conclusion. When you rewrite the statement in ifthen form, you may need to reword the hypothesis or conclusion. a. A conditional statement is a statement of logic that combines two statements or facts and can be written in ifthen form. The part of the statement that follows if is the hypothesis, and the part that follows then is the conclusion. 2. Hypothesis: The measure of an angle is 90 o. 3. Conclusion: It is a right angle. 4. Next, use the hypothesis and conclusion to write the conditional statement as an ifthen statement. After the if you write the hypothesis and after the then you write the conclusion. a. If the measure of an angle is 90 o, then it is a right angle. This example is a model to help solve Practice problem 9. Which of the following is a counterexample of this statement? "All mammals have legs. A. Whales are mammals that do not have legs B. Cats are mammals that have legs C. People only have two legs D. Chairs are not mammals and have legs A counterexample is an example that can be found for which the hypothesis is true, but the conclusion is false. So the hypothesis in this statement would be if an animal is a mammal and we want this to remain true, but our conclusion, the animal will have legs needs to be false. The counterexample would be A because the hypothesis of the statement remains true, a whale is a mammal, and the conclusion is false, a whale does not have legs. 4
19 Practice Support for Lesson 32 Continued: This example is a model to help solve Practice problem 10. EXAMPLE Identify the hypothesis and the conclusion of the statement: If today is Friday, then tomorrow is Saturday. The letter p and q are often used to represent the hypothesis and conclusion, respectively, in a conditional statement. The basic form of an ifthen statement would then be, if p, then q. So the hypothesis is represented by p and follows the word if. The conclusion is represented by q and follows the word then. If today is Friday, then tomorrow is Saturday. p (hypothesis) q (conclusion) Hypothesis: today is Friday Conclusion: tomorrow is Saturday Tip: The words if and then are not included when giving the hypothesis and conclusion. This example is a model to help solve Practice problems 11 and 12. Christina says that correct? Explain. is a counterexample that shows the following conditional statement is false. Is Christina If, then. Christina is correct because the hypothesis of the conditional is true, but the conclusion is false. Since, this counterexample shows that the conditional statement is false. A counterexample must show that the conclusion can be false when the hypothesis is true. In Christina s example, the hypothesis is true and the conclusion is false. Once you have found one counterexample for a condition statement the conditional statement is false. 5
20 Additional Practice 32: 1. Write each statement in ifthen form. a. The only time I wake up early is when I set my alarm clock. b. I eat breakfast at a restaurant only if it is a weekend. c. An obtuse angle has a measure between 90 and State or describe a counterexample for each conditional statement. 3. a. If then. b. If three points A, B, and C are collinear, then B is between A and C. Answers to additional practice: a. If I wake up early, then I set my alarm clock. b. If I eat breakfast at a restaurant, then it is a weekend. c. If an angle is obtuse, then its 3. D measure is between 90 and 180. a. A counterexample is. b. Two counterexamples are a line with A between B and C and a line with C between A and B. 6
21 Learning Targets for lesson are found on page 32. Main Ideas for success in: Support for Lesson 33 Write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement. Write and interpret biconditional statements. Identify logically equivalent statements. Vocabulary used in this lesson includes: converse, inverse, contrapositive, interchange, negate, truth values, and biconditional statement. Standards for Mathematical Practice to be demonstrated are: make use of structure, critique the reasoning of others, and reason abstractly. Practice Support for Lesson 33 This example is a model to help solve Practice problems Write the converse, inverse, and contrapositive of the following conditional statement. If it thunders then it is raining. 1. Identify the hypothesis (p) and conclusion (q) a. Hypothesis (p) it thunders b. Conclusion (q) it is raining 2. Converse: If q, then p. Interchange the hypothesis and conclusion a. If it is raining, then it thunders. 3. Inverse: If not p, then not q. Negate the hypothesis and negate the conclusion. a. If it is not thundering then it is not raining. 4. Contrapositive: If not q, then not p. Interchange and negate both the hypothesis and conclusion. Or you could think about it as negating the converse. a. If it is not raining then it is not thundering. This example is a model to help solve Practice problem 14. Write the definition of coplanar lines as a biconditional statement. A biconditional statement is a conditional statement and its converse. It typically includes the words if and only if as in a definition. For a biconditional to be a true statement, it must be true both forward and backward. Writing a biconditional statement is equivalent to writing a conditional statement and its converse. Conditional: If three lines are coplanar, then they lie in the same plane. Converse: If three lines lie in the same plane, then they are coplanar. Biconditional: Three lines are coplanar if and only if they lie in the same plane. 7
22 Practice Support for Lesson 33 Continued: This example is a model to help solve Practice problem 15. EXAMPLE Give an example of a false statement that has a true converse. You will want a statement that has a false conditional, but the converse is true when you interchange the hypothesis and conclusion. Statement: If a polygon has four sides, then the figure is a square. (false) Converse: If a polygon is a square, then it has four sides. (true) This example is a model to help solve Practice problem 16. What conclusions can be drawn from the given statements? Given: (1) If you exercise regularly, then you have a healthy body. (2) You do not have a healthy body. Conclusions: If you do not have a healthy body then you do not exercise regularly. If you exercise regularly, then you have a healthy body. The first statement is the conditional statement and the second statement is the beginning of the contrapositive. Since the conditional and the contrapositive are logically equivalent then the negation of the hypothesis must be true if the negation of the conclusion is true. 8
23 Additional Practice 33: Answers to additional practice: 1. a. Inverse: If it is not raining, then I do not stay indoors; Contrapositive: If I do not stay indoors, then it is not raining. b. Inverse: If I do not have a hammer, then I do not hammer in the morning; Contrapositive: If I do not hammer in the morning, then I do not have a hammer. 2. If people have the same ZIP code, then they live in the same neighborhood. If people live in the same neighborhood, then they have the same ZIP code. 3. D 9
24 Learning Targets for lesson are found on page 39. Main Ideas for success in lesson # here: Support for Lesson 41 Apply the Segment Addition Postulate to find lengths of segments. Use the definition of midpoint to find lengths of segments. Use a ruler to measure the length of a line segment. Vocabulary used in this lesson includes: axiom, postulate, distance along a line, ruler postulate, segment addition postulate, midpoint, congruent ( ), and bisect. Standards for Mathematical Practice to be demonstrated are: attend to precision, reason abstractly, reason quantitatively, and use appropriate tools strategically. Practice Support for Lesson 41 This example is a model to help solve Practice problem 19. Given: Point O is between points D and G,,, and. Find the value of. 1. First, draw a picture to help visualize the information given. Draw a segment where D and G are the endpoints and O is between them. 2. Next, add in the information given about each part of the segment. 3. The Segment Addition Postulate tells us that if point O is between points D and G then. So we will use this to write an equation and solve for. Input the values of each segment into the equation. Combine like terms Subtract 18 from both sides Divide both sides by 6 4. The value of is 7. 1
25 Practice Support for Lesson 41 Continued: This example is a model to help solve Practice problems 20 and 23. EXAMPLE If M is the midpoint of, and, find the value of and the length of. 1. First, draw a picture to help visualize the information given. Draw a segment where P and Q are the endpoints and M is in the middle of them. 2. Next, add in the information given about each part of the segment. 3. Because M is the midpoint of, you know that and. Therefore 4. Now use the Segment Addition Postulate to write an equation where. Use this equation to solve for. Input the values of each segment into the equation. Combine like terms Subtract from both sides Subtract 16 from both sides Divide 4 on both sides The value of is Once we have the value of, we need to find the length of. Substitute 10 for in the expression for. has a length of 58 units. 2
26 Practice Support for Lesson 41 Continued: This example is a model to help solve Practice problem 21. Point D is between points A and B. The distance between points A and D is of AB. What is the coordinate of point D? 1. First we need to find the distance between A and B. Point A is at and point B is at. 2. Since the distance between A and D is of AB, we need to find of AB. This tells us that point D is 9 units away from A. 3. To find the coordinate of D we need to add 9 units to the coordinate of A. the coordinate of D is 3 This example is a model to help solve Practice problem 22. EXAMPLE Explain how to use a ruler to measure the length of a line segment. To measure a certain line segment, place the ruler with its edge along the line segment such that the zero mark of the ruler coincides with an endpoint. Now we read the mark on the ruler which is against the other endpoint. That mark would be length of the line segment. If the line segment does not start at the zero on the ruler, to find the length of the segment we either count the number of units between the ends of the line segment or take the absolute value of the difference between the endpoints. This example is a model to help solve Practice problem 24. Compare and contrast a postulate and a theorem. A postulate is understood as true without proof, while a theorem must be proven. A theorem is a proposition that can be deduced from postulates. Usually postulates provide the starting point for the proof of a theorem. We make a series of logical arguments using postulates to prove a theorem. 3
27 Additional Practice 41: 1. Suppose point T is between points R and V on a line. If RT = 6.3 units and RV = 13.1 units, then what is TV? A. 2.5 units B. 6.8 units C. 7.8 units D units 2. Suppose P is between M and N. a. If,, and, what is the value of? b. If,, and, what is the value of? 3. Use the centimeter ruler shown. a. What is the length of b. What number on the ruler represents the midpoint of 4. Points P, M, and T are on a line and PT  PM = MT. Which point is between the other two? Explain your answer. Answers to additional practice: 1. B a. 5 b. 3 a. 9cm b M is between P and T. Starting with PT  PM = MT, add PM to each side to get PT = MT = PM or PT = PM + MT. That equation satisfies the situation that M is a point between P and T. 4
28 Learning Targets for lesson are found on page 45. Main Ideas for success in lesson #here: Support for Lesson 42 Apply the Angle Addition Postulate to find angle measures. Use the definition of angle bisector to find angle measures. Use a protractor to measure the degree of an angle. Vocabulary used in this lesson includes: protractor postulate, angle addition postulate, bisector of an angle, adjacent angles, congruent angles, vertical angles, perpendicular ( ), complementary, and supplementary. Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, express regularity in repeated reasoning, construct viable arguments, and critique the reasoning of others Capitalize the first letter Practice Support for Lesson 42 This example is a model to help solve Practice problem 12. Point K is in the interior of m ABC, m ABC =, m ABK = 42, and m KBC =. What is m ABC? 1. First, draw an angle to help visualize the information given. Draw an angle whose vertex is B and the rays creating the angle are and as shown below. Point K needs to be in the interior of ABC as shown below. 2. Next, add in the information given about each part of the angle. 3. The Angle Addition Postulate tells us that. Angle Addition Postulate Substitute the values of each angle into the equation Combine Like Terms Subtract from both sides Subtract 8 from both sides Divide both sides by 5 4. Once you have the value of, substitute it into the equation of ABC. The measure of angle ABC is
29 Practice Support for Lesson 42 Continued: This example is a model to help solve Practice problems 13 and 15. EXAMPLE bisects. If and, then what is? 1. First, draw an angle to help visualize the information given. Draw an angle whose vertex is Q and the rays creating the angle are and as shown below. bisects the angle which means it goes through the middle of the angle, creating two congruent angles. 2. Next, add in the information given about each part of the angle. 3. Because bisects, you know that and. Therefore Add this to the diagram. Include that the congruent symbol means =. 4. Use the Angle Addition Postulate to write an equation where. Substitute the values of each angle and solve for y. Angle Addition Postulate Substitute the values of each angle into the equation Combine Like Terms Subtract from both sides Add 2 to both sides Divide both sides by 2 5. Substitute the value of y into the equation for 6
30 This example is a model to help solve Practice problem 14. Practice Support for Lesson 42 Continued: EXAMPLE C and D are supplementary. If and, what is the measure of each angle? 1. If C and D are supplementary this means that the sum of their angle measures is 180 o. Write an equation where C + D = 180 and solve for. Supplementary angles are angles whose measure sums to 180 Substitute the values of each angle into the equation Combine like terms Subtract 8 from both sides Divide both sides by 4 The value of is Substitute the value of into the equations for C and D. 3. Check to make sure your answers give supplementary angles; they add to 180. This example is a model to help solve Practice problems 16. Kevin knows that point B is in the interior of and he knows that. What can Kevin conclude from this information? Explain. Make a sketch that supports your answer. 1. Draw an angle with point F being the vertex and the rays creating the angle are and. 2. Add so that B is in the interior of the angle and 3. Since B is in the interior of and, we can conclude that is the angle bisector of because the bisector of an angle is a ray that divides the angle into two congruent adjacent angles,. 7
31 Additional Practice 42: 1. Make sense of problems. Suppose that bisects MPN. What conclusion can you make? 2. Suppose bisects CAR. If and, what is? 3. D and E are complementary. If and, what is? Answers to additional practice: 1. m MPQ = m QPN
32 Learning Targets for lesson are found on page 51. Main Ideas for success in lesson# here: Support for Lesson 51 Use the Pythagorean Theorem to derive the Distance Formula. Use the Distance Formula to find the distance between two points on the coordinate plane. Vocabulary used in this lesson includes: derive, Pythagorean Theorem, hypotenuse, and Venn diagram. Standards for Mathematical Practice to be demonstrated are: model with mathematics, attend to precision, express regularity in repeated reasoning, reason abstractly, and reason quantitatively Capitalize the first letter Practice Support for Lesson 51 This example is a model to help solve Practice problems 21 and 22. Find the distance between the points with the given coordinates. and 1. The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by. 2. Label the points as follows and Therefore,, and. Hint: is read as x sub one and it refers to the first xvalue. 3. Substitute the values into the Distance Formula. Distance Formula, and Substitute into formula 4. Simplify the equation. Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add explain what squaring is Add 16 and 49 before you take the square root explain square root **justify why the answer is reasonable for your example, modeling 23 separately, you may need an additional pg The distance between and is which is about 8.06 units. This example is a model to help solve Practice problems 23 and 24. Besides the Distance Formula, what other method could you use to find the distance between two points in the coordinate plane? The structure of the Distance Formula and the Pythagorean Theorem are nearly identical. When using the Pythagorean Theorem it involves the relationship between the lengths of the legs and the hypotenuse of a right triangle. 9
33 This example is a model to help solve Practice problem 25. EXAMPLE Practice Support for Lesson 51 Continued: Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 5). What is the perimeter of the triangle? 1. First, we need to find the length of each side of the triangle. Use the Distance Formula to find these lengths. 2. Use the Distance Formula to find the length of AB. Distance Formula Substitute into formula:, and Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add refer to ex#_ for how to square # s Add 9 and 0 before you take the square root Take the square root of 9. The length of AB is 3 units. 3. Use the Distance Formula to find the length of AC. Refer to ex#_for finding sq. roots Distance Formula Substitute into formula:, and Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add refer to ex#_... Add 0 and 16 before you take the square root 4. Use the Distance Formula to find the length of CB. Take the square root of 16. The length of AC is 4 units. Distance Formula Substitute into formula, and Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add Add 9 and 16 before you take the square root Refer to ex#_for finding sq. roots Take the square root of 25. The length of CB is 5 units. 5. The perimeter of a triangle is the sum of the lengths of its 3 sides. In this case we will add. Perimeter = units 10
34 This example is a model to help solve Practice problem 26. EXAMPLE Use the Distance Formula to show that. Practice Support for Lesson 51 Continued: 1. If, this means that the length of AB is the same as the length of CD. To find the length of the segments we use the Distance Formula. 2. Use the Distance Formula to find the length of AB. Distance Formula Substitute into formula:, and Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add Refer to ex# for Add 25 and 1 before you take the square root 3. Use the Distance Formula to find the length of CD. Take the square root of 26. Refer to ex#_for finding sq. roots The length of AB is about 5.1 units. Distance Formula Substitute into formula:, and Use order of operations to simplify the equation. Simplify inside the parenthesis first. Square and before you add Add 1 and 25 before you take the square root Take the square root of So, by the definition of congruent segments, because AB=CD. The length of AB is about 5.1 units. 11
35 Additional Practice 51: 1. Which expression represents the distance between points and? 2. The coordinates of the vertices of triangle are, and. Find the perimeter of the triangle. 3. The coordinates of the vertices of a triangle are, and. a. Find AB. b. Find BC. c. Find AC. d. Bases on the lengths of the sides, what kind of triangle is? Answers to additional practice: 1. D
36 Learning Targets for lesson are found on page 56. Main Ideas for success in lesson# here: Support for Lesson 52 Use inductive reasoning to determine the Midpoint Formula. Use the Midpoint Formula to find the coordinates of the midpoint of a segment on the coordinate plane. Vocabulary used in this lesson includes: midpoint. Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, make use of structure, reason quantitatively, and make sense of problems capitalize the first letter Practice Support for Lesson 52 This example is a model to help solve Practice problems 9, 10, 11 and 12. Find the coordinates of the midpoint of the segment with the given endpoints. and The midpoint of a segment is the point that divides the segment into two congruent segments. In order to find the midpoint of a line segment you use the Midpoint Formula:. You are finding the average of the xcoordinates and the average of the ycoordinates. 1. Label the points as follows and Therefore,, and. Hint: is read as x sub one and it refers to the first xvalue. 2. Substitute the values into the Midpoint Formula and simplify. Midpoint Formula and Substitute into the formula. Simplify the numerator. Simplify the fractions, if possible. 3. The midpoint is located at 13
37 Practice Support for Lesson 52 Continued: This example is a model to help solve Practice problem 13. EXAMPLE Find and explain the errors that were made in the following calculation of the coordinates of a midpoint. Then fix the errors and determine the correct answer. Find the coordinates of the midpoint M of the segment with endpoints X(4, 2) and Y(0, 8). The midpoint can be found by averaging the xcoordinates of the two different points and averaging the ycoordinates of the two different points. The mistakes made was instead of adding the two xcoordinates and adding the two y coordinates, the second coordinate was subtracted from the first. The midpoint can be found correctly by changing the subtraction sign to an addition sign and simplifying. The correct coordinates of the midpoint are. This example is a model to help solve Practice problem 14. AB is graphed on a coordinate plane. Explain how you would determine the coordinates of the point on the segment that is of the distance from A to B. One way to find the coordinates would be to find the midpoint M of AB. Below is an example of a line segment and its midpoint graphed on a coordinate plane. Next, you could find the midpoint MB since that would be of the distance from A to B. So the coordinates of the point on the segment that is ¾ the distance from A to B would be located halfway between the midpoint of AB and B. 14
38 Additional Practice 52: 1. has endpoints and. Find the coordinates of the midpoint of. 2. Which expression represents the midpoint of the line segment with endpoints and? 3. For the coordinates (5, 8) and (9, 14), one is an endpoint of a line segment and the other is the midpoint. How many possibilities are there for the other endpoint? Find each one. Explain your method. Answers to additional practice: 1. (4.5, 3.5) 2. C 3. 15
39 Learning Targets for lesson are found on page 63. Main Ideas for success in 61: Support for Lesson 61 Students use definitions, properties, and theorems to justify a statement. Write twocolumn proofs to prove theorems about lines and angles. Vocabulary used in this lesson includes: properties, postulates, and midpoint of a line segment Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and Critique the reasoning of others Practice Support for Lesson 61 Lines CF, DH, and EA intersect at point B. Use this figure for Items 4 8. Write the definition, postulate, or property that justifies each statement. This example is a model to help solve Practice problems 4. If 6 is supplementary to ABF, then m 6 + m ABF = 180. (meaning that two angles have a value that adds up to 180 degrees.) Definition of supplementary angles (meaning that two angles have a value that adds up to 180 degrees.) This example is a model to help solve Practice problems 5. If 5 4, then bisects ABG. Definition of angle bisector (meaning a ray or line that has split an angle into two equal parts.) This example is a model to help solve Practice problems 6. EXAMPLE DB + BH = DH Segment Addition Postulate (meaning that you add two segments together to get a larger segment) This example is a model to help solve Practice problems 7. If CBH is a right angle, then. Definition of perpendicular lines (meaning two lines that intersect at a 90 degree or right angle) 1
40 Practice Support for Lesson Continued: This example is a model to help solve Practice problems 8. EXAMPLE If m 5 = m 4, then m 5 + m 6 = m 4 + m 6. Addition Property of Equality (meaning you can replace an equal angle in an addition problem and get the same total angle value.) This example is a model to help solve Practice problems 9. Write a statement related to the figure above that can be justified by the Angle Addition Postulate. (meaning the addition of two angles to get a larger angle value.) m CBH = m CBA + m ABH Because the two angles combined complete the angle CBH. 2
41 Additional Practice 61: 1. Use the diagram shown. Write a statement that can be justified by each of the following: a. definition of angle bisector b. Angle Addition Postulate 2. Use the diagram shown. 3
42 Support for Lesson 62 Learning Targets for lesson are found on page 66. Main Ideas for success in 62: Complete twocolumn proofs to prove theorems about segments. Complete twocolumn proofs to prove theorems about angles. Vocabulary used in this lesson includes: Vertical angles theorem, and Two Column Proofs Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and Attend to precision. Practice Support for Lesson 62 This example is a model to help solve Practice problems 4. Given: BAC, CAD, DAE, EAF, Ray AC bisects BAD, Ray AE bisects DAF. Prove: m BAC, + m EAF = m CAE Statement Reason Ray AC bisects Ray AE bisects BAD DAF 1. Given m DAE = m EAF m BAC = m CAD 2. Bisected angles are equal in measure. (Angle bisector) m BAC + m EAF = m 3. Angle Addition CAD + m DAE (referred to in 61) m CAD + m DAE = m 4. Angle Addition CAE m BAC + m EAF = m CAE 5. Substitution 4
43 Practice Support for Lesson 62 Continued: This example is a model to help solve Practice problems 5. (Keep in mind that the proof is about the use of the segment addition postulate not the actual value of the segments.) Given: ABCD on AD Prove: AB + BC + CD = AD Statement Reason Given: ABCD on AD AB + BD = AD BC + CD = BD AB + BC + CD = AD 1. Given 2. Segment Addition 3. Segment Addition 4. Substitution This example is a model to help solve Practice problems 6. EXAMPLE 5
44 Practice Support for Lesson 62 Continued: This example is a model to help solve Practice problems 7. Statements Reasons Given Each angle has one unique angle bisector An angle bisector is a ray whose endpoint is the vertex of the angle and which divides the angle into two congruent angles Reflexive Property. A quantity is congruent to itself SAS  If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent CPCTC  Corresponding parts of congruent triangles are congruent. This example is a model to help solve Practice problems 8. EXAMPLE How do you know that the triangle is item 7 is a Isosceles triangle? Since Isosceles triangles are by definition triangles that have two sides and two angle s congruent, and the proof asked us to prove two angles congruent we know that the triangle must be isosceles. 6
45 Additional Practice: Additional Practice Answers: D 2. a. Definition of angle Bisector b. Given c. Substitution d. is supplementary to. e. Definition of supplementary angles. 3. a. m 1 =37; m PTR =53 b. Angle Addition Postulate c. m 2 =16 d. Subtraction Property of Equality 7
46 Learning Targets for lesson are found on page 73. Main Ideas for success in 71: Support for Lesson 71 Make conjectures about the angles formed by a pair of parallel lines and a transversal. Students need to prove theorems about these angles. Vocabulary used in this lesson includes: parallel, transversal, corresponding angles, sameside interior angles, alternate interior angles, confirms, and means parallel. Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Construct viable arguments, Express regularity in repeated reasoning, and Use appropriate tools strategically Practice Support for Lesson 71 In the diagram, a b (meaning that line a is parallel to line b, also meaning that the lines run side by side and do not touch.)use the diagram for Items Determine whether each statement in Items is true or false. Justify your response with the appropriate postulate or theorem. This example is a model to help solve Practice problem is supplementary to 6. True; SameSide Interior Angles Theorem (meaning that the angles are located on the inside of the two lines and are on the same side of the transversal c). This example is a model to help solve Practice problem True, Corresponding Angles Theorem (meaning that the angles are in the same general location in there pod of four angles grouping) This example is a model to help solve Practice problem supplementary False, Vertical angles like 2 and 6 are congruent. 1
47 Practice Support for Lesson Continued: This example is a model to help solve Practice problems True, alternate interior angles theorem This example is a model to help solve Practice problems 19. If m 1 = 2x 20, and m 4 = 4x + 5, what is m 1? What is m 4? m 1 + m 4=180 same side exterior angles are supplementary (as stated in ex 15) (2x  20) + (4x + 5)=180 Substitution 6x  15=180 6x = Combine like terms Addition property of equality 6x=195 Add = Division property of equality X=32.5 m 1=2x  20 so 2(32.5)  20= 45 and m 4=4x + 5 so 4(32.5) + 5=135 finally m 1=45 and m 4=135 This example is a model to help solve Practice problems 20. Based on your answer to example 19, what are the measures of the other numbered angles in the diagram? Explain your reasoning. m 3 = m 5 = m 7 = 45 ; m 2 = m 6 = m 8 = 135. Sample explanation: 1 3 and 4 2 by the Corresponding Angles Postulate. 5 3, 6 4, 7 1, and 8 2 by the Vertical Angles Theorem. 2
48 Additional Practice 71: Answers to additional practice:
49 Learning Targets for lesson are found on page 79. Main Ideas for success in 72: Support for Lesson 72 Student need to develop theorems to show that lines are parallel. Determine whether lines are parallel. Vocabulary used in this lesson includes: Converse of a theorem or postulate. Standards for Mathematical Practice to be demonstrated are: Make use of structure, Use appropriate tools strategically, Reason abstractly, Attend to precision, and Model with mathematics Practice Support for Lesson 72 For Items 12 14, use the diagram to answer each question. Then justify your answer. This example is a model to help solve Practice problem 12. Given that m 12= 152 and m 6 = 152, is m p? Yes. I know that I also know that 6 8 because these angles are vertical Angles that are created by the intersection of two lines or line segments. So, by the Transitive Property (meaning if a = b, and b = c, then a = c), and 8 are congruent since they are corresponding angles, (meaning that the angles are in the same general location in there pod of four angles grouping) so m p by the Converse of the Corresponding Angles Postulate. This example is a model to help solve Practice problem 13. Given that m 9 = 52 and m 5 = 56, is m n? No. Sample justification: 9 and 5 are alternate interior angles formed by lines m and n and a transversal. Because these angles are not congruent, line m is not parallel to line n. This example is a model to help solve Practice problem 14 EXAMPLE Given that m 6 = 124 and m 3 = 52, is n p? No. Sample justification: 6 and 8 are vertical angles, so they are congruent and m 8 = and 9 are vertical angles, so they are congruent and m 9 = and 9 are sameside interior angles formed by lines n and p and a transversal Because these angles are not supplementary ( ), line n is not parallel to line p. 4
50 Practice Support for Lesson 72 Continued: This example is a model to help solve Practice problem 15. Two lines are cut by a transversal such that a pair of corresponding angles are right angles. Are the two lines parallel? Explain. Yes. Sample explanation: All right angles are congruent, so the corresponding angles are congruent. Thus, the lines are parallel by the Converse of the Alternate Interior Angles Theorem This example is a model to help solve Practice problem 16. Describe how a construction worker can determine whether two power lines painted on the ground are parallel. Assume that the worker has a protractor, string, and two stakes. The worker can place the stakes outside the lines and tie the string to them so that the string crosses the power lines like a transversal. Then the worker can measure 2 corresponding angles formed by the power lines and the string. If the corresponding angles are congruent, then the power lines are parallel. 5
51 Additional Practice 72: Answers to additional practice:
52 Learning Targets for lesson are found on page 84. Main Ideas for success in 73: Support for Lesson 73 Students need to develop theorems to show that lines are perpendicular. Determine whether lines are perpendicular. Vocabulary used in this lesson includes: perpendicular, perpendicular lines, Perpendicular Postulate, Parallel Postulate, Perpendicular Transverse Theorem, and Perpendicular bisector. Standards for Mathematical Practice to be demonstrated are: Make use of structure, Reason abstractly, Critique the reasoning of others, and Use appropriate tools strategically. Practice Support for Lesson 73 In the diagram, l m, m 1 = 90, and 5 is a right angle. Use the diagram for Items This example is a model to help solve Practice problem 9. Explain how you know that m p. 4 measures 90, so it is a right angle. Because lines m and p intersect to form a right angle, they are perpendicular. This example is a model to help solve Practice problem 10. Show that m 10 = 90 I know that l m and m p, and l p, so line l must also be perpendicular to line p. Perpendicular lines intersect to form right angles, so 10 must be a right angle and measure 90. This example is a model to help solve Practice problem 11. Show that l n 2 and 6 are both right angles, so they are congruent. 2 and 6 are congruent corresponding angles formed by lines l and n and a transversal, so l n by the Converse of the Corresponding Angles Postulate 7
53 Practice Support for Lesson 73 Continued: This example is a model to help solve Practice problem 12. The perpendicular bisector of intersects at point B. If =15 what is Since =15 and it is the point of intersection of the perpendicular bisector (meaning intersecting a line at a right or 90 degree angle and then splitting that line in half,) the value is doubled and is 30. This example is a model to help solve Practice problem 13. An angle formed by the intersection of two lines is Acute. Could the lines be perpendicular? Explain No. An acute angle measures less than 90, so it is not a right angle. Because one of the angles formed by the intersection of the lines is not a right angle, the lines are not perpendicular. 8
54 Additional Practice 73: Additional Practice answers: 1. C
55 Learning Targets for lesson are found on page 89. Main Ideas for success in 81: Support for Lesson 81 Make conjectures about the slopes of parallel and perpendicular lines. Use slope to determine whether lines are parallel or perpendicular. Vocabulary used in this lesson includes: Slope, Parallel, Perpendicular, Conjecture, and opposite reciprocals. Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Express regularity in repeated reasoning, and Model with mathematics Practice Support for Lesson 81 This example is a model to help solve Practice problem 10 Is? (meaning is line TQ parallel to line SR?) Yes, the slope of is  which reduces to 2, and has a slope of  which also reduces to 2. Since the slopes are equal they are parallel. This example is a model to help solve Practice problem 11 Is? (Meaning is line QR meeting line SR at a right angle?) Yes, The slope (the rise and run of the line) of is and the slope of is 2. Since the slopes are opposite reciprocals (meaning is the fraction flipped and the power reversed) they are perpendicular (meaning that the lines meet at a right or 90 degree angle) This example is a model to help solve Practice problem 12 EXAMPLE Is? No, The slope of is  and the slope of is 2. Since they are not opposite reciprocals (meaning is the fraction flipped and the power reversed) the lines are not parallel. 1
56 Practice Support for Lesson 81 Continued: This example is a model to help solve Practice problems 13. Line m passes through the origin (0,0) and the point (3, 4). What is the slope of a line parallel to line m? If line m passes through the origin (0,0), and through (3,4) the slope would be. Because: Following the slope formula learned in Algebra 1. Therefore the slope of the perpendicular line would have to be an opposite reciprocal. That slope would be . This example is a model to help solve Practice problems 14. EXAMPLE RST is a right angle, if has a slope of 5 what is the slope of? If the slope of one leg of the angle is 5 then the other leg would have to be perpendicular to create a right angle. SO the slope would have to be opposite reciprocals (meaning is the fraction flipped and the power reversed.) so the other leg (ST) would have a slope of. 2
57 Additional Practice 81: Answers to additional practice: C 3. 3
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