Math Foundations 20 Unit 1 Lesson 1 PDF

Math Foundations 20 | Unit 1 | Lesson 1 Unit 1 Lesson 1 Inductive Reasoning Purpose ▪ Identify examples of inductive reasoning. ▪ Make conjectures by observing patterns. ▪ Discuss the limitations of inductive reasoning. ▪ Use a counterexample to prove that a conjectur...

Math Foundations 20 | Unit 1 | Lesson 1 Unit 1 Lesson 1 Inductive Reasoning Purpose ▪ Identify examples of inductive reasoning. ▪ Make conjectures by observing patterns. ▪ Discuss the limitations of inductive reasoning. ▪ Use a counterexample to prove that a conjecture is false. Inductive Reasoning Inductive reasoning is a type of logical reasoning that involves making specific observations, recognizing patterns, and drawing a general conclusion. Since inductive reasoning is based on a limited number of specific observations, it can be quite unreliable. Typically, a greater number of observations leads to a more reliable conclusion. Inductive reasoning can have drawbacks. For example, in daily life, stereotypes are often formed as a result of inductive reasoning with few observation. Making a Conjecture Inductive reasoning involves the making of conjectures. A conjecture is a statement that appears to be true based on available data and observations but has not been conclusively proven. A conjecture cannot be proven through inductive reasoning. If it has been proven through another, more rigorous, form of reasoning, it is no longer called a conjecture. 1  Math Foundations 20 | Unit 1 | Lesson 1 Make a conjecture about the sum of two odd integers. To begin, we can list several examples: 1+3=4 5 + 9 = 14 11 + 23 = 34 As we generate examples, it is important to be aware of the possibility that a pattern arises as a result of the choice of examples. The first three examples suggest that the sum of two odd integers ends in 4. We then try to find examples that do not end in 4. 23 + 3 = 26 −15 + 7 = −8 −117 + (−19) = −136 A possible conjecture is: “The sum of two odd integers is an even integer.” Example 1: Make a Conjecture a) Use inductive reasoning to make a conjecture about the sum of the squares of any two consecutive even integers. 2  Math Foundations 20 | Unit 1 | Lesson 1 b) Use inductive reasoning to make a conjecture about the product of an odd integer and an even integer. Using a Counterexample to Disprove a Conjecture It is not possible to prove a conjecture using inductive reasoning. However, it is possible to prove a conjecture false by finding an example that contradicts the conjecture. An example that meets the conditions of the conjecture but does not lead to the conjecture’s conclusion is called a counterexample. The existence of a counterexample does not necessarily mean that the conjecture must be discarded completely. The conjecture can often be revised based on the new evidence. The following examples were used to develop a conjecture about the square of a number: 𝟐𝟐 = 𝟒 Conjecture: 𝟏𝟏𝟐 = 𝟏𝟐𝟏 The square of a number is greater than the number itself. (−𝟑)𝟐 = 𝟗 Find a counterexample to disprove the conjecture, then rewrite the conjecture so that it cannot be disproven by the counterexample. One counterexample for this conjecture is the square of the number 0.5. (0.5)2 = 0.25 In this case, the square of the number is less than the number itself. This is true for numbers between 0 and 1. 3  Math Foundations 20 | Unit 1 | Lesson 1 We can restrict the conjecture by changing the word “numbers” to “integers”, but we must consider whether there are still obvious exceptions. Both 0 and 1 are integers, and both are examples where the square of the number is equal to the number itself. 02 = 0 12 = 1 The revised conjecture is as follows: The square of an integer is greater than or equal to the integer itself. Example 2: Find a Counterexample to a Conjecture a) Find a counterexample to the following conjecture. “The sum of any two numbers is greater than the larger of the two numbers.” b) Steve claims that the difference between any two positive odd integers is always a positive even integer. Do you agree or disagree? Justify your decision. 4  Math Foundations 20 | Unit 1 | Lesson 2 Unit 1 Lesson 2 - Deductive Reasoning – Using Logic Worksheet Example 1: Use a Venn Diagram in Deductive Reasoning A fishing derby is open to entrants who are 𝟏𝟓 years of age or older. Draw a Venn diagram to represent this statement, then use the Venn diagram to determine what conclusion can be made based on each of the additional statements below. a) Monty is 𝟗 years old. b) Ari is not entered in the fishing derby. c) Zoe is 𝟐𝟏 years old. d) Luis is entered in the fishing derby.  Math Foundations 20 | Unit 1 | Lesson 2 Example 2: Explain Why a Proof is Not Valid a) All of Dana’s soccer games were on very hot days. He says this proves the following statement: “If it is a hot day, Dana has a soccer game.” Use a Venn diagram to explain why the statement cannot be proven. b) Amy, Phil, and Penny are all registered in both chemistry and physics. Dan is registered in chemistry. Therefore, Dan is also registered in physics. Use a Venn diagram to explain why the conclusion cannot be proven.  Math Foundations 20 | Unit 1 | Lesson 2 Example 3: Solve a Logic Puzzle Michael has a side gig making birthday cakes. Using only the clues below, determine the cake type and decoration theme for each date. Clues: 1. The chocolate cake is to be delivered before the superhero cake. 2. The white cake is ordered for two days before the video game cake. 3. The sports cake has been ordered for March 3rd. 4. The cake ordered for March 3rd is the carrot cake.  Math Foundations 20 | Unit 1 | Lesson 2 Example 4: Explain Reasoning Used to Solve a Sudoku Puzzle a) For the sudoku below, explain why the 𝟑 is being placed in the cell shown. b) For the sudoku below, explain why the 𝟑 is being placed in the cell shown.  Math Foundations 20 | Unit 1 | Lesson 2 Unit 1 Lesson 2 Deductive Reasoning – Using Logic Purpose ▪ Identify examples of deductive reasoning. ▪ Use a Venn diagram to develop or test a conclusion. ▪ Solve a logic puzzle using deductive reasoning. ▪ Identify and describe errors in deductive reasoning. Deductive Reasoning Deductive reasoning is a type of reasoning that starts with statements that are known to be true, and uses logic to reach a specific conclusion. Recall that inductive reasoning is used to form a conjecture, which is a statement that is believed to be true based on limited information. Inductive reasoning cannot be used to prove a conjecture. In contrast, conclusions reached by deductive reasoning are considered to be proven. Venn Diagrams A Venn diagram uses circles (or other shapes) to show the relationship between two or more sets of items. Each circle represents the set of items that possess a common characteristic. When two or more circles overlap, the area of overlap represents items that possess the characteristics of both or all sets. Below are three basic examples of Venn diagrams: The circles overlap, meaning there are one or more numbers that are both even and prime. even prime 2 numbers numbers There is actually only one element in both sets: 2 is the only even prime number. 5  Math Foundations 20 | Unit 1 | Lesson 2 The circles do not overlap, meaning there are no numbers that are both perfect squares and prime. perfect prime A perfect square has at least three factors (1, its squares numbers square root, and itself), so it cannot be prime. The exception is the number 1, which has only one factor, but 1 is not considered prime. integers The circle representing prime numbers is completely within the circle representing integers, meaning all prime prime numbers are integers, but not all integers are numbers prime. When a Venn diagram has been drawn to accurately represent the relationships between sets of items, it can be used in deductive reasoning to prove conjectures or reach new conclusions about the sets of items. All snakes are reptiles, and all reptiles are animals. Using the Venn diagram shown, explain why Jill’s pet snake, Leon, is an animal. Leon is a snake, so he can be represented animals by the snake image in the smallest circle. The entire circle representing snakes is reptiles within the circle representing reptiles, and also within the circle representing animals. snakes Leon must be both a reptile and an animal. 6  Math Foundations 20 | Unit 1 | Lesson 2 You meet five students from Super Elite and Exclusive Basketball Academy at a pickup basketball game, and you note that all of them are very good basketball players. Using inductive reasoning, you make the conjecture that, if someone attends SEEBA, they are a very good basketball player. Use a Venn diagram to deductively prove your conjecture. Start by drawing a circle representing very good basketball players. You may also include this circle in a rectangle representing all people. The area outside the circle represents people that are not very good basketball players. Where should you draw the circle representing students from Super Elite and Exclusive Basketball Academy? Here are some possibilities and what they signify: Some people are SEEBA students and some are not. Some of the SEEBA students are very good basketball players and some are not. All very good basketball players are students at SEEBA. 7  Math Foundations 20 | Unit 1 | Lesson 2 Some people are SEEBA students, some people are very good basketball players, and some are neither. No SEEBA students are very good basketball players, and no very good basketball players are SEEBA students. Some people are SEEBA students, some people are very good basketball players, and some are both. Some SEEBA students are very good basketball players, and some are not. Some very good basketball players are SEEBA students, and some are not. Some people are very good basketball players, and some are not. Some very good basketball players are SEEBA students, and some are not. All SEEBA students are very good basketball players. 8  Math Foundations 20 | Unit 1 | Lesson 2 In order to choose, we need to know this fact that we may have inferred from the name of the school: the Super Elite and Exclusive Basketball Academy only admits a select few very good basketball players as students. The correct Venn diagram is: Any student that is in the circle representing SEEBA students is also in the circle representing very good basketball players. This proves the statement below: “If a student attends the Super Elite and Exclusive Basketball Academy, that student is a very good basketball player.” 9  Math Foundations 20 | Unit 1 | Lesson 2 Example 1: Use a Venn Diagram in Deductive Reasoning A fishing derby is open to entrants who are 15 years of age or older. Draw a Venn diagram to represent this statement, then use the Venn diagram to determine what conclusion can be made based on each of the additional statements below. a) Monty is 9 years old. b) Ari is not entered in the fishing derby. c) Zoe is 21 years old. d) Luis is entered in the fishing derby. 10  Math Foundations 20 | Unit 1 | Lesson 2 Errors in Deductive Reasoning Deductive reasoning uses facts and logic to reach a conclusion or prove a conjecture. If statements that are treated as facts are not actually facts, or the logic is not sound, the proof is not valid. Consider the following statement, which is used to prove that all Albertans love hockey: “All Canadians love hockey. All Albertans are Canadians. Therefore, all Albertans love hockey.” Explain the error in reasoning. There is nothing wrong with the logic. If all Albertans are Canadians, and all Canadians love hockey, then all Albertans must love hockey. This is confirmed by the Venn diagram on the right. The problem is that “All Canadians love hockey” is not a fact. If this statement is not true, it cannot be used to prove another statement. Consider the following statement that is used to prove that all rhombuses are rectangles. “All squares are rhombuses. All squares are rectangles. Therefore, all rhombuses are rectangles.” Explain the error in reasoning. The two initial statements are true facts: a rhombus is a quadrilateral with four equal sides, so a square is always a rhombus; and a rectangle is a quadrilateral with four right angles, so a square is always a rectangle. The error is in the logic. Each of the following Venn diagrams agrees with the two given statements. Based only on the two statements, there is no way of knowing which one is accurate, so the statement has not been proven. 11  Math Foundations 20 | Unit 1 | Lesson 2 The diagram that accurately shows the relationship between rhombuses, squares, and rectangles is shown below. The circle around “squares” has been removed because all figures represented by this overlap region are squares. 12  Math Foundations 20 | Unit 1 | Lesson 2 Example 2: Explain Why a Proof is Not Valid a) All of Dana’s soccer games were on very hot days. He says this proves the following statement: “If it is a hot day, Dana has a soccer game.” Use a Venn diagram to explain why the statement cannot be proven. b) Amy, Phil, and Penny are all registered in both chemistry and physics. Dan is registered in chemistry. Therefore, Dan is also registered in physics. Use a Venn diagram to explain why the conclusion cannot be proven. 13  Math Foundations 20 | Unit 1 | Lesson 2 Solving a Logic Puzzle A logic puzzle is a puzzle in which facts about a situation are given as clues, and the solver must use logic to discover new facts, continuing until the solution to the puzzle is obtained. Teams from the West, South, Central, and North regions of a city participated in volleyball playoffs. Each team wore a different colour uniform, and the colours worn were red, green, yellow, and blue. From the clues below, determine each team’s uniform colour and the place that they finished in the competition. A grid is provided to help in solving the puzzle. Clues: 1. West was either the team that finished in second place or the team that finished in first place. 2. Central finished in first place. 3. South was either the team that finished in first place or the team that wore a yellow uniform. 4. South finished two places behind the team that wore a red uniform. 5. The team that wore a green uniform, the team that finished in first place and the team that finished in fourth place were all different teams. 6. South finished in third place. 14  Math Foundations 20 | Unit 1 | Lesson 2 Start with the first clue: 1. West was either the team that finished in second place or the team that finished in first place. - This doesn’t tell us what place West finished, but it does tell us it can’t be third or fourth. We can mark these with x’s. Move on to the second clue: 2. Central finished in first place. - We can fill in the box where Central meets first place. - If Central finished in first place, it can’t have finished in any other place, and no other school can have finished in first place. - If West finished first or second, but Central finished first, West must be second. - If West finished second, no other school can have finished second. 15  Math Foundations 20 | Unit 1 | Lesson 2 Move on to the third clue: 3. South was either the team that finished in first place or the team that wore a yellow uniform. - First place is already taken by Central, so South must have worn a yellow uniform. - This means no other school could have worn a yellow uniform. - Since South did not finish first or second, the team wearing yellow did not finish first or second. Move on to the fourth clue: 4. South finished two places behind the team that wore a red uniform. - South is in third or fourth place, so the first or second place team wore red. This means either West or Central wore red, so North and South did not. - If either West (second place) or Central (first place) wore red, red was not worn by the teams that finished in third or fourth place. 16  Math Foundations 20 | Unit 1 | Lesson 2 Move on to the fifth clue: 5. The team that wore a green uniform, the team that finished in first place and the team that finished in fourth place were all different teams. - The team that wore a green uniform did not place first or fourth. Move on to the sixth clue: 6. South finished in third place. - If South finished in third place, it can’t have finished in any other place, and no other school can have finished in third place. - North must have finished in fourth place. - If South finished third, and South wore yellow, then the team wearing yellow finished third. - No other colour was worn by the third- place team, and yellow did not finish in any other place. All of the clues have now been used. We can use the grid to complete the puzzle. It may be necessary to go through the clues again. 17  Math Foundations 20 | Unit 1 | Lesson 2 - The only place still available for the team wearing green is second place. - No other team can have finished second. - Now, the only place still available for the team wearing red is first place. - Also the only team that could have finished fourth is blue. Even though the bottom part of the grid is not filled in, the puzzle is solved: In first place is the Central team, wearing red. In second place is the West team, wearing green. In third place is the South team, wearing yellow. In fourth place is the North team, wearing blue. If this solution is correct, it will not be contradicted by any of the clues. 1. West was either the team that finished in second place or the team that finished in first place. 2. Central finished in first place. 3. South was either the team that finished in first place or the team that wore a yellow uniform. 4. South finished 2 places behind the team that wore a red uniform. 5. The team that wore a green uniform, the team that finished in first place and the team that finished in fourth place were all different teams. 6. South finished in third place. 18  Math Foundations 20 | Unit 1 | Lesson 2 Example 3: Solve a Logic Puzzle Michael has a side gig making birthday cakes. Using only the clues below, determine the cake type and decoration theme for each date. Clues: 1. The chocolate cake is to be delivered before the superhero cake. 2. The white cake is ordered for two days before the video game cake. 3. The sports cake has been ordered for March 3rd. 4. The cake ordered for March 3rd is the carrot cake. 19  Math Foundations 20 | Unit 1 | Lesson 2 Solving a Sudoku Puzzle A sudoku is a logic puzzle in which the objective is to fill a 9 × 9 grid so that each row, column, and 3 × 3 section contains each of the digits from 1 to 9. As with other logic puzzles, we use known facts and logic to discover new facts until the puzzle is solved. Solve the sudoku puzzle below. It might take some time to find the first digit that can be filled in. Examine different digits until you find one that has can only be placed into one possible cell. For example, a 2 must be added into the top row. There is already a 2 in the first and second 3 × 3 sections. Therefore, the 2 in the top row must be in the third 3 × 3 section. There is only one space available, so the 2 can be filled in. 20  Math Foundations 20 | Unit 1 | Lesson 2 There may be several different options for what to do next, so the following is just one possibility. The top row also requires a 3 to be added. It cannot be in the second column because there is already a 3 in that column. It cannot be in the second 3 × 3 section because there is a 3 there as well. There is only one possible space for the 3 in the top row. It won’t always work to start with the top row. Often the possible moves will be found elsewhere in the puzzle. For example, there is a 1 in each of the top middle 3 × 3 section and the bottom middle 3 × 3 section. Since these are in the fourth and sixth columns of the grid, there is only one possible space for a 1 in the middle 3 × 3 section. 21  Math Foundations 20 | Unit 1 | Lesson 2 A possible next step would be to place a 6 in the bottom middle section as shown. Can you explain why? A 6 must be added to the bottom middle section, but it cannot be in the eighth or ninth rows. There is only one space left in the seventh row and the bottom middle section. Using similar logic to that shown above, the grid can be completely filled in. 22  Math Foundations 20 | Unit 1 | Lesson 2 Example 4: Explain Reasoning Used to Solve a Sudoku Puzzle a) For the sudoku below, explain why the 𝟑 is being placed in the cell shown. b) For the sudoku below, explain why the 𝟑 is being placed in the cell shown. 23  Math Foundations 20 | Unit 1 | Lesson 3 Unit 1 Lesson 3 Deductive Reasoning – Using Algebra Purpose ▪ Identify examples of deductive reasoning. ▪ Prove conjectures about numbers using deductive reasoning. ▪ Identify errors in proofs that lead to incorrect conclusions. Representing Unknown Numbers Algebraically When using deductive reasoning to prove conjectures about numbers, you must be able to write expressions to represent the numbers. This step must be done correctly in order for the reasoning to be valid. The following table includes examples of expressions you might use in a proof: A possible To represent: expression is: Notes: “an integer” 𝑥 “an even integer” 2𝑚 All even integers are multiples of 2, meaning they have 2 as a factor. If we write the expression as a product of 2 and any integer 𝑚, we can be sure that the expression represents an even number. “an odd integer” 2𝑎 + 1 Since integers alternate between odd and even, adding one to an even integer always gives an odd integer. Students often make the mistake of using 3𝑎 to represent an odd integer. This is incorrect, because if 𝑎 is even, 3𝑎 is even. 24  Math Foundations 20 | Unit 1 | Lesson 3 “a multiple of 3” 3𝑚 In the same way that 2𝑚 is an even number because it has 2 as a factor, 3𝑚 is a multiple of 3 because it has 3 as a factor. “any two 𝑥 and 𝑦 Use two different variables so that the value of the integers” second integer is not influenced by the value of the first integer. “two consecutive 𝑎 and 𝑎 + 1 Choose any variable to represent the smaller of the two integers” integers, then add 1 to the same variable represent the next consecutive integer. “two consecutive 2𝑏 and 2𝑏 + 2 Use 2𝑏 to represent the first even integer (the factor of even integers” two ensures that it is even), then add 2 to skip to the next even integer. “two consecutive 2𝑎 + 1 and Use 2𝑎 + 1 to represent the first odd integer, then add 2 odd integers” 2𝑎 + 3 to skip to the next odd integer. “any two-digit 𝑎𝑏 Here, 𝑎 represents the first digit of the integer (the digit integer” in the 10s place) and 𝑏 represents the second digit of the integer (the digit in the 1s place). The boxes are drawn around the variables so that it is not read as 𝑎𝑏, which would be the product of the two integers. 25  Math Foundations 20 | Unit 1 | Lesson 3 Deductive Reasoning Using Algebra Deductive reasoning is the use of accepted facts and logical reasoning to reach a conclusion. When using algebra in deductive reasoning, we use mathematical facts and sound mathematical reasoning to prove a conjecture. Prove that the sum of two odd integers is an even integer. Let 2𝑎 + 1 and 2𝑏 + 1 represent the odd integers. Write an expression for their sum, then simplify: (2𝑎 + 1) + (2𝑏 + 1) = 2𝑎 + 1 + 2𝑏 + 1 = 2𝑎 + 2𝑏 + 2 Next, we need to show that this expression represents an even integer. Recall that an even integer always has 2 as a factor. Therefore, if we can take a common factor of 2 out of the expression, we know it represents an even integer. 2𝑎 + 2𝑏 + 2 = 2(𝑎 + 𝑏 + 1) The expression for the sum represents an even integer. Since we began with expressions that could represent any two odd integers, we can conclude that the sum of any two odd integers will always be an even integer. Prove that a three-digit number is divisible by 𝟗 if the sum of its digits is divisible by 𝟗. Let the digit in the hundreds place be 𝑎, the digit in the tens place be 𝑏, and the digit in the ones place be 𝑐. The value of the three-digit number can be written as 100𝑎 + 10𝑏 + 𝑐. We can rewrite this expression to separate out terms that are divisible by 9. 99𝑎 + 9𝑏 + 𝑎 + 𝑏 + 𝑐 = 0 Both 99𝑎 and 9𝑏 are divisible by 9. If 𝑎 + 𝑏 + 𝑐 is also divisible by 9, the three-digit number is divisible by 9. This proves the divisibility rule. 26  Math Foundations 20 | Unit 1 | Lesson 3 Example 1: Use Deductive Reasoning to Prove a Conjecture a) Prove that the product of any two odd integers is an odd integer. b) Use inductive reasoning to develop a conjecture about the sum of any three consecutive positive integers, then prove the conjecture by deductive reasoning. 27  Math Foundations 20 | Unit 1 | Lesson 3 Errors in Deductive Reasoning Using Algebra Errors in deductive reasoning usually result from one of the following mistakes: the expression used to represent the numbers does not ensure that the numbers satisfy the conditions (for example, let 𝑥 = any even number) the proof begins with an incorrect assumption the proof uses circular reasoning, which uses the statement to be proven as the assumption upon which the proof is based there is an error in mathematical reasoning (for example, dividing by an expression or variable that is equal to 0) there is an error in arithmetic (i.e. an error made while performing mathematical operations) Identify the error in the following proof. 𝟏=𝟏+𝟏 𝟐(𝟏) = 𝟐(𝟏 + 𝟏) 𝟐(𝟏) + 𝟑 = 𝟐(𝟏 + 𝟏) + 𝟑 𝟐+𝟑= 𝟒+𝟑 𝟓=𝟕 All the mathematical steps in the proof are valid, but this proof begins with an incorrect assumption. It is not true that 1 = 1 + 1, so a valid proof cannot begin with this assumption. The proof below shows that 𝟐 = 𝟏. Find the error. let 𝑎 = any integer let 𝑏 = an integer equal to 𝑎 𝑎=𝑏 𝑎2 = 𝑏 2 𝑎2 − 𝑏 2 = 𝑎𝑏 − 𝑏 2 (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑏(𝑎 − 𝑏) 𝑎+𝑏 =𝑏 𝑏+𝑏 =𝑏 2𝑏 = 𝑏 2=1 28  Math Foundations 20 | Unit 1 | Lesson 3 All of the above math looks correct, but we know the conclusion can’t be correct, so we must look more clearly at the steps to try to find one that is not valid. 𝑎=𝑏 Square both sides of the equation. 𝑎2 = 𝑏 2 Subtract 𝑏 2 from both sides of the equation. 𝑎2 − 𝑏 2 = 𝑎𝑏 − 𝑏 2 Factor both sides. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑏(𝑎 − 𝑏) Divide both sides by 𝑎 − 𝑏. 𝑎+𝑏 =𝑏 Substitute 𝑏 for 𝑎 on the left. 𝑏+𝑏 =𝑏 Add the left side. 2𝑏 = 𝑏 Divide both sides by 𝑏. 2=1 The mistake is in dividing by 𝑎 − 𝑏. We know that 𝑎 = 𝑏, which means that 𝑎 − 𝑏 = 0. In the step where we divide by 𝑎 − 𝑏, we are dividing by 0, which is not a valid step. Every step after this is also not valid, so the proof itself is not valid. Example 2: Identify the Error in a Proof a) Identify the error in the following proof. 𝟐=𝟐 𝟒(𝟐) = 𝟒(𝟏 + 𝟏) 𝟒(𝟐) + 𝟑 = 𝟒(𝟏 + 𝟏) + 𝟑 𝟖+𝟑=𝟔+𝟑 𝟏𝟏 = 𝟗 29  Math Foundations 20 | Unit 1 | Lesson 3 b) The following proof was used to demonstrate that −𝟐 = 𝟐. Find the error in reasoning. −𝟐 = 𝟐 (−𝟐)𝟐 = (𝟐)𝟐 𝟒=𝟒 Since the steps lead to a correct statement, −𝟐 must be equal to 𝟐. 30  Math Foundations 20 | Unit 1 | Lesson 4 Unit 1 Lesson 4 Angles Formed by Parallel Lines Purpose ▪ Explain relationships between angles formed by parallel lines and a transversal. ▪ Use angle relationships to determine whether two lines are parallel. ▪ Use relationships between angles formed by parallel lines and a transversal to determine the measures of unknown angles. Useful Angle Relationships Before we begin to explore angles formed by parallel lines, we need to know the following two angle relationships. Two adjacent angles that add to make a straight line are called a linear pair. These have a sum of 180°. 𝑚∠1 + 𝑚∠2 = 180° When two lines intersect, two pairs of vertically opposite angles are formed. The measures of these two angles are the same. 𝑚∠3 = 𝑚∠4 𝑚∠5 = 𝑚∠6 Angles Formed by Parallel Lines When parallel lines are intersected by a third line, called a transversal, specific angle relationships exist. In order to discuss these angle relationships and use them to solve problems or construct proofs, you must know the terms used to describe them. 31  Math Foundations 20 | Unit 1 | Lesson 4 Two angles that are in the same position relative to an intersection point are called corresponding angles. For example, ∠1 and ∠5 are both up and to the left from the intersection points, so they are corresponding angles. Corresponding angles have the same measure. 𝑚∠1 = 𝑚∠5 There are three other pairs of corresponding angles: ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8. Angles that are between the parallel lines are called interior angles. Two interior angles that are on the same side of the transversal are called same-side interior angles. For example, ∠3 and ∠5 are same-side interior angles. What is the relationship between the measures of these two angles? We know that 𝑚∠1 = 𝑚∠5, because they are corresponding angles. We know that 𝑚∠3 + 𝑚∠1 = 180°, because they are a linear pair. We can substitute 𝑚∠5 for 𝑚∠1 to obtain the equation 𝑚∠3 + 𝑚∠5 = 180°. Same-side interior angles are supplementary. Two interior angles that are on opposite sides of the transversal, and on different vertices, are called alternate interior angles. For example, ∠3 and ∠6 are alternate interior angles. What is the relationship between the measures of these two angles? We know that 𝑚∠6 = 𝑚∠2, because they are corresponding angles. We know that 𝑚∠2 = 𝑚∠3, because they are vertically opposite angles. Using the transitive property, we know that if 𝑚∠6 = 𝑚∠2 and 𝑚∠2 = 𝑚∠3, then 𝑚∠6 = 𝑚∠3. Alternate interior angles have the same measure. 32  Math Foundations 20 | Unit 1 | Lesson 4 Angles that are above or below both parallel lines are called exterior angles. Two exterior angles that are on the same side of the transversal are called same-side exterior angles. For example, ∠1 and ∠7 are same-side exterior angles. Same-side exterior angles are supplementary. Two exterior angles that are on opposite sides of the transversal are called alternate exterior angles. For example, ∠1 and ∠8 are alternate exterior angles. Alternate exterior angles have the same measure. In summary, when parallel lines are intersected by a tranversal, corresponding angles are equal in measure alternate interior angles are equal in measure same-side interior angles are supplementary alternate exterior angles are equal in measure same-side exterior angles are supplementary The above statements are only true if the lines are parallel. The converse is also true: the lines are only parallel if the above statements are true. Therefore, we can use these angle relationships to determine whether two lines are parallel. 33  Math Foundations 20 | Unit 1 | Lesson 4 Determine which of the following pairs of lines are parallel. The known angles are corresponding angles. If the lines were parallel, these angles would 110° be equal in measure. 115° Corresponding angles are not equal, so the lines are not parallel. The two 70° angles are alternate interior angles. 110° 70° Since alternate interior angles are equal in 70° measure, the lines are parallel. The known angles are only formed by the 120° 60° 60° intersection of the transversal with one of the lines. It is not possible to determine whether the lines are parallel. 34  Math Foundations 20 | Unit 1 | Lesson 4 Using Parallel Lines to Find Unknown Angle Measures Relationships between angles formed by parallel lines and a tranversal can be used to determine the measures of unknown angles. Find the values of 𝒙 and 𝒚 in the diagram. Explain your reasoning. 𝑦° The sum of the angles in a triangle is always 180°. The 100° 𝑥° triangle on the right has a 90° angle and a 50° angle. 𝑥° = 180° − 90° − 50° = 40° Since the lines are parallel, same-side interior angles are supplementary. The 100° and the angle that is the sum of the 𝑥° and the 𝑦° angles are supplementary. 50° 100° + 𝑥° + 𝑦° = 180° 100° + 40° + 𝑦° = 180° 𝑦° = 40° 𝑥 = 40, 𝑦 = 40 Find the values of 𝒙 and 𝒚 that make ̅̅̅̅ 𝑫𝑭 and ̅̅̅̅ 𝑨𝑪 parallel to ̅̅̅̅ 𝑨𝑬 parallel to ̅̅̅̅ 𝑩𝑭. Explain your reasoning. If ̅̅̅̅ 𝐴𝐶 is parallel to ̅̅̅̅ 𝐷𝐹 , alternate interior angles formed when they are cut by the transversal will be equal in measure. 3𝑥 + 20 = 𝑥 + 50 2𝑥 = 30 𝑥 = 15 A B C. (3𝑥 + 20)° Calculate the measures of these two alternate.D interior angles: (2𝑦 − 5)° (𝑥 + 50)° 3𝑥 + 20 𝑥 + 50 E F = 3(15) + 20 = 15 + 50 = 45 + 20 = 65 = 65 35  Math Foundations 20 | Unit 1 | Lesson 4 The diagram now shows the angles calculated in the previous step. If ̅̅̅̅ ̅̅̅̅ , 𝐴𝐸 is parallel to 𝐵𝐹 same-side interior angles formed when they are cut by the transversal will be supplementary.. (2𝑦 − 5) + 65 = 180 A B C 2𝑦 − 5 + 65 = 180 65° 2𝑦 = 120 𝑦 = 60 As a check, calculate the measure of ∠𝐴𝐸𝐹:.D E (2𝑦 − 5)° 65° F 2𝑦 − 5 = 2(60) − 5 = 120 − 5 = 115 This angle is supplementary to its same-side interior angle, making ̅̅̅̅ 𝐴𝐸 parallel to ̅̅̅̅ 𝐵𝐹. The ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ values that make 𝐴𝐶 parallel to 𝐷𝐹 and 𝐴𝐸 parallel to 𝐵𝐹 are 𝑥 = 15 and 𝑦 = 60. Example 1: Determine the Measure of an Angle a) Find the values of 𝒙 and 𝒚 in the diagram. Explain your reasoning. 𝑥° 𝑦° 30° 45° 36  Math Foundations 20 | Unit 1 | Lesson 4 b) Find the measure of ∠𝑹𝑺𝑻. Explain your reasoning. (Hint: Draw a third parallel line that passes through S.) R 60° S 50° T a) Determine the measures of ∠𝒂 and ∠𝒃. Explain your reasoning. (The other angles are marked so that you can refer to them in your explanation.) 60° 𝑓 𝑔 𝑑 𝒂 𝑐 𝑒 𝑖 140° ℎ 𝒃 37  Math Foundations 20 | Unit 1 | Lesson 5 Unit 1 Lesson 5 Angle Relationships In Triangles and Polygons Purpose ▪ Prove and apply the relationship relating the sum of the angles in a triangle. ▪ Prove and apply the relationship between the sum of the interior angles and the number of sides in a polygon. ▪ Solve problems involving angles in triangles and polygons. Reasoning About Angle Relationships in Triangles Recall that deductive reasoning involves starting with accepted facts, and using logical reasoning to arrive at a conclusion. We can use what we know about angle relationships and parallel lines to prove triangle properties and to calculate unknown angle measures in triangles. Use the angles formed by parallel lines to prove that the sum of the measures of the angles of a triangle is 𝟏𝟖𝟎°. Draw a triangle, then draw a line through one side and a parallel line through the opposite vertex. Number the angles of the triangle and the two angles adjacent to the angle at the top. 4 1 5 2 3 Angles 4, 1, and 5 add together to make a straight line, so we know that 𝑚∠1 + 𝑚∠4 + 𝑚∠5 = 180° 38  Math Foundations 20 | Unit 1 | Lesson 5 There are two pairs of alternate interior angles, ∠2 and ∠4, and ∠3 and ∠5. 𝑚∠2 = 𝑚∠4 and 𝑚∠3 = 𝑚∠5 Substitute ∠2 for ∠4 and ∠3 for ∠5. 𝑚∠1 + 𝑚∠2 + 𝑚∠3 = 180° The sum of the measures of the angles of a triangle is 180°. Example 1: Develop a Proof Related to Triangles Prove the Exterior Angle Theorem, which states that the size of an exterior angle is equal to the sum of the two remote interior angles. 39  Math Foundations 20 | Unit 1 | Lesson 5 Reasoning About Angle Relationships in Polygons We can use deductive reasoning involving angle relationships in triangles to prove angle relationships in polygons. In the following examples, we use inductive reasoning to develop a conjecture about angle relationships in polygons, the use deductive reasoning to prove the conjecture. Use inductive reasoning to develop an equation for the sum of the interior angles in a convex polygon. We could begin by measuring the interior angles in a number of convex polygons and recording the sum of interior angles for different polygons in a table. number sum of polygon of sides interior angles triangle 3 180° quadrilateral 4 360° pentagon 5 540° hexagon 6 720° heptagon 7 900° Since inductive reasoning involves looking for patterns, we look for a pattern in the sums. Each time the number of sides increases by 1, the sum increases by 180°. We can also notice that the sum of the interior angles is always a multiple of 180°. We can rewrite the sum in the table as a product of an integer multiplier and 180°. number sum of polygon of sides interior angles triangle 3 1(180°) = 180° quadrilateral 4 2(180°) = 360° pentagon 5 3(180°) = 540° hexagon 6 4(180°) = 720° heptagon 7 5(180°) = 900° 40  Math Foundations 20 | Unit 1 | Lesson 5 Again, we can try to figure out the pattern. When 𝑛 = 3, the multiplier is 1. When 𝑛 = 4, the multiplier is 2. When 𝑛 = 5, the multiplier is 3. The multiplier is always 2 less than the number of sides. In general, for 𝑛 sides, the multiplier is 𝑛 − 2. The equation for the sum of the measures of the interior angles of a polygon, 𝑆(𝑛), is 𝑆(𝑛) = 180°(𝑛 − 2) Use deductive reasoning to develop an equation for the sum of the interior angles in a convex polygon. To begin this proof, we can draw a polygon and divide the interior of the polygon into triangles by drawing all possible diagonals from one of the vertices. From any one vertex of a polygon with 𝑛 sides, there will always be 𝑛 − 3 diagonals. The vertex does not form a diagonal with the two adjacent vertices or with itself. The number of triangles that are formed when all diagonals are drawn is 1 more than the number of diagonals. Since there are 𝑛 − 3 diagonals, there are 𝑛 − 2 triangles formed. The sum of the interior angles in the polygon is equal to the sum of the all the angles in these 𝑛 − 2 triangles. Since the sum of the measures of the angles of a triangle is 180°, we can multiply the number of triangles by 180° to obtain the sum of the interior angles. The equation for the sum of the measures of the interior angles of a polygon, 𝑆(𝑛), is 𝑆(𝑛) = 180°(𝑛 − 2) This proves the conjecture developed through inductive reasoning. 41  Math Foundations 20 | Unit 1 | Lesson 5 A regular polygon is a polygon for which all interior angle measures are the same and all side lengths are the same. The measure of each interior angle of a regular polygon can be obtained by dividing the sum of all 𝑛 angles by 𝑛. 180°(𝑛 − 2) 𝑛 Example 2: Develop an Equation Related to Angles in a Polygon a) Use inductive reasoning to develop an equation for the measure of each exterior angle in a regular polygon. number measure of polygon of sides exterior angles triangle 3 120° quadrilateral 4 90° pentagon 5 72° hexagon 6 60° b) Use deductive reasoning to develop an equation for the measure of each exterior angle in a regular polygon. 42  Math Foundations 20 | Unit 1 | Lesson 5 Solving Problems Involving Angles in Polygons We have the following relationships that are true for all polygons: 𝑠𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = 180°(𝑛 − 2) 𝑠𝑢𝑚 𝑜𝑓 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = 360° and the following relationships that are true for regular polygons: 180°(𝑛 − 2) 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 = 𝑛 360° 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 = 𝑛 We can apply these relationships to situational questions about polygons. Kaden is making a wooden puzzle in the shape of a regular pentagon. What angle 𝜽 must he cut to make the piece shown below? A regular pentagon has all sides equal in length and all angles equal in measure. 𝜃 Calculate the measure of each internal angle: 180°(𝑛 − 2) 𝑛 180°(5 − 2) = 5 180°(3) = 5 540° = 5 = 108° 43  Math Foundations 20 | Unit 1 | Lesson 5 The triangular piece is an isosceles triangle because two sides are equal. This means their opposite angles are also equal. Use the sum of the measures of the angles in a triangle to find the value of 𝜃. 108° + 𝜃 + 𝜃 = 180° 108° + 2𝜃 = 180° 2𝜃 = 72° 𝜃 = 36° Kaden must cut the piece at a 36° angle. Determine whether it is possible to tile a floor with regular pentagon tiles. It will only be possible to tile a floor with pentagon tiles if they can be placed with sides together without leaving a gap between tiles. The sum of the interior angles that share a vertex must be 360°. The interior angles of a regular pentagon measure 108°. If three pentagons share a vertex, the total will be 324°, and there will be a gap. Four pentagons would have a total of 432°, so the pentagons would overlap. It is not possible to tile a floor with regular pentagon tiles. 44  Math Foundations 20 | Unit 1 | Lesson 5 Example 3: Solve Problems Involving Angles in Polygons a) Determine the values of 𝒂 and 𝒃 in the regular polygon shown below. 𝑏 𝑎 b) Determine the measure of each angle in the polygon shown below. (5𝑥 + 18)° (6𝑥 + 10)° (4𝑥 + 15)° (5𝑥 + 12)° (4𝑥 + 5)° 45