Geometry Topic 5: Conditional statements and converses page 1 Student Activity Sheet 5.1; use with Overview

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1 Geometry Topic 5: Conditional statements and converses page 1 Student Activity Sheet 5.1; use with Overview 1. REVIEW Complete this geometric proof by writing a reason to justify each statement. Given: B is the midpoint of AC ; AD = EC Prove: BE = DB Statements Reasons a. B is the midpoint of AC. a. Given b. AD = EC b. Given c. AB = BC c. Definition of midpoint d. AD + DB = AB d. Segment Addition Postulate e. BE + EC = BC e. Segment Addition Postulate f. AD + DB = BC f. Substitution Property g. AD + DB = BE + EC g. Substitution Property h. AD + DB = BE + AD h. Substitution Property i. DB = BE i. Subtraction Property j. BE = DB j. Symmetric Property 2. You already proved that the Vertical Angle Theorem is true. This theorem states: If two lines intersect to form vertical angles, then the angles have the same measure. What about the following statement? Is this statement true or false? [OV, pages 1-2] False "If two angles have the same measure, then they are vertical angles." 3. Fill in the blank to complete the definition of a conditional statement. [OV, page 4] A conditional statement is a statement that can be written in a(n) if-then format. Page 1 of 1

2 Geometry Topic 5: Conditional statements and converses page 2 Student Activity Sheet 5.2; use with Exploring Conditional statements 1. A statement is a declarative sentence that is either true or false but not both. For each sentence below, decide whether it qualifies as a statement according to the definition. Then write "Statement" or "Not a statement" beside the sentence. [EX1, page 2] a. Statement It's cold outside today. b. Statement I'm going to France if I can earn enough money. c. Not a statement Will you go to France with me? d. Statement All pigs can fly. e. Not a statement Find your missing socks. f. Statement If 3x + 17 = 29, then x = Joe says, I always lie. Is Joe s claim true or false? Explain your answer. [EX1, page 3] If Joe's claim, "I always lie!" is a lie, then he's telling the truth. But if his claim is the truth, then he's telling a lie. So, this sentence cannot be true or false. 3. Is the sentence below true or false? Explain your answer. [EX1, page 3] "This sentence is false." If the sentence is true, then it is false; but if it is false, then it is true. So, this sentence is neither true nor false. 4. What do you call a sentence that cannot be interpreted as either true or false? [EX1, page 3] A sentence that cannot be interpreted as either true or false is called a logical paradox. Page 1 of 5

3 Geometry Topic 5: Conditional statements and converses page 3 Student Activity Sheet 5.2; use with Exploring Conditional statements 5. For each statement below, determine whether it is true or false and explain your answer. [EX1, page 4] a. If 3x 2 = 13, then 7 2x = -3. True. 3x 2 = 13 3x = 15 x = 5 So, 7 2x = 7 2(5) = 7 10 = -3. b. If x 2 = x, then x = 1 is the only solution to the equation. False. Since x 2 = x, x 2 x = 0. Factor to get x(x 1) = 0. So, x = 0 or x - 1 = 0. This gives two solutions: x = 0 or x = 1. c. If a and b are odd numbers, then a + b is an even number. True. Since a and b are odd numbers, let a = 2m 1 and b = 2n + 1, where m and n are any integers. Then: a + b = (2m 1) + (2n + 1) = 2m 1 + 2n + 1 = 2m + 2n = 2(m + n) 2(m + n) is an even number, so a + b is even. Page 2 of 5

4 Geometry Topic 5: Conditional statements and converses page 4 Student Activity Sheet 5.2; use with Exploring Conditional statements 6. Using the answers provided, write the hypothesis and conclusion for each statement. [EX1, page 8] a + b + c is divisible by 3. The wind is gusting at 50 mph. a, b, and c are consecutive odd numbers. There is a dust storm in the desert. m 1+ m 2 + m 3 = 180 o. 1, 2, and 3 are the angles of a triangle. Statement Hypothesis: If Conclusion: Then Wind gusts of 50 mph cause dust storms in the desert. The sum of any three consecutive odd numbers is always divisible by 3. The measures of the angles of a triangle sum to 180. The wind is gusting at 50 mph. a, b, and c are consecutive odd numbers. 1, 2, and 3 are the angles of a triangle. There is a dust storm in the desert. a + b + c is divisible by 3. m 1 + m 2 + m 3 = 180 o. Page 3 of 5

5 Geometry Topic 5: Conditional statements and converses page 5 Student Activity Sheet 5.2; use with Exploring Conditional statements 7. Fill in the boxes to label the conditional statement and complete the Euler diagram. [EX1, page 7] 8. REINFORCE For each statement below, determine whether it is true or false and explain your answer. a. If I am in standing at the top of the Empire State Building, then I am in New York City. True. The Empire State Building is in New York City. b. If I stay home from school today, then I am sick. False. You might be home from school for a family emergency or a doctor s appointment. Being sick is not the only reason for missing school. Or, it might be a school holiday. c. If x = 3, then x(x 2 3x + 5) = 15. True. Given that x = 3, 3( ) = 3 5 = 15. d. If the sum of the measures of two angles is 180, then the two angles form a linear pair. False. The two angles must also be adjacent to form a linear pair. Page 4 of 5

6 Geometry Topic 5: Conditional statements and converses page 6 Student Activity Sheet 5.2; use with Exploring Conditional statements 9. REINFORCE Sketch Euler diagrams for the three conditional statements in question 8, showing the locations of hypotheses and conclusions, and using appropriate logic notation. q q dust storm in the desert p a + b + c is divisible by 3 p wind is gusting at 50 mph a, b, and c are consecutive odd numbers q m 1 + m 2 + m 3 = 180 p 1, 2, and 3 are the angles of a triangle Page 5 of 5

7 Geometry Topic 5: Conditional statements and converses page 7 Student Activity Sheet 5.3; use with Exploring Converses "If an animal is a dog, then it is a mammal." p à q 1. Given the above conditional statement and its notation, write the statement represented by the following notation. [EX2, page 1] If an animal is a mammal, then it is a dog. q à p 2. For a given conditional with the form p à q, what is the statement with the form q à p called? [EX2, page 2] The statement is called the converse of the conditional. 3. Write the converse of the following conditional statement. [EX2, page 2] If Cleo chases birds, then Cleo is a cat. "If Cleo is a cat, then Cleo chases birds." 4. Is the converse of a true conditional always true? Look at the following example to help answer this question. [EX2, page 3] Conditional: If an animal is a dog, then it is a mammal. Converse: If an animal is a mammal, then it is a dog. No, the converse of a true conditional is not always true. In this case, the converse is not true because not all mammals are dogs. Page 1 of 5

8 Geometry Topic 5: Conditional statements and converses page 8 Student Activity Sheet 5.3; use with Exploring Converses 5. A counterexample to a statement is any example that shows the statement is false. Name a counterexample for the converse in question 4. Then sketch an Euler diagram of the statement and show where your counterexample should be placed in the diagram. [EX2, pages 3-4] Answers will vary. Answer could be Treo the squirrel or any other mammal that is not a dog. 6. Write the converse of the Vertical Angle Theorem. [EX2, page 5] Vertical Angle Theorem: If two lines intersect forming vertical angles, then those angles have the same measure. Converse of the Vertical Angle Theorem: If two angles have the same measure, then they are vertical angles formed by two intersecting lines. Page 2 of 5

9 Geometry Topic 5: Conditional statements and converses page 9 Student Activity Sheet 5.3; use with Exploring Converses 7. In the conditional statement below, underline the hypothesis with a single line and the conclusion with a double line. Then use this statement to answer questions [EX2, pages 6-7] If two angles are vertical angles, then they have the same measure. 8. Represent the statement using logic notation. [EX2, pages 6-7] p à q 9. Represent the statement by sketching an Euler diagram. [EX2, pages 6-7] 10. Write the converse of the statement. [EX2, pages 6-7] If two angles have the same measure, then they are vertical angles. 11. Represent the converse using logic notation. [EX2, pages 6-7] q à p 12. Decide if the converse is true or false. If false, provide a counterexample. [EX2, pages 6-7] False. Counterexamples will vary. Sample solution: A counterexample is two 30 angles that are not vertical angles. Page 3 of 5

10 Geometry Topic 5: Conditional statements and converses page 10 Student Activity Sheet 5.3; use with Exploring Converses 13. Below are some true conditional statements. Using the answer choices provided, fill in the blanks to form the converse of each conditional. Also tell whether each converse is true or false. [EX2, page 8] you are 18 years old False you can vote x is an even number AC = CB True C is on the bisector of AB x is divisible by 4 Conditional: Converse: If you are 18 years old, then you can vote. If you can vote, then you are 18 years old. True or false? Conditional: False If x is a number divisible by 4, then x is an even number. Converse: If x is an even number, then x is divisible by 4. True or false? Conditional: False If C is on the bisector of AB, then AC = CB. Converse: If AC = CB, then C is on the bisector of AB. True or false? True Page 4 of 5

11 Geometry Topic 5: Conditional statements and converses page 11 Student Activity Sheet 5.3; use with Exploring Converses 14. When the statements "If p, then q" and "if q, then p" are both true, we can combine the conditional and its converse into one statement and say "p if and only if q." What is a statement in this format called? [EX2, page 9] A statement in this format is called a biconditional. 15. Combine the following conditional and its converse to form a biconditional statement. [EX2, page 9] If C is on the bisector of AB, then AC = CB. If AC = CB, then C is on the bisector of AB. C is on the bisector of AB if and only if AC = CB. Page 5 of 5

12 Geometry Topic 5: Conditional statements and converses page 12 Student Activity Sheet 5.4; use with Exploring Indirect proof 1. If a statement is true, then what must be the truth value of its negation? [EX3, page 1] A true statement has a false negation. 2. If a statement is false, then what must be the truth value of its negation? [EX3, page 1] A false statement has a true negation. 3. What is the name for a situation in which a statement and its negation have the same truth value? [EX3, page 2] A situation in which a statement and its negation have the same truth value is called a contradiction. 4. Consider the following everyday example of using indirect proof to reach a logical conclusion. Suppose that last weekend your friend Bill drove to see his girlfriend, Elena, who lives 125 miles away. He made the trip in 1.5 hours. You want to show that Bill exceeded the 70 mph speed limit. First, assume that Bill did not exceed the 70 mph speed limit. Then your assumption would mean that, in 1.5 hours, Bill would have traveled at most 105 miles. But in 1.5 hours, Bill traveled all the way to Elena's, 125 miles. In 1.5 hours, Bill cannot have traveled both 105 miles or less and also 125 miles! Your assumption has led to a contradiction. Therefore, your assumption is false, making its negation true. What conclusion does the contradiction logically imply? [EX3, page 3] The assumption that Bill did not exceed the 70 mph speed limit is false. Therefore, Bill exceeded the 70 mph speed limit. Page 1 of 4

13 Geometry Topic 5: Conditional statements and converses page 13 Student Activity Sheet 5.4; use with Exploring Indirect proof 5. Complete the table below by writing a beginning statement you might use to construct an indirect proof for each of the conditionals. [EX3, page 5] Conditional Beginning statement for indirect proof If an animal is a dog, then it is a mammal. Assume there is an animal that is not a mammal. If you don't wash your hands, then you will catch a cold. Assume you will not catch a cold. If 2x = 12, then x = 6. Assume x 6. If you are younger than 18, then you cannot vote. Assume that you can vote. Page 2 of 4

14 Geometry Topic 5: Conditional statements and converses page 14 Student Activity Sheet 5.4; use with Exploring Indirect proof 6. Complete the indirect proof of the Vertical Angle Theorem by filling in the blanks in the steps below. [EX3, page 6] Step 1: Assume the negation of the prove statement. Write your assumption below. Since we are to prove m DEC = m AEB, assume m DEC m AEB. Step 2: Reason until you reach a contradiction. Adding m DEA to both sides of the assumption tells us that: m DEC + m DEA m AEB + m DEA. But the angles in each sum form a linear pair, so m DEC + m DEA and m AEB + m DEA both equal 180. Thus, instead of m DEC + m DEA m AEB + m DEA we have m DEC + m DEA = m AEB + m DEA. Step 3: Because it results in a logical contradiction, the original assumption must be false. Step 4: Therefore, the assumption's negation must be true. Write the negation of the assumption. m DEC = m AEB Page 3 of 4

15 Geometry Topic 5: Conditional statements and converses page 15 Student Activity Sheet 5.4; use with Exploring Indirect proof Using the answer choices provided, fill in the blanks to complete the following indirect proof. [EX3, page 7] Given: Quadrilateral ABCD with m A = 80 Prove: ABCD is not a rectangle. ABCD is a rectangle ABCD is not a rectangle false Assume ABCD is a rectangle. This means that all four angles are right angles. But, by the Given, m A = The assumption that ABCD is a rectangle is false. 11.Therefore, ABCD is not a rectangle. 12.For each number provided below, determine if it is a prime number or a composite number and write it in the appropriate column in the table. [EX3, page 8] Prime numbers Composite numbers Page 4 of 4