Provide (write or draw) a counterexample to show that the statement is false.
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1 Geometry SOL G.1 G.3a Study Guide Name: Date: Block: SHOW ALL WORK. Use another piece of paper as needed. SECTION 1: G.1 1. Provide (write or draw) a counterexample to show that the statement is false. a. If the quotient of two numbers is positive, then the two numbers must both be positive. b. When solving x 2 = a, where a is a positive real number, the value of x must be positive. c. The sum of any two prime numbers is always even. d. All intersecting lines are perpendicular. 2. Write the following sentence in symbolic notation if p: it is summer, q: we do not have school, and r: it is a school holiday. a. If it is summer, then we do not have school. b. If we do not have school, then it is summer or it is a school holiday. c. If it is a school holiday then it is not summer. d. If it is not summer and it is not a school holiday, then we do have school. e. If we have school, then it is not summer. f. If it is not a school holiday and we have school, then it is not summer. 3. Underline the hypothesis and circle the conclusion in the conditional statements. a. If negotiations fail, then the baseball strike will not end. b. If an animal is a panther, then it lives in the forest. c. If an angle measures 30 degrees, then it is acute. d. If two angles are supplementary, then their sum is 180 degrees. e. If I don't play sports then I have to get a job. 4. Write the sentence as each of the following statements (if possible). A poet is a writer. a. Conditional statement: b. Converse: c. Inverse: d. Contrapositive: e. Can we make a valid biconditional out of the statement A poet is a writer? Explain your answer.
2 Geometry SOL G.1 G.3a Study Guide Page 2 5. Assuming the statement A poet is a writer used in question 2 is true, write the truth values (true or false) of the logical statements you wrote above: a. Converse: b. Inverse: c. Contrapositive: 6. Write the sentence as each of the following statements (if possible). A congressman is a politician. a. Conditional statement: b. Converse: c. Inverse: d. Contrapositive: e. Can we make a valid biconditional out of the statement A congressman is a politician? Explain your answer. 7. Assuming the statement A congressman is a politician. used in question 2 is true, write the truth values (true or false) of the logical statements you wrote above: a. Converse: b. Inverse: c. Contrapositive: Use the Laws of Logic to determine a conclusion from the information given in each example. State the Law of Logic used or write "None" if no valid conclusion can be reached. 8. If you save $30 then you will buy a CD player. You saved $ If a quadrilateral is a square, then it has four right angles. If a quadrilateral has four right angles, then it is a rectangle. 10. If you live in Ashburn, then you live in Virginia. Mary lives in Virginia. 11. If you understand the law of detachment, then you will do well on the test. If you do well on the test, then you will get a passing grade.
3 Geometry SOL G.1 G.3a Study Guide Page Construct a Venn/Euler diagram for the statement. a. If you drive 80 miles per hour, then you are breaking the law. b. If an animal is a husky, then it is a dog. If an animal is a dog, then it is a mammal. c. Some students at Eagle Ridge take keyboarding. Some Eagle Ridge students take band. Some students take keyboarding and band. 13. Write a conditional statement that corresponds to each of the following Euler diagrams. a. b. 14. Use the following Venn diagram to answer the following questions. a. How many total cars are in the lot? b. How many total trucks are in the lot? c. How many vehicles in the lot are not trucks? d. How many vehicles in the lot are not red? e. How many vehicles in the lot are not trucks and are not red? f. How many vehicles are red trucks? SECTION 2: G.3a 1. The town of Appleton is between Bakersfield and Cartersville. The distance from Appleton to Bakersfield is 27 miles and the distance from Bakersfield to Cartersville is 78 miles. How far is it from Appleton to Cartersville? 2. Find the lengths of all segments.
4 Geometry SOL G.1 G.3a Study Guide Page 4 3. Find the midpoint of the segments with the following endpoints. Endpoint 1 Midpoint Endpoint 2 a. (5, 2) (-7, -4) b. (2, -1) (-6, 0) c. (-1, -6) (-4, 5) d. (-4, 4) (5, -1) 4. Find the other endpoint of the line segment with the given endpoint and midpoint. Endpoint 1 Midpoint Endpoint 2 a. (-1, 9) (-9, 10) b. (2, 5) (5, 1) c. (-6, 4) (4, 8) d. (5, 2) (-10, -2) 5. The endpoints of two segments are given. Find the exact length of each segment in simplest radical form. 6. Endpoint 1 Endpoint 2 Distance a. A(0, -2) B(-5, -1) b. C(3, 8) D(9, 10) c. E(10, 1) F(9, -4) d. G(-8, 10) H(-6, 7) e. Tell whether any of the segments are congruent. Find the distance in simplest radical form between points: a) A and B b) C and F c) D and E 7. In the figure shown, m 2 = (6x + 2) o and m 3 = (8x - 10) o. a. What is m 1, m 2, and m 3? b. What is the measure of m 1 + m 2 + m 3?
5 Geometry SOL G.1 G.3a Study Guide Page 5 8. a. Tell whether angle 1 and 2 are vertical angles, adjacent angles, a linear pair, complementary, or supplementary. b. If m<1 = 25 o, what is m<2? 9. Use the information to find the value of x and the measures of the angles. a. m HGF =16x+ 4, m EGF = 110, and m HGE =3x+ 11. Find m HGE and m HGF. b. m ABC =17x+ 8, m ABK = 42, and m KBC =12x- 4. Find m ABC and m KBC. c. If ZYX = 5x - 1, DEF = 6x+ 3, and ZYX is complementary to DEF, find m DEF and m ZYX. d. If LMN = 3x -10, STV = 12x + 25, and LMN is supplementary to STV, find m STV and m LMN. e. 1 and 2 form a vertical pair. 1= 5x o and 2 = (9x 4) o. Find m 1 and m 2. f. Two angles form a linear pair. The measure of one angle is four times greater than the measure of the other angle. Find the measure of each angle.
6 Geometry SOL G.1 G.3a Study Guide Page 6 STUDY GUIDE ANSWERS SECTION 1: G.1 1. a. Answers will vary: (-6)/(-3) = 2 (The quotient of 2 negative numbers is positive) b. Answers will vary: If x = -3, then x 2 = (-3) 2 = 9 (Taking the square of any negative number results in an answer that is positive) c. Answers will vary: 2 and 3 are prime numbers and their sum is 5 which is odd. d. Not all lines have perpendicular intersections. 2. a. p q b. q p r c. r p d. p r q e. q p f. r q p 3. a. If negotiations fail, then the baseball strike will not end. b. If an animal is a panther, then it lives in the forest. c. If an angle measures 30 degrees, then it is acute. d. If two angles are supplementary, then their sum is 180 degrees. e. If I don't play sports then I have to get a job. 4. a. Conditional statement: If you are a poet, then you are a writer. b. Converse: If you are a writer, then you are a poet. c. Inverse: If you are not a poet, then you are not a writer. d. Contrapositive: If you are not a writer, then you are not a poet. e. To write a valid biconditional statement, both the conditional statement and converse must both be true. Since the conditional statement is true and the converse is false, you cannot write a valid biconditional for this statement. 5. a. Converse: False b. Inverse: False c. Contrapositive: True 6. a. Conditional statement: If you are a congressman, then you are a politician. b. Converse: If you are a politician, then you are a congressman. c. Inverse: If you are not a congressman, then you are not a politician. d. Contrapositive: If you are not a politician, then you are not a congressman. e. To write a valid biconditional statement, both the conditional statement and converse must both be true. Since the conditional statement is true and the converse is false, you cannot write a valid biconditional for this statement. 7. a. Converse: False b. Inverse: False c. Contrapositive: True 8. By the Law of Detachment: You will buy a CD player. 9. By the Law of Syllogism: If a quadrilateral is a square, then it is a rectangle. 10. No valid conclusion (Mary could live in Ashburn but she doesn't necessarily live in Ashburn)
7 Geometry SOL G.1 G.3a Study Guide Page By the Law of Syllogism: If you understand the law of detachment, then you will get a passing grade. 12. a. b. c. 13. a. Possible correct statements: If you live in Virginia, then you live in the USA. If you are a resident of Loudoun County, then you live in Virginia. If you are a resident of Loudoun County, then you live in the USA. b. If you are allergic to nuts, then you do not eat peanut butter. 14. a. 87 b. 33 c. 54 d. 50 e. 31 f. 14 SECTION 2: G.3a miles 2. x = 3 3. Midpoint 4. Endpoint 2 a. (-1, -1) a. (-17, 11) b. 1 b. (8, -3) 2, 2 c. (14, 12) c. d. (-25, -6) 5 1, 2 2 d. 1 3, Distance 6. a. 2 5 a. 26 b b. 10 c. 26 d. 13 c. 17 e. AB EF 7. a. m 1 = 90 o m 2 = 44 o m 3 = 46 o b. m 1 + m 2 + m 3 = 180 o 9. a. m HGE = 38 o ; m HGF = 148 o b. m ABC = 110 o ; m KBC = 68 o c. m DEF = 51 o ; m ZYX = 39 o d. m STV = 157 o ; m LMN = 23 o e. m 1 = 5 o ; m 2 = 5 o f. 36 o and 144 o 8. a. Adjacent angles b. m 2 = 55 o
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