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Chapter 2 Review Packet Geometry ANSWER KEY If a polygon has three sides, then it is a triangle. TV: True 1. Identify the hypothesis: A polygon has three sides 2. Identify the conclusion: It is a triangle 3. What is the converse: If a polygon is a triangle, then it has three sides TV: True Determine the truth value of each conditional. If it is true, draw a venn diagram. If it is false, give a counterexample. 4. If a number is divisible by 5, then it is odd. TV: False; 10 Wisconsin 5. If Susan lives in Milwaukee, then she lives in Wisconsin. TV: True Milwaukee Susan 6. Write the following biconditional as two separate true conditionals. Two angles are complementary iff the measures of the angles have a sum of 90 . Conditional #1: If two angles are complementary, then the measures of the angles have a sum of 90 Conditional #2: If two angles measures have a sum of 90 , then they are complementary. Determine whether each statement is a good or bad definition. If it is a good, then write it as a biconditional. If it is bad, give (or draw) a detailed counterexample. 7. An angle bisector is a ray that divides an angle into two congruent angles. Bad definition; does not have to be a ray 8. A bird is a creature that flies. Bad definition; insect Use the following statements …. A. If Cindy goes to the mall she will buy a pair of jeans. B. If Cindy buys a pair of jeans, she will spend at least $30. C. Cindy went to the mall. 9. What can you conclude from A and B and which Law(s) did you use to draw this conclusion? If Cindy goes to the mall, then she will spend at least $30; Law of Syllogism 10. What can you conclude from A and C and which Law(s) did you use to draw this conclusion? Cindy will buy a pair of jeans; Law of Detachment 11. What can you conclude from all three statements and which Law(s) did you use to draw this conclusion? Cindy will spend at least $30; Law of Syllogism and Law of Detachment 12. If an animal is a fish, then it lives in the water. TV: True Converse: If an animal lives in the water, then the animal is a fish. TV: False Inverse: If an animal is NOT a fish, then the animal does NOT live in the water. TV: False Contrapositive: If an animal does NOT live in the water, then the animal is NOT a fish. TV: True 13. If a circle has an area of 16π cm2, then its radius is 4 cm. Converse: If a circle has a radius of 4 cm, then the circle has an area of 16π cm2. Biconditional: A circle has an area of 16π cm2 if an only iff its radius is 4 cm. Why can you write the biconditional? It is a good definition because both the Conditional and Converse have a truth value that is TRUE. 14. Using the information below, what can you conclude? Which law did you use to draw your conclusion? If the tire is flat, then Jack changes his tire. The tire is flat. Jack changes his tire; Law of Detachment 15. Using the information below, what can you conclude? What law did you use to draw your conclusion? If it is summer, then Tara is not in school. If Tara is not in school, then she gets to sleep in. It is July. What can you conclude? What law(s) did you use? Tara gets to sleep in Law of Syllogism and Law of Detachment 16. Name the property that justifies each: a) 𝐼𝑓 𝑥 = 𝑦, 𝑦 = 7𝑡ℎ𝑒𝑛 𝑥 = 7 a) Transitive Property of Equality b) If QR ST then ST QR b) Symmetric Property of Congruency c) If x = 3 then x 3 2 2 d) 4(x – 3) = 4x – 12 c) Division Property of Equality d) Distributive Property 17. Solve for x . 4 x 12 3 x 34 3 x 3x x 12 34 12 12 x 22 18. Given ∠A is three times as large as its supplement ∠B, find the measure of ∠A and ∠B. 23.______________ a b 180 a 3b 3b b 180 4b 180 b 45 a 135 19. 18. A 135 and B 45 Given: 𝑚∠𝑀𝐴𝐻 = 𝑥 + 5, 𝑚∠𝑇𝐴𝐻 = 𝑥 − 3, and 𝑚∠𝑀𝐴𝑇 = 12 H M Prove: 𝑥 = 5 A Statements 1. mMAH x 5, mTAH x 3, mMAT 12 Reasons 1. Given 2. mMAH mTAH mMAT 2. Angle Addition Postulate 3. x 5 x 3 12 3. Substitution Property 4. 2x 2 12 4. Simplify 5. 2 x 10 5. Subtraction Property of Equality 6. x 5 6. Division Property of Equality T 20. Write a two column proof. You may not do this in 2-steps. Given: 1 and ∠2 are supplementary angles. Give a∠paragraph proof for the Congruent Supplements Theorem. Prove: ∠2 ∠3 1 2 3 1. 1 and 2 are supplementary 1. Given 2. 1 and 3 are supplementary 2. Definition of Linear Pair 3. m1 m2 180; m1 m3 180 3. Definition of Supplementary Angles 4. m1 m2 m1 m3 4. Transitive Property of Equality or Substitution Property 5. m2 m3 5. Subtraction Property of Equality 6. 6. 2 3 21. Solve for x and write a two-column proof. Given: KL 2 x 5, LM 2 x, and KM 55 Prove: x 15 55 K 2x 5 L 1. KL 2 x 5, LM 2 x, and KM 55 1. Given 2. KL LM KM 2. Segment Addition Postulate 3. 2x 5 2x 55 3. Substitution Property 4. 4x 5 55 4. Simplify 5. 4x 60 5. Addition Property of Equality 6. x 15 6. Division Property of Equality 2x M 22. Solve for 𝑞 and write a two-column proof. Given: 𝑚∠𝐴𝐵𝐷 = 7𝑞 − 46 and 𝑚∠𝐶𝐵𝐷 = 3𝑞 + 6 Prove: 𝑚∠𝐶𝐵𝐷 = 72 1. mABD 7q 46 and mCBD 3q 6 1. Given 2. ABD and CBD are supplementary 2. Definition of Linear Pair 3. mABD mCBD 180 3. Definition of Supplementary Angles 4. 7q 46 3q 6 180 4. Substitution Property 5. 10q 40 180 5. Simplify 6. 10q 220 6. Addition Property of Equality 7. Division Property of Equality 7. q 22 8. Substitution Property 8. mCBD 3(22) 6 9. Simplify 9. mCBD 72 23. Write a two-column proof. Given: ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 1. 2 4 1. Given 2. 2 1 2. Vertical angles are 3. 1 4 3. Transitive Property of Congruency 4. 3 4 4. Vertical angles are 5. 1 3 5. Transitive Property of Congruency 24. Write a two column proof. You may not do this in 2-steps. Given: ∠1 & ∠2 are complementary ∠3 & ∠2 are complementary Prove: ∠1 ≅ ∠3 1. 1 and 2 are complentary 3 and 2 are complentary 2. m1 m2 90 m3 m2 90 1. Given 2. Definition of Linear Pair 3. m1 m2 m3 m2 4. m1 m3 5. 1 3 3. Transitive Property of Equality or Substitution Property 4. Subtraction Property of Equality 5. 25. Solve for 𝑥. Write a two-column proof. Given: 2(3𝑥 − 4) + 11 = 𝑥 − 27 Prove: 𝑥 = −6 1. 2(3x 4) 11 x 27 1. Given 2. 6x 8 11 x 27 2. Distributive Property 3. 6x 3 x 27 3. Simplify 4. 5x 3 27 4. Subtraction Property of Equality 5. 5x 30 5. Subtraction Property of Equality 6. x 6 6. Division Property of Equality