Answer each of the following problems. Make sure to show your work. 1. What is a conjecture? 2. What does it mean if there is no counterexample for a conjecture? 3. What purpose would be served by a counterexample of the conjecture: All geometry students take classes online? 4. What does formal proof of a conjecture look like? 5. What is the conditional form of this statement? The intersection of two planes form a line. 6. Consider the statement: a = 4 if and only if a^2 = 16. Why is it a bi-conditional statement?
7. Rewrite the bi-conditional statement below as a conditional statement and as its converse. Statement: A touchdown is scored if and only if the football crosses the goal line. 8. How do you write the converse of a conditional statement like this one? You will pass geometry class if you pass all of the exams? 9. How does a conjecture differ from a proof? 10. Write a conjecture for the square below.
11. Based on the diagram below, Kate has made a few conjectures. Which of her conjectures are true? Conjecture 1: Line m bisects line JCH. Conjecture 2: Line FB is perpendicular to GH. Conjecture 3: Angles DCJ and DCH are supplementary. 12. Is the conjecture All prime numbers are odd true or false? Explain your answer. 13. What is the converse of this conditional statement? If I get 8 hours of sleep, then I will feel rested? 14. What was the original statement if the contrapositive is the statement below? If I do not get a raise, then I will not work hard at my job.
15. Write the converse, inverse, and contrapositive of the following conditional statement. If I snack too much, I will not want dinner. 16. What is true if the conditional statement If it rains, I will get wet is true? 17. Given the following statements, P and Q, write the syllogism. P = If I study hard, then I will pass geometry. Q = If I pass geometry, then I will get credit. 18. Decide if the following is a properly formed syllogism. If not, correct the statement(s) as needed. If it snows today, then I will wear my boots. If I wear my boots, then I need my socks. Therefore, if I wear my socks then it will snow today.
19. Given statements P, Q, and R, how does the Law of Syllogism differ from the Law of Detachment? The Law of Syllogism presents: whereas the Law of Detachment presents: 20. Apply the Law of Detachment to the following conditional statement: If you went to law school, then you are a lawyer. 21. Shortly after opening your email, you noticed a virus on your computer. The same happened to two of your classmates. You conclude that emails cause computer viruses. What type of reasoning is this an example of? Why? 22. What is the difference between inductive reasoning and deductive reasoning?
23. What is the next number in the pattern? 1, 8, 27, 64,??? Explain your answer. 24. Which word makes the conclusion in the scenario below true? Campus apartments do not allow pets. Stevie lives at the campus apartments. So, Stevie (must not, may) have pets. 25. Show this conjecture is false by finding a counterexample. Conjecture: Every real number squared is greater than or equal to the number itself. 26. What is the purpose of a counterexample? 27. What is the problem with the statement All 9 th graders are 16 years old? 28. What is a counterexample to the conjecture: All acute angles measure 50 degrees.
29. How is a paragraph proof different from a formal proof? 30. Which statement could be made from the diagram below? 31. Given the diagram below, what statement could you make about the relationship about angles 3 and 6?
32. Using the diagram below as reference, write a paragraph proof to prove that the angles 1 and 2 are congruent. Given: 1 and 2 are right angles. Prove: 1 and 2 are congruent. 33. If XYZ is an equilateral triangle, write a statement that represents the sides. 34. Which statement follows from the statement: X is complementary to Y?
35. Which theorem justifies the sum of 3 and 5 equal to 180 degrees? 36. If you are told that line FH is a bisector of angle EFG, what do you know from that statement?
37. Complete the two-column proof table below by filling in all of the blanks. Given: Q is the midpoint of line PR. Prove: PQ = ½ PR and QR = ½ PR Statements Reason Q is the midpoint of line PR. Given. PQ = QR (a) PQ + QR = PR (b) PQ + PQ = PR Substitution Property 2 PQ = PR Distributive Property (c) Division Property QR = ½ PR (d) 38. How would you prove that the triangle below is equilateral?
39. What statement could be obtained from the fact that angle E measures 50 degrees? Justify your statement. 40. If two angles are supplementary to the same angle then the angles are congruent. Given: 5 and 6 are linear pairs. 6 and 7 are linear pairs. Prove: 5 is congruent to 7