Chapter 2 Review - Formal Geometry
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1 *This packet is due on the day of the test:. It is worth 10 points. ALL WORK MUST BE SHOWN FOR FULL CREDIT!!! Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine if the conjecture is true or false. If false, give a counterexample. If n is a composite number, then n+1 is also a composite number. A. True. B. False; 8 is a composite number and 8+1 is not a composite number. C. False; 2 is a composite number and 2+1 is not a composite number. D. False; 4 is a composite number and 4+1 is not a composite number. 2. Which of the following are logically equivalent? A. A statement and its converse B. A statement and its inverse C. A statement and its contrapositive D. A statement, its converse, its inverse and its contrapositive 3. A four legged animal is called a dog. Which of the following best describes a counterexample to the assertion above? A. German Shepherd B. Bull Dog C. A Shark D. A Cat 4. Determine which statement follows logically from the given statements. If your book falls, then you must pick it up. Fallen books are most likely damaged. A. If I drop my book, then I am a klutz. B. If I damage my book, it is because I left it outside. C. Sometimes people drop their textbooks. D. If I drop my book, then it will most likely be damaged. 1
2 5. Determine whether the conjecture is true or false. Give a counterexample for any false conjecture. Given: Two angles are congruent. Conjecture: They are both right angles. A. False; two acute angles can have the same degree measure. B. True C. False; two adjacent angles can be congruent. D. False; they must be vertical angles. 6. Write the statement in if-then form. A counterexample invalidates a statement. A. If there is a counterexample, then it invalidates the statement. B. If there is a counterexample, then it is true. C. If it is true, then there is a counterexample. D. If it invalidates the statement, then there is a counterexample. 7. Determine whether a valid conclusion can be reached. If so, state the conclusion. Explain your reasoning. Given: If you practice basketball every day, then you will make it to the NBA! If you make it to the NBA, then you will make a lot of money. A. Valid; conclusion: If you practice basketball every day, then you can buy a big house because you will have a lot of money. Law of Syllogism. B. Invalid. I am not given the hypothesis. C. Valid; conclusion: If you practice basketball every day, then you will make a lot of money. Law of Syllogism. D. Valid; conclusion: If you practice basketball every day, then you will make a lot of money. Law of Detachment. 2
3 8. Determine whether a valid conclusion can be reached. If so, state the conclusion. Explain your reasoning. Given: Old Navy is selling sweaters on Monday at 15% off. Vanessa buys a sweater for 15% off. A. Valid; conclusion: Vanessa bought a sweater from Old Navy on Monday. Law of Detachment. B. Invalid; Vanessa bought the sweater on Tuesday. C. Valid; conclusion: Vanessa bought a sweater from Old Navy on Monday. Law of Detachment. D. Invalid; I am not given the hypothesis. 9. Find the value of x and AC if B is between points A and C. AB = x 1, BC = 5x, AC = 4x + 5 A. x = 1; AC = 1 B. x = 3; AC = 17 C. x = 2; AC = 13 D. x = 9; AC = 41 Use the figure below to answer questions In the figure, XA and XE are opposite rays, and AXC is bisected by XB. 10. If m AXC = 50x + 16 and m AXB = 30x 2, find m AXC. A. m AXC = 80x B. m AXC = 2 C. m AXC = 116 D. m AXC = If m DXC = 41, m DXE = 6x 1, and m CXE = 15x + 4, find m DXE. A. m DXE = 64 B. m DXE = 4 C. m DXE = 20 D. m DXE = 23 3
4 Given: 2(x 5) = 2x + 7 Prove: x = 3 4 Statements Reasons 2(x 5) = 2x + 7 Given 2x + 10 = 2x Addition Property of equality 3 = 4x Subtraction Property of equality 3 4 = x 14. x = 3 4 Symmetric Property of equality 12. Choose one of the following to complete the proof. A. Distributive Property B. Multiplication Property of Equality C. Addition Property of Equality D. Substitution Property of Equality 13. Choose one of the following to complete the proof. A. 2x = 2x 3 B. 4x = 17 C. 10 = 4x + 7 D. x 10 = Choose one of the following to complete the proof. A. Distributive Property B. Multiplication Property of Equality C. Addition Property of Equality D. Division Property of Equality 4
5 Given: H is the midpoint of GK K is the midpoint of HL Prove: GH KL Statements H is the midpoint of GK Given K is the midpoint of HL GH HK 15. Reasons 16. If K is the midpoint of HL, then HK KL. Midpoint Theorem. GH KL Transitive property of congruence. 15. Choose one of the following to complete the proof. A. If a point passes through the midpoint of a segment, then the segment is bisected at that point. Definition of segment bisector. B. If H is the midpoint of GK, then GH HK. Midpoint Theorem. C. If GH = HK, then GH HK. Definition of Congruence. D. If H is the midpoint of GK, then H bisects the segment. Definition of segment bisector. 16. Choose one of the following to complete the proof. A. HK KL B. GK HL C. HK HK D. GH KL 5
6 Given: FB bisects CFA Prove: m 1 = m 3 Statements Reasons FB bisects CFA Given Definition of Angle Congruence m 2 = m m 1 = m 3 Substitution property of equality 17. Choose one of the following to complete the proof. A. If part of a line has one endpoint and extends infinitely in one direction, then it is a ray. Definition of ray. B. If a line passes through the midpoint of a segment, then it intersects that segment. Definition of segment bisector. C. If a ray divides an angle into two congruent angles, then the ray is an angle bisector. Definition of angle bisector. D. If a point is the endpoint of two collinear rays, then the rays are opposite rays. Definition of opposite rays. 18. Choose one of the following to complete the proof. A. 1 = 2 B. 2 3 C. 1 2 D. m 1 = m Choose one of the following to complete the proof. A. If two angles are supplementary to a third angle or to congruent angles, then the two angles are congruent. Congruent Supplements Theorem. B. If two angles have a sum of 90 degrees, then the angles are complementary. Definition of complementary angles. C. If two angles are vertical angles, then they have equal measures. Vertical Angle Theorem. D. If two angles have a sum of 180 degrees, then the angles are supplementary. Definition of supplementary angles. 6
7 Given: 1 and 2 are supplements 3 and 4 are supplements 1 4 Prove: and 2 are supplements 3 and 4 are supplements 1 4 Given m 1 = m m 1 + m 2 = 180 m 3 + m 4 = 180 m 1 + m 2 = m 3 + m 4 m 1 + m 2 = m 3 + m 1 m 2 = m Definition of supplementary angles. Substitution Property of Equality Substitution Property of Equality Subtraction Property of Equality 20. Choose one of the following to complete the proof. A. If two angles have a sum of 180 degrees, then the angles are supplementary angles. Definition of supplementary angles. B. If two angles are congruent, then they have the same degree measure. Definition of angle congruence. C. Substitution Property of Equality. D. If two angles are supplementary to a third angle or to congruent angles, then the two angles are congruent. Congruent Supplements Theorem. 21. Choose one of the following to complete the proof. A. If two angles are supplementary to a third angle or to congruent angles, then the two angles are congruent. Congruent Supplements Theorem. B. If two angles are congruent, then they have the same degree measure. Definition of angle congruence. C. If two angles are vertical angles, then they have equal measures. Vertical Angle Theorem. D. If two angles have a sum of 180 degrees, then the angles are supplementary angles. Definition of supplementary angles. 7
8 PROOFS. There will be 2 proofs on the Chapter 2 Test. The proofs may come from this packet or from homework assignments (2.5, 2.6, 2.7, 2.8). They are worth 8 points apiece. Partial credit will be awarded. 22. Given: AC = BD Prove: AB = CD 23. Given: KX bisects SKI Prove: Prove YKX EKX 8
9 24. Given: 1 4, AFC EFC Prove: Given: CD DE Prove: CDF is comp FDE C F D E 9
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