1 Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No, the three points are not collinear. b. Yes, they lie on the line MP. c. Yes, they lie on the line NP. d. Yes, they lie on the line MO.. Name the plane represented by the front of the box. a. FBC b. BAD c. FEC d. FKG 3. Are points B, J, and C collinear or noncollinear? a. collinear b. noncollinear c. impossible to tell. Name the line and plane shown in the diagram. R U S T a. c. and plane RSU b. line R and plane RSU d. and plane UR and plane UT
2 5. What is the intersection of plane TUYX and plane VUYZ? a. b. c. d.. Name the intersection of plane BPQ and plane CPQ. a. c. b. d. The planes need not intersect. 7. Name a fourth point in plane TUW. a. Y b. Z c. W d. X 8. two points are collinear. a. Any b. Sometimes c. No 9. Plane ABC and plane BCE be the same plane. a. must b. may c. cannot 10. and be coplanar. a. must b. may c. cannot 11. Which diagram shows plane PQR and plane QRS intersecting only in?
3 a. c. b. d. 1. Name the ray in the figure. A B a. b. c. d. 13. Name the ray that is opposite C D B A a. b. c. d. 1. Name the four labeled segments that are skew to
4 a.,,, c.,,, b.,,, d.,,, 15. Name the three labeled segments that are parallel to a.,, b.,, c.,,, d.,, 1. How many pairs of skew lines are shown? a. b. 1 c. 8 d. 17. Which plane is parallel to plane EFHG? a. plane ABDC b. plane ACGE c. plane CDHG d. plane BDHF 18. Find the distance between points P(8, ) and Q(3, 8) to the nearest tenth. a. 11 b. 7.8 c. 1 d The Frostburg-Truth bus travels from Frostburg Mall through the City Center to Sojourner Truth Park. The mall is 3 miles west and miles south of the City Center. Truth Park is miles east and 5 miles north of the Center. How far is it from Truth Park to the Mall to the nearest tenth of a mile? a. 9.9 miles b. 3. miles c. 3. miles d.. miles
5 0. A high school soccer team is going to Columbus to see a professional soccer game. A coordinate grid is superimposed on a highway map of Ohio. The high school is at point (3, ) and the stadium in Columbus is at point (7, 1). The map shows a highway rest stop halfway between the cities. What are the coordinates of the rest stop? What is the approximate distance between the high school and the stadium? (One unit. miles.) a. c., 5 miles, 3 miles b., 10 miles d., 1 miles 1. Each unit on the map represents 5 miles. What is the actual distance from Oceanfront to Seaside? y 8 Seaside 8 8 x Landview Oceanfront 8 a. about 10 miles c. about 8 miles b. about 50 miles d. about 0 miles. Find the coordinates of the midpoint of the segment whose endpoints are H(8, ) and K(, 10). a. (7, ) b. (1, ) c. (1, 1) d. (, 8) 3. Find the midpoint of 10 y P x 5 Q 10 a. ( 3, 1) b. (, 0) c. (, 1) d. ( 3, 0)
6 . M(9, 8) is the midpoint of The coordinates of S are (10, 10). What are the coordinates of R? a. (9.5, 9) b. (11, 1) c. (18, 1) d. (8, ) 5. M is the midpoint of for the points C(3, ) and F(9, 8). Find MF. a. 13 b. 13 c. d. 13. Write the two conditional statements that make up the following biconditional. I drink juice if and only if it is breakfast time. a. I drink juice if and only if it is breakfast time. It is breakfast time if and only if I drink juice. b. If I drink juice, then it is breakfast time. If it is breakfast time, then I drink juice. c. If I drink juice, then it is breakfast time. I drink juice only if it is breakfast time. d. I drink juice. It is breakfast time. 7. One way to show that a statement is NOT a good definition is to find a. a. converse c. biconditional b. conditional d. counterexample 8. Use the Law of Detachment to draw a conclusion from the two given statements. If two angles are congruent, then they have equal measures. and are congruent. a. + = 90 c. is the complement of. b. = d. 9. Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible. I can go to the concert if I can afford to buy a ticket. I can go to the concert. a. I can afford to buy a ticket. b. I cannot afford to buy the ticket. c. If I can go to the concert, I can afford the ticket. d. not possible 30. Which statement is the Law of Detachment? a. If is a true statement and q is true, then p is true. b. If is a true statement and q is true, then is true. c. If and are true, then is a true statement. d. If is a true statement and p is true, then q is true. 31. If possible, use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible. Statement 1: If x = 3, then 3x = 5. Statement : x = 3 a. 3x = 5 c. If 3x = 5, then x = 3. b. x = 3 d. not possible 3. Use the Law of Syllogism to draw a conclusion from the two given statements. If a number is a multiple of,then it is a multiple of 8.
7 If a number is a multiple of 8, then it is a multiple of. a. If a number is a multiple of, then it is a multiple of. b. The number is a multiple of. c. The number is a multiple of 8. d. If a number is not a multiple of, then the number is not a multiple of. 33. Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements. If an elephant weighs more than,000 pounds, then it weighs more than Jill s car. If something weighs more than Jill s car, then it is too heavy for the bridge. Smiley the Elephant weighs,150 pounds. a. Smiley is too heavy for the bridge. b. Smiley weighs more than Jill s car. c. If Smiley weighs more than 000 pounds, then Smiley is too heavy for the bridge. d. If Smiley weighs more than Jill s car, then Smiley is too heavy for the bridge. 3. Which statement is the Law of Syllogism? a. If is a true statement and p is true, then q is true. b. If is a true statement and q is true, then p is true. c. if and are true statements, then is a true statement. d. If and are true statements, then is a true statement. 35. Find the values of x, y, and z. The diagram is not to scale x z y a. c. b. d. 3. Find the value of x. The diagram is not to scale x a. 33 b. 1 c. 17 d Find the value of the variable. The diagram is not to scale.
8 11 x 7 a. b. 19 c. 9 d Find the missing values of the variables. The diagram is not to scale x y 5 a. x = 1, y = 15 c. x = 11, y = 5 b. x = 5, y = 11 d. x = 5, y = Graph y = 3 x 1. a. y c. y x x b. y d. y x x
9 0. Graph. a. y c. 8 8 y 8 8 x x 8 b. 8 y d. 8 y 8 8 x x 8 1. Write an equation in point-slope form of the line through point J( 5, ) with slope. a. c. b. d.. Write an equation in point-slope form of the line through points (, ) and (1, ). Use (, ) as the point (x 1, y 1 ). a. (y ) = (x + ) c. (y + ) = (x ) b. (y ) = (x + ) d. (y + ) = (x ) 3. Write an equation for the horizontal line that contains point E( 3, 1). a. x = 1 b. x = 3 c. y = 1 d. y = 3. Graph the line that goes through point ( 5, 5) with slope 1 5.
10 a. y c. y x x b. y d. y x x 5. Write an equation in slope-intercept form of the line through point P( 10, 1) with slope 5. a. y = 5x 9 c. y 10 = 5(x + 1) b. y 1 = 5(x + 10) d. y = 5x + 1. Write an equation in slope-intercept form of the line through points S( 10, 3) and T( 1, 1). a. 9 x + 13 c. 9 9 x 13 9 b. y = 9 x 13 9 d. y = 9 x Write an equation for the line parallel to y = 7x + 15 that contains P(9, ). a. x + = 7(y 9) c. y = 7(x 9) b. y + = 7(x 9) d. y + = 7(x 9) 8. Is the line through points P(1, 9) and Q(9, ) perpendicular to the line through points R(, 0) and S( 9, 8)? Explain. a. Yes; their slopes have product 1. b. No, their slopes are not opposite reciprocals. c. No; their slopes are not equal. d. Yes; their slopes are equal.
11 9. Write an equation for the line perpendicular to y = x 5 that contains ( 9, ). a. y = (x + 9) c. y 9 = 1 (x + ) b. x = (y + 9) d. y = 1 (x + 9) 50. Are the lines y = x and x + y = 1 perpendicular? Explain. a. Yes; their slopes have product 1. b. No; their slopes are not opposite reciprocals. c. Yes; their slopes are equal. d. No; their slopes are not equal 51. In each pair of triangles, parts are congruent as marked. Which pair of triangles is congruent by ASA? a. c. b. d. 5. Can you use the ASA Postulate, the AAS Theorem, or both to prove the triangles congruent? a. either ASA or AAS c. AAS only b. ASA only d. neither 53. What else must you know to prove the triangles congruent by ASA? By SAS?
12 ( ( ( A B ( D C a. ; c. ; b. ; d. ; 5. Based on the given information, what can you conclude, and why? Given: I K J H L a. by ASA c. by ASA b. by SAS d. by SAS 55. Find the values of x and y. A y x 7 B D C Drawing not to scale a. c. b. d. 5. Find the value of x. The diagram is not to scale.
13 0 x a. 3 b. 50 c. d The length of G is shown. What other length can you determine for this diagram? D 1 F E a. EF = 1 c. DF = b. DG = 1 d. No other length can be determined. 58. bisects Find FG. The diagram is not to scale. E n + 8 F 3n ) ) D G a. 15 b. 1 c. 19 d Find the center of the circle that you can circumscribe about the triangle.
14 ( 3, 3) 5 y 5 5 x ( 3, ) (1, ) 5 a. ( 1, 1) b. ( 1, 1 ) c. ( 3, 1 ) d. ( 1, ) 0. Where is the center of the largest circle that you could draw inside a given triangle? a. the point of concurrency of the altitudes of the triangle b. the point of concurrency of the perpendicular bisectors of the sides of the triangle c. the point of concurrency of the bisectors of the angles of the triangle d. the point of concurrency of the medians of the triangle 1. Where can the perpendicular bisectors of the sides of a right triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle a. I only b. II only c. I or II only d. I, II, or II. Where can the bisectors of the angles of an obtuse triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle a. I only b. III only c. I or III only d. I, II, or II 3. In ACE, G is the centroid and BE = 9. Find BG and GE. C B G D A F E a. BG = 1, GE = 3 c. b. d. BG = 1, GE = 1
15 . Name a median for A D E ) ) C F B a. b. c. d. 5. For a triangle, list the respective names of the points of concurrency of perpendicular bisectors of the sides bisectors of the angles medians lines containing the altitudes. a. incenter circumcenter centroid orthocenter b. circumcenter incenter centroid orthocenter c. circumcenter incenter orthocenter centroid d. incenter circumcenter orthocenter centroid. Where can the lines containing the altitudes of an obtuse triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle a. I only b. I or II only c. III only d. I, II, or II 7. Which diagram shows a point P an equal distance from points A, B, and C? a. c. b. d. 8. Name the second largest of the four angles named in the figure (not drawn to scale) if the side included by and is 11 cm, the side included by and is 1 cm, and the side included by and is 1 cm.
16 1 3 a. b. c. d. 9. Name the smallest angle of C The diagram is not to scale. 5 A 7 B a. b. c. Two angles are the same size and smaller than the third. d. 70. Which three lengths can NOT be the lengths of the sides of a triangle? a. 3 m, 17 m, 1 m c. 5 m, 7 m, 8 m b. 11 m, 11 m, 1 m d. 1 m, m, 10 m 71. Two sides of a triangle have lengths 7 and 15. Which inequalities represent the possible lengths for the third side, x? a. c. b. d. 7. Two sides of a triangle have lengths 10 and 15. What must be true about the length of the third side? a. less than 5 b. less than 10 c. less than 15 d. less than 5 Short Answer 73. Construct the perpendicular bisector of the segment.
17 7. Construct the bisector of 75. Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, state not possible. Explain. Statement 1: If two lines intersect, then they are not parallel. Statement : do not intersect. 7. For the given statements below, write the first statement as a conditional in if-then form. Then, if possible, use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible. Explain. A straight angle has a measure of 180. is a straight angle. 77. Write the missing reasons to complete the flow proof. Given: Prove: are right angles, B A ) D ( C
18 78. Can you conclude the triangles are congruent? Justify your answer. Essay 79. Write a paragraph proof to show that. Given: and B D C A E 80. Write a two-column proof. Given: and Prove:
19 B D C A E 81. Write a two-column proof: Given: Prove: B C A D Other 8. Use indirect reasoning to explain why a quadrilateral can have no more than three obtuse angles. 83. Keegan knows that the statement if a figure is a rectangle, then it is a square is false, but he thinks the contrapositive is true. Is he correct? Explain. 8. Explain why P, Q, and R are three different points. PQ = 3x +, QR = x, and RP = x +, and.. List the angles of PQR in order from largest to smallest and justify your response. 8. Two sides of a triangle have lengths and 8. What lengths are possible for the third side? Explain.
20 Semester Exam Review Answer Section MULTIPLE CHOICE 1. A. A 3. A. A 5. A. A 7. B 8. A 9. B 10. B 11. C 1. A 13. A 1. A 15. A 1. A 17. A 18. B 19. A 0. C 1. D. A 3. C. D 5. A. B 7. D 8. B 9. D 30. D 31. A 3. A 33. A 3. C 35. D 3. A 37. B 38. C 39. D 0. C
21 1. D. D 3. C. D 5. A. D 7. D 8. B 9. D 50. B 51. B 5. C 53. B 5. A 55. D 5. C 57. A 58. B 59. B 0. C 1. B. A 3. B. D 5. B. C 7. A 8. A 9. D 70. D 71. D 7. A SHORT ANSWER 73.
22 Not possible. Explanations may vary. Sample: To use the Law of Detachment, you must know that the hypothesis is true. 7. If an angle is straight, then its measure is 180. Conclusion: is a. Definition of right triangles b. Converse of Isosceles Triangle Theorem c. Reflexive property d. Angle-Angle-Side Congruence Theorem 78. Yes, the diagonal segment is congruent to itself, so the triangles are congruent by SAS. ESSAY 79.  Answers may vary. Sample: You are given that and. Vertical angles BCA and ECD are congruent, so by SAS.  correct idea, some details inaccurate  correct idea, not well organized  correct idea, one or more significant steps omitted 80.  Statement Reason 1. and 1. Given.. Vertical angles are congruent SAS.. CPCTC  correct idea, some details inaccurate  correct idea, not well organized  correct idea, one or more significant steps omitted 81.  Statement 1. and Reason 1. Given.. Reflexive Property ASA.. CPCTC  correct idea, some details inaccurate  correct idea, not well organized
23  correct idea, one or more significant steps omitted OTHER 8. Assume a quadrilateral has more than three obtuse angles. Then it has four angles, each with a measure greater than 90. Their sum is greater than 30, which contradicts the fact that the sum of the measures of the angles of a quadrilateral is 30. Thus a quadrilateral can have no more than three obtuse angles. 83. No; a statement and its contrapositive are equivalent so they have the same truth value. 8. The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. 85. R, Q, P. Sample: Since x = QR > 0, x < x + < 3x +, so QR < RP < PQ. The largest angle ( R) is opposite PQ, the next largest angle ( Q) is opposite RP. 8. Let x be the length of the third side. By the Triangle Inequality Theorem, + x > 8, + 8 > x, and 8 + x >. Solving each inequality, x >, x < 1, and x >, respectively, or < x < 1.
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GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology firstname.lastname@example.org 1 iameter of incircle T Lemma 1. Let the incircle of triangle
summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of mid-point and segment bisector M If a line intersects another line segment
47 Similar Triangles An overhead projector forms an image on the screen which has the same shape as the image on the transparency but with the size altered. Two figures that have the same shape but not
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
Sec 1.6 CC Geometry Triangle Proofs Name: POTENTIAL REASONS: Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point