# Formal Geometry S1 (#2215)

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1 Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Formal Geometry S1 (#2215) When used as test practice, success on the Instructional Materials does not guarantee success on the district math common final. Students can use these Instructional Materials to become familiar with the format and language used on the district common finals. Familiarity with standards and vocabulary as well as interaction with the types of problems included in the Instructional Materials can result in less anxiety on the part of the students. The length of the actual final exam may differ in length from the Instructional Materials. Teachers can use the Instructional Materials in conjunction with the course guides to ensure that instruction and content is aligned with what will be assessed. The Instructional Materials are not representative of the depth or full range of learning that should occur in the classroom. *Students will be allowed to use a non-programmable scientific calculator on Formal Geometry Semester 1 and Formal Geometry Semester 2 final exams.

2 Formal Geometry Reference Sheet Note: You may use these formulas throughout this entire test. Linear Quadratic Slope m = y 2 y 1 x 2 x 1 Vertex-Form y = a(x h) 2 + k Midpoint M = ( x 1 + x 2 2, y 1 + y 2 ) Standard Form y = ax 2 + bx + c 2 Distance d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Intercept Form y = a(x p)(x q) Slope-Intercept Form y = mx + b Exponential Probability (h, k) Form y = ab x h + k P(A and B) = P(A) P(B) P(A and B) = P(A) P(B A) P(A or B) = P(A) + P(B) P(A and B) Volume and Surface Area V = πr 2 h SA = 2(πr 2 ) + h(2πr) V = 4 3 πr3 SA = 4πr 2 V = 1 3 πr2 h SA = πr (2πr l) 2 V = 1 3 Bh SA = B (Pl) Where B =base area and P =base perimeter

3 Multiple Choice: Identify the choice that best completes the statement or answers the question. Figures are not necessarily drawn to scale. 1. Identify which of the following is the best name for the figure formed by the coordinates: A( 1, 4), B(1, 1), C(2, 2). A. scalene triangle C. equilateral triangle B. isosceles triangle D. obtuse triangle 2. A pilot is flying an airplane on a straight path from Norfolk to Madison. On the trip, the pilot stops to refuel exactly halfway in between at Columbus and decides to program the autopilot for the rest of the trip. The pilot knows the coordinates for Norfolk are (36.9, 76.3) and the coordinates for Columbus are (39.9, 83.0). What coordinates should the pilot use for Madison? A. ( 1.5, 3.3) C. (33.9, 69.6) B. (61.5, 56.6) D. (42.9, 89.7) 3. In the diagram below, R is the midpoint of AB. T is the midpoint of AC. S is the midpoint of BC. Find the area of RST and AB. A. Area of RST = 4; AB 4 5 B. Area of RST = 8; AB 4 5 C. Area of RST = 4; AB 8 5 D. Area of RST = 8; AB 8 5

4 4. Given the coordinates below, compare RS and XY and determine which of the following statements is true: R( 2, 7) S(5, 1) X( 3, 3) Y(6, 1) A. The midpoints of RS and XY have the same x-coordinate. B. The midpoints of RS and XY have the same y-coordinate. C. The length of RS and the length of XY are the same. D. The length of RS is longer than the length of XY. 5. Given the following: B is a complement of A C is a supplement of B D is a supplement of C E is a complement of D F is a complement of E G is a supplement of F Then which angle is congruent to G? A. B C. E B. C D. F 6. Which diagram below shows a correct mathematical construction using only a compass and a straightedge to bisect an angle? A. C. B. D.

5 7. A line is constructed through point P parallel to a given line m. The following diagrams show the steps of the construction: Step 1 Step 2 Step 3 Step 4 Which of the following justifies the statement PS A. PS B. PS C. PS D. PS QR QR QR QR QR? because TPS and PQR are congruent corresponding angles. because TPS and PQR are congruent alternate interior angles. because PS does not intersect QR. because a line can be drawn through point P not on QR. 8. Find the values of x and y in the diagram below. A. x = 18, y = 94 B. x = 18, y = 118 C. x = 74, y = 94 D. x = 74, y = 88

6 9. Which of the following are logically equivalent? A. A conditional statement and its converse B. A conditional statement and its inverse C. A conditional statement and its contrapositive D. A conditional statement, its converse, its inverse and its contrapositive 10. Two lines that do NOT intersect are always parallel. Which of the following best describes a counterexample to the assertion above? A. coplanar lines B. parallel lines C. perpendicular lines D. skew lines 11. Determine which statement follows logically from the given statements. If I am absent on a test day, I will need to make up the test. Absent students take the test during their lunch time or after school. A. If I am absent, it is because I am sick. B. If I am absent, I will take the test at lunch time or after school. C. Some absent students take the test at lunch time. D. If I am not absent, the test will not be taken at lunch time or after school. 12. Determine whether the conjecture is true or false. Give a counterexample if the conjecture is false. Given: Two angles are supplementary. Conjecture: They are both acute angles. A. False; either both are right or they are adjacent. B. True C. False; either both are right or one is obtuse. D. False; they must be vertical angles.

7 13. Write the statement in if-then form. A counterexample invalidates a statement. A. If it invalidates the statement, then there is a counterexample. B. If there is a counterexample, then it invalidates the statement. C. If it is true, then there is a counterexample. D. If there is a counterexample, then it is true Which statement is true based on the figure? A. a b b B. b c C. a c a c D. d e d 120 e 15. In the diagram below, MQ = 30, MN = 5, MN = NO, and OP = PQ. Which of the following statements is not true? A. NP = MN + PQ C. MQ = 3 PQ B. MP = OQ D. NQ = MP

8 For #16-17 use the following: Given: KM bisects JKL Prove: m 2 = m 3 Statements Reasons KM bisects JKL Given m 1 = m m 1 = m 3 m 2 = m Choose one of the following to complete the proof. Definition of Congruence Definition of Congruence Substitution Property of Equality A. Definition of angle bisector- If a ray is an angle bisector, then it divides the angle into two congruent angles. B. Definition of opposite rays- If a point on the line determines two rays are collinear, then the rays are opposite rays. C. Definition of ray- If a line begins at an endpoint and extends infinitely, then it is ray. D. Definition of segment bisector- If any segment, line, or plane intersects a segment at its midpoint then it is the segment bisector. 17. Choose one of the following to complete the proof. A. Definition of complementary angles- If the angle measures add up to 90, then angles are supplementary B. Supplemental Angle Theorem- If two angles are supplementary to a third angle then the two angles are congruent C. Definition of supplementary angles- If the angles are supplementary, then the angle s measures add to 180. D. Vertical Angle Theorem- If two angles are vertical angles, then they have congruent angle measures.

9 18. What are the coordinates of the point P that lies along the directed segment from L( 5, 7) to M(4, 8) and partitions the segment in the ratio of 1 to 4? A. ( 3.2, 4) C. (1.8, 3) B. ( 2.5, 3) D. (2, 5) 19. An 80 mile trip is represented on a gridded map by a directed line segment from point M(3, 2) to point N(9, 14). What point represents 50 miles into the trip? Round your answers to the nearest hundredth. A. (2.31, 4.62) C. (5.31, 6.62) B. (3.75, 7.50) D. (6.75, 9.50) 20. The equations of four lines are given. Identify which lines are parallel. I. 3x + 2y = 10 II. 9x 6y = 8 III. y + 1 = 3 (x 6) 2 IV. 5y = 7.5x A. I, II, and IV C. III and IV B. I and II D. None of the lines are parallel 21. Which equation of the line passes through (4, 7) and is perpendicular to the graph of the line that passes through the points(1, 3) and ( 2, 9)? A. y = 2x 1 C. y = 1 2 x 5 B. y = 1 2 x + 5 D. y = 2x + 15

10 22. Which equation of the line passes through (29, 8) and is perpendicular to the graph of the line y = 1 x + 17? 13 A. y = 385x C. y = 13x B. y = 1 13 x D. y = 13x Solve for x and y so that a b c. Round your answer to the nearest tenth if necessary. A. x = 17.6, y = 3.1 C. x = 54.3, y = 8.5 B. x = 17.6, y = 5.5 D. x = 54.3, y = 26.9

11 For #24-25 use the following: Given: p q Prove: m 3 + m 6 = 180 Statements p q 24. m 3 = m 5 5 and 6 are supplementary Reasons Given m 5 + m 6 = m 3 + m 6 = 180 If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. Definition of Congruence If two angles form a linear pair, then they are supplementary. Substitution Property of Equality 24. Choose one of the following to complete the proof. A. 4 5 B. 2 8 C. 3 6 D Choose one of the following to complete the proof. A. Vertical Angle Theorem- If two angles are vertical angles, then they have congruent angle measures B. Supplemental Angle Theorem- If two angles are supplementary to a third angle then they are congruent C. Definition of supplementary angles- If two angles are supplementary, then their angle measures add to 180. D. Definition of complementary angles- If two angles are a complementary, then their angle measures add to 90

12 26. Line k is represented by the equation, y = 2x + 3. Which equation would you use to determine the distance between the line k and point (0, 0)? A. y = 2x C. y = 1 2 x + 3 B. y = 1 2 x D. y = 1 2 x 27. Which of the following is true? A. All triangles are congruent. B. All congruent figures have three sides. C. If two figures are congruent, there must be some sequence of rigid transformations that maps one to the other. D. If two triangles are congruent, then they must be right angles. 28. Describe the transformation M: ( 2, 5) ( 2, 5). A. A reflection across the y-axis B. A reflection across the x-axis C. A clockwise rotation of 270 with center of rotation (0, 0) D. A counterclockwise rotation of 90 with center of rotation (0, 0) 29. The endpoints of AB have coordinates A(1, 3) and B( 4, 5). After a translation A is mapped on to A ( 1, 7). What are the coordinates of B after the translation? A. ( 6, 1) C. ( 6, 1) B. (6, 1) D. (1, 6)

13 30. Figure ABC is rotated 90 counterclockwise about the point ( 2, 3). What are the coordinates of A after the rotation? A. A ( 4, 5) B. A ( 1, 6) C. A ( 3, 0) D. A (4, 5) 31. Point A is reflected over the line BC. Which of the following is not true of line BC? A. line BC B. line BC C. line BC D. line BC is perpendicular to line AA is perpendicular to line AB bisects line segment AB bisects line segment AA 32. A graphic designer is creating a cover for a geometry textbook by reflecting a design across line p and then reflecting the image across line n. Describe a single transformation that moves the design from its starting position to its final position. A. clockwise rotation of 180 about the origin B. clockwise rotation of 90 about the origin C. translation along the line p = n D. reflection across the line p = n

14 33. What are the coordinates for the image of GHK after a rotation 90 clockwise about the origin and a translation of (x, y) (x + 3, y + 2)? A. G ( 3, 2), H ( 5, 1), K ( 1, 2) B. G (0, 4), H ( 2, 1), K (2, 0) C. G (1, 2), H (5, 1), K (2, 1) D. G (6, 0), H (8, 3), K (4, 5) 34. Which composition of transformations maps ABC into the third quadrant? A. Reflection across the line y = x and then a reflection across the y-axis. B. Clockwise rotation about the origin by 180 and then a reflection across the y-axis. C. Translation of (x 5, y) and then a counterclockwise rotation about the origin by 90. D. Clockwise rotation about the origin by 270 and then a translation of (x + 1, y). 35. The point P( 2, 5) is rotated 90 counterclockwise about the origin, and then the image is reflected across the line x = 3. What are the coordinates of the final image P? A. (1, 2) C. ( 2, 1) B. (11, 2) D. (2, 11)

15 36. Describe the rigid motion(s) that would map ABC on to XYC to satisfy the SAS congruence criteria. A. Rotation B. Translation C. Rotation and Reflection D. Translation and Reflection 37. In the figure below, DE = EH, GH DF, and F G. Is there enough information to conclude DEF HEG? If so, state the congruence postulate that supports the congruence statement. A. Yes, by SSA Postulate B. Yes, by SAS Postulate C. Yes, by AAS Theorem D. No, not enough information 38. If ABC DEF, which of the following is true? A. A D, BC EF, C F B. A D, AB DF C E C. A F, BC AC, C D D. A E, DF EF, C F 39. In the figure GAE LOD and AE DO. What information is needed to prove that AGE OLD by SAS? A. GE LD B. AG OL C. AGE OLD D. AEG ODL

16 40. You are given the following information about GHI and EFD. I. G E II. H F III. I D IV. GH EF V. GI ED Which combination cannot be used to prove that GHI EFD? A. V, IV, II B. II, III, V C. III, V, I D. All of the above prove GHI EFD 41. In the figure DE EH and GH DF. Which theorem can be used to conclude that DEF HEG? A. SSA B. AAA C. SAS D. HL 42. In the figure, ABC AFD. What is the m D? A. m D = 57 B. m D = 42 C. m D = 30 D. m D = 25

17 43. Given MNP, Anna is proving m 1 + m 2 = m 4. Which statement should be part of her proof? A. m 1 = m 2 B. m 1 = m 3 C. m 1 + m 3 = 180 D. m 3 + m 4 = 180 For #44 use the following: Given: Q is the midpoint of MN ; MQP NQP Prove: MQP NQP Statements Reasons Q is the midpoint of MN ; MQP NQP Given [1] Definition of Midpoint MQP NQP Given QP QP Reflexive property of congruence MQP NQP [2] 44. Choose one of the following to complete the proof. A. [1] MQ NQ [2] AAS Congruence B. [1] MP NP [2] Linear Pair Theorem C. [1] MQ NQ [2] SAS Congruence D. [[1] MN QP [2] SAS Congruence

18 45. In the figure, MON NPM. What is the value of y? A. y = 8 B. y = 10 C. y = 42 D. y = In the figure, AC AB. Find the value of y in terms of x. A. y = 3x B. y = 6x 140 C. y = 6x + 40 D. y = 3x

19 For #47 use the following: Given: AB AC and 1 2 Prove: BC ED Statements AB AC Reasons Given BC ED Given Transitive property of congruence If two coplanar lines are but by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. 47. Choose one of the following to complete the proof. A. Isosceles Triangle Symmetry Theorem- If the line contains the bisector of the vertex angle of an isosceles triangle, then it is a symmetry line for the triangle. B. Isosceles Triangle Coincidence Theorem- If the bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base, then the median to the base is the same line C. Isosceles Triangle Base Angle Converse Theorem- If two angles of a triangle are congruent, the sides opposite those angles are congruent D. Isosceles Triangle Base Angle Theorem- If two sides of a triangle are congruent, then the angles opposite those sides are congruent 48. Which of the following best describes the shortest distance from a vertex of a triangle to the opposite side? A. altitude B. diameter C. median D. segment

20 49. EB is the angle bisector of AEC. What is the value of x? A. x = 35 B. x = 51.5 C. x = 70.5 D. x = In DOG, line m is drawn such that it is perpendicular to DO at point X and DX OX. Which of the following best describes line m? A. altitude C. angle bisector B. median D. perpendicular bisector 51. Reflect point H across the line FG A. HF FG B. HF H G C. HG H G D. FG H G to form point H, which of the following is true? 52. The vertices of JKL are located at J( 5, 3), K(3, 9), and L(7, 2). If LM is an altitude of JKL, what are the coordinates of M? A. M(7, 3) C. M( 1, 3) B. M(1, 6) D. M( 2, 2)

21 53. On the graph below, PQR is reflected over QR PQP. What are the coordinates of P? so that QR is an angle bisector of A. P (5, 3) B. P (1, 5) C. P ( 9, 1) D. P (1, 7) 54. A segment has endpoints T( 4, 5) and U(6, 1). Find the equation of the perpendicular bisector of TU. A. x = 1 C. y = 5 2 x B. y = 2 5 x + 4 D. y = 5 2 x 21 2

22 For #55-56 use the following: Given: GF is a median of isosceles GIJ with base IJ Prove: JGF IGF Statements GF is a median Reasons Given F is a midpoint of IJ 55. FI FJ Definition of midpoint 56. Definition of isosceles triangle FG FG Reflexive property of congruence JGF IGF SSS Congruence 55. Choose one of the following to complete the proof. A. Definition of angle bisector- If a ray divides an angle into two congruent angles, then it is an angle bisector. B. Definition of segment bisector- If any segment, line, or plane intersects a segment at its midpoint, then it is a segment bisector. C. Definition of isosceles triangle- If a triangle has at least two congruent sides, then it is an isosceles triangle. D. Definition of median- If a segment is a median, then it has endpoints at the vertex of a triangle and the midpoint of the opposite side. 56. Choose one of the following to complete the proof. A. GI GH B. GI GJ C. KG HG D. KI HJ

23 57. Which of the following indirect proofs is correct given the following? Given: ABC Prove: ABC has no more than one right angle Assume: ABC has more than one right angle A. Assume that A and B are both obtuse angles. So by definition of an obtuse angle, m A = 120 and m B = 120. According to the Triangle Angle-Sum Theorem, m A + m B + m C = 180. By substitution, m C = 180. Combining like terms give the equation m C = 180. Subtracting 240 from both sides of the equation gives m C = 60. This contradicts the fact that an angle in a triangle has to be more than 0. Therefore, the assumption ABC has more than one right angle is false. The statement ABC has no more than one right angle is true. B. Assume that A and B are both right angles. So by definition of a right angle, m A = 180 and m B = 180. According to the Triangle Angle-Sum Theorem, m A + m B + m C = 180. By substitution, m C = 180. Combining like terms give the equation m C = 180. Subtracting 360 from both sides of the equation gives m C = 180. This contradicts the fact that an angle in a triangle has to be more than 0. Therefore, the assumption ABC has more than one right angle is false. The statement ABC has no more than one right angle is true. C. Assume that A and B are both right angles. So by definition of a right angle, m A = 90 and m B = 90. According to the Triangle Angle-Sum Theorem, m A + m B + m C = 180. By substitution, m C = 180. Combining like terms give the equation m C = 180. Subtracting 180 from both sides of the equation gives m C = 0. This contradicts the fact that an angle in a triangle has to be more than 0. Therefore, the assumption ABC has more than one right angle is false. The statement ABC has no more than one right angle is true. D. Assume that A and B are both acute angles. So by definition of an acute angle, m A = 60 and m B = 60. According to the Triangle Angle-Sum Theorem, m A + m B + m C = 180. By substitution, m C = 180. Combining like terms give the equation m C = 180. Subtracting 120 from both sides of the equation gives m C = 60. This contradicts the fact that an angle in a triangle has to be 90. Therefore, the assumption ABC has more than one right angle is true.

24 58. If a triangle has two sides with lengths of 8 cm and 14 cm. Which length below could not represent the length of the third side? A. 7 cm C. 15 cm B. 13 cm D. 22 cm 59. Find the range of values containing x. A. 2 < x < 5 B. x < 5 C. 0 < x < 9 D. x > The captain of a boat is planning to travel to three islands in a triangular pattern. What is the possible range for the number of miles round trip the boat will travel? A. between 32 and 75 miles B. between 43 and 107 miles C. between 139 and 182 miles D. between 150 and 214 miles

25 Formal Geometry Semester 1 Instructional Materials Answers 1. B 11. B 21. B 31. C 41. D 51. C 2. D 12. C 22. C 32. A 42. B 52. B 3. B 13. B 23. B 33. B 43. D 53. D 4. A 14. D 24. D 34. C 44. C 54. C 5. B 15. D 25. C 35. A 45. B 55. D 6. C 16. A 26. D 36. C 46. B 56. B 7. A 17. D 27. C 37. D 47. D 57. C 8. A 18. A 28. B 38. A 48. A 58. D 9. C 19. D 29. C 39. B 49. A 59. A 10. D 20. A 30. B 40. A 50. D 60. D

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