# Geometry S1 (#2211) Foundations in Geometry S1 (#7771)

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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 courses: Geometry S1 (#2211) Foundations in Geometry S1 (#7771) 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 Geometry Semester 1 and Geometry Semester 2 final exams.

2 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. Describe the transformation M: ( 2, 5) (2, 5). A. A reflection across the y-axis B. A reflection across the x-axis C. A rotation 180 with center of rotation (0, 0) D. A rotation 90 with center of rotation (0, 0) 2. Find the coordinates of the image of the point ( 5, 7) when it is reflected across the line y = 11. A. ( 5, 18) C. ( 5, 4) B. ( 5, 15) D. ( 5, 7) 3. What are coordinates for the image of quadrilateral ABCD after the translation of (x, y) (x + 7, y 2)? A. A (8, 5), B (2, 1), C (12, 4), D (10, 7) B. A ( 6, 1), B ( 5, 1), C ( 2, 0), D ( 4, 3) C. A ( 6, 5), B (2, 1), C (12, 4), D (10, 7) D. A (8, 5), B (9, 3), C (12, 4), D (10, 7)

4 4. Figure ABC is rotated 90 clockwise about the point (2, 0). What are the coordinates of A after the rotation? A. A ( 1, 4) B. A (3, 6) C. A (5, 4) D. A (6, 3) 5. What are the coordinates for the image of GHK after a rotation 90 counterclockwise about the origin and a translation of (x, y) (x + 3, y + 2)? A. K ( 2, 0), H (2, 1), G (0, 4) B. K (2, 0), H ( 2, 1), G (0, 4) C. K ( 4, 4), H ( 8, 3), G ( 6, 0) D. K (4, 4), H (8, 3), G (6, 0) 6. 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)

5 7. Draw line FG and point H not on the line. Reflect point H across the line FG to form point H. Based on your diagram, which of the following is true? A. HF FG B. HF H G C. FG H G D. HG H G 8. Which composition of transformations maps DEF into the first quadrant? A. Reflection across the line y = x and then a translation of (x + 2, y + 5). B. Translation of (x 1, y 6) and then a reflection in the line y = x. C. Clockwise rotation about the origin by 180 and then reflection across the y-axis. D. Counterclockwise rotation about the origin by 90 and then a translation of (x 1, y + 6).

6 9. Which of the following options correctly explains how to find the value of x in the figure below? A. Since EA and ED are opposite rays, and EB and EC are opposite rays, AEB and DEB are vertical angles. Vertical angles are congruent and have the same measure, so m AEB = m DEC. Using substitution gives the equation, 2x = 30. Dividing by 2 on both sides gives x = 15. B. Since EA and ED are opposite rays, and EB and EC are opposite rays, AEB and DEB are vertical angles. Vertical angles are complimentary meaning their measures add up to 90, so m AEB + m DEC = 90. Using substitution gives the equation, 2x + 30 = 90. Subtracting both sides by 30, and then dividing both sides by 2 gives x = 30. C. Since AED is a straight angle, its measure is 180. By the Angle Addition Postulate, m AEB + m BEC + m CED = m AED. Using substitution gives the equation, 2x + 4x + 30 = 180. After combining like terms the equation becomes 6x + 30 = 180. Subtracting both sides by 30, and then dividing both sides by 6 gives x = 25. D. Since AED is a straight angle, its measure is 90. By the Angle Addition Postulate, m AEB + m BEC + m CED = m AED. Using substitution gives the equation, 2x + 4x + 30 = 90. After combining like terms the equation becomes 6x + 30 = 90. Subtracting both sides by 30, and then dividing both sides by 6 gives x = In the figure m FGI = (2x + 9) and m HGI = (4x 15). Find m FGI and m HGI. A. m FGI = 71 and m HGI = 109 B. m FGI = 45 and m HGI = 45 C. m FGI = 33 and m HGI = 33 D. m FGI = 41 and m HGI = 49

7 11. 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 = Based on the figure below, which statements are true if l m and r s? I. m 1 = m 5 II. m 2 + m 3 = 180 III. m 4 = m 8 IV. m 6 + m 9 = 180 V. m 1 + m 9 = 180 A. All of the above C. II, III, and IV B. I, III, and IV D. III, IV, and V 13. In the figure, m ABC = m ACB = x. Find the m EAB in terms of x. A x B. 2x C. x D. x + 90

8 For #14-15 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 an 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. 15. 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 For #16 use the following: Given: 1 and 2 are supplementary, and m 1 = 135 Prove: m 2 = 45 Statements 1 and 2 are supplementary Reasons Given [1] Given m 1 + m 2 = 180 [2] m 2 = 180 Substitution Property of Equality m 2 = 45 [3] 16. Fill in the blanks to complete the two column proof: A. [1] m 2 = 135 [2] Definition of Supplementary Angles [3] Subtraction Property of Equality B. [1] m 1 = 135 [2] Definition of Supplementary Angles [3] Substitution Property of Equality C. [1] m 1 = 135 [2] Definition of Supplementary Angles [3] Subtraction Property of Equality D. [1] m 1 = 135 [2] Definition of Complementary Angles [3] Subtraction Property of Equality 17. Find CE given the following information: AC is the perpendicular bisector of DB EA = 16 DC = 13 AB = 20 A. CE = 15.9 B. CE = 5 C. CE = 17.7 D. CE = 16

10 18. 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 19. Which statement is true based on the figure? A. a b B. b c C. a c D. d e 20. Draw two lines and a transversal such that 1 and 2 are alternate interior angles, 2 and 3 are corresponding angles, and 3 and 4 are alternate exterior angles. What type of angle pair is 1 and 4? A. 1 and 4 are supplementary angles C. 1 and 4 are corresponding angles B. 1 and 4 are vertical angles D. 1 and 4 are alternate exterior angles

11 For #21-22 use the following: Given: p q Prove: m 3 + m 6 = 180 Statements p q 21. 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 21. 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 23. Solve for x and y so that a b c. A. x = 9, y = 10.1 C. x = 29, y = 3.2 B. x = 9, y = 9.9 D. x = 29, y = In the diagram below, line p is parallel to line j and line t is perpendicular to AB. What is the measure of BAC? A. m BAC = 37 C. m BAC = 45 B. m BAC = 53 D. m BAC = What are the coordinates of the point P that lies along the directed segment from C( 3, 2) to D(6, 1) and partitions the segment in the ratio of 2 to 1? A. (0, 3) C. (1.5, 0.5) B. (3, 0) D. (4.5, 1.5)

13 26. An 80 mile trip is represented on a gridded map by a directed line segment from point M(3, 2) to point N(9, 13). What point represents 60 miles into the trip? A. (8, 14.6) C. (9, 11.25) B. (3.75, 9.75) D. (7.5, 10.25) 27. Given the two lines below, which statement is true? Line 1: x 3y = 15 and Line 2: y = 3(x + 2) 1 A. The lines are parallel. B. They are the same line. C. The lines are perpendicular. D. The lines intersect but are not perpendicular. 28. Which equation of the line passes through (5, 8) and is parallel to the graph of the line y = 2 x 4? 3 A. y = 2 3 x 34 3 C. y = 2 3 x 8 B. y = 3 2 x 1 2 D. y = 3 2 x 8

14 29. 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 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 motions that maps one to the other. D. If two triangles are congruent, then they must be right triangles. 31. If CED QRP, which of the following is true? A. C Q, E R, D P B. C Q, E P, D R C. C P, E R, D Q D. C R, E Q, D P

15 32. 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 33. Which conclusion can be drawn from the given facts in the diagram? A. TQ bisects PTS B. TQS RQS C. PT RS D. TS = PQ 34. In the figure H L and HJ = JL. Which of the following statements is about congruence is true? A. HIJ LKJ by ASA B. HIJ KLJ by SSS C. HIJ KLJ by SAS D. HIJ LKJ by SAS

16 35. Refer To the figure to complete the congruence statement, ABC. A. ACE B. EDC C. EAD D. EDA 36. Which theorem can be used to conclude that CAB CED? A. SAA B. SAS C. SSS D. AAA 37. In the figure, identify which congruence statement is true. Then find m TYW. A. WTY WXY by HL m TYW = 22 B. WTY XYW by HL m TYW = 78 C. YWT WXY by HL m TYW = 22 D. WTY WXY by HL m TYW = 68

17 38. Determine which postulate or theorem can be used to prove the pair of triangles congruent. A. AAS B. SAS C. ASA D. SSS 39. Which of the following sets of triangles can be proved congruent using the AAS Theorem? A. C. B. D. 40. 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

18 For #41-42 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 41. Definition of Midpoint MQP NQP Given QP QP Reflexive property of congruence MQP NQP Choose one of the following to complete the proof. A. MN QP B. MQ NQ C. MP NP D. QP QP 42. Choose one of the following to complete the proof. A. Reflexive property of equality B. SSA Congruence C. SAS Congruence D. AAS Congruence

19 43. In the figure, MON NPM. What is the value of y? A. y = 8 B. y = 10 C. y = 42 D. y = 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 45. Find the range of values containing x. A. 2 < x < 5 B. x < 5 C. 0 < x < 9 D. x > 0

20 46. Reflecting over which line will map the rhombus onto itself? A. y = 2x B. y = 0 C. y = 1 4 x D. y = x 47. What is the measure of HJ in Parallelogram FGHJ, given the following: FG = x + 7 GH = 5x + 3 m F = 46 m H = (3x + 10) A. HJ = 63 B. HJ = 19 C. HJ = 12 D. HJ = What is the value of x in the rectangle? A. x = 42 B. x = 24 C. x = 8 D. x = 4

21 49. Which of the following is not always true of Parallelogram ABCD? A. AB BC, DC BC B. AB DC, BC AD C. m A + m B = 180 D. AB + BC = AD + DC 50. JKLM is a rhombus. If m JML = 70 and m JKM = (5x + 16), find the value of x. A. x = 18.8 B. x = 10.8 C. x = 7.8 D. x = Based on the figure below, which statements are true? I. The figure is a rectangle II. The figure is a parallelogram III. 6x 4 = 9x + 3 IV. 9x + 3 = 10x 2 V. x = 8 VI. The longest side has a length of 60. A. I, III, and V C. II, IV, and VI B. I, IV, and VI D. II, III, and V

22 52. A wooden frame has screws at A, B, C, and D so that the sides of it can be pressed to change the angles occurring at each vertex. AB CD and AB CD, even when the angles change. Why is the frame always a parallelogram? A. The angles always stay the same, so ABCD is a parallelogram. B. All sides are congruent, so ABCD is a parallelogram. C. One pair of opposite sides is congruent and parallel, so ABCD is a parallelogram. D. One pair of opposite sides is congruent, so ABCD is a parallelogram. 53. Which statement is true? A. All quadrilaterals are rectangles. B. All rectangles are parallelograms. C. All parallelograms are rectangles. D. All quadrilaterals are squares.

23 54. A student is given the following information and then asked to write a paragraph proof. Determine which statement would correctly complete the student s proof. Given: Parallelogram PRST and Parallelogram PQVU Prove: V S Proof: We are given Parallelogram PRST and Parallelogram PQVU. Since opposite angles of a parallelogram are congruent, P V and P S.. A. Therefore, V S by the Transitive Property of Congruence. B. Therefore, V S by the Transformative Property of Congruence. C. Therefore, V S by the Reflective Property of Congruence. D. Therefore, V S by the Reflexive Property of Congruence. 55. Archeologists use coordinate grids on their dig sites to help document where objects are found. While excavating the site of a large ancient building, archeologists find what they believe to be a wooden support beam that extended across the entire building. One end of the beam was located at the coordinate ( 13, 12) and markings on the beam indicate that the midpoint of the beam was located at (2, 4). Assuming the beam is still intact, where should the archeologists begin digging to find the other end of the beam? A. ( 17, 4) B. ( 7.5, 4) C. ( 5.5, 8) D. (17, 4)

24 56. Use slope or the distance formula to determine the most precise name for the figure: A( 1, 4), B(1, 1), C(4, 1), D(2, 2). A. Kite B. Rhombus C. Trapezoid D. Square 57. Find the distance between the line y = 2x 3 and the point (3, 7). A. 4.5 units C. 10 units B. 5 units D units 58. Given points B ( 3,3), C(3, 4), and D(4, 2). Which of the following points must be point A in order for the quadrilateral ABCD to be a parallelogram? A. A( 2, 1) B. A( 3, 2) C. A( 2, 3) D. A( 1, 2) 59. Find the area of the polygon with vertices A( 8, 1), B( 5, 6), C(4, 2), and D( 2, 2). A units B. 46 units C. 48 units D units

25 60. A researcher finds that there are 242 bacteria evenly distributed on a 1-square inch sample of a table surface. How many bacteria would the scientist expect to find on the table top mapped below where each unit of the graph is equivalent to 1 foot? A. 7,744 bacteria B. 92,928 bacteria C. 557,568 bacteria D. 1,115,136 bacteria 61. Given the lines y = 1 x + 2 and y = x 4, find the equation for a third line that 3 would complete a right triangle in the coordinate plane. A. y = x + 4 B. y = x 5 C. x = 3 D. y = 3x + 5

26 Geometry Semester 1 Instructional Materials Answers 1. C 11. A 21. D 31. A 41. B 51. C 2. B 12. B 22. C 32. B 42. C 52. C 3. D 13. B 23. D 33. C 43. B 53. B 4. C 14. A 24. A 34. A 44. D 54. A 5. D 15. D 25. B 35. B 45. A 55. D 6. A 16. C 26. D 36. B 46. A 56. B 7. D 17. B 27. D 37. D 47. B 57. A 8. A 18. C 28. A 38. A 48. B 58. C 9. C 19. D 29. B 39. B 49. A 59. D 10. D 20. C 30. C 40. D 50. D 60. D 61. B

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