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Formal Geometry Semester 1 Instructional Materials 2013-2014 2013-2014 Formal Geometry Semester 1 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for this course. 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 vocabulary and interaction with the types of problems included in the Instructional Materials can result in less anxiety on the part of the students. 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 Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 Multiple Choice: Identify the choice that best completes the statement or answers the question. Figures are not necessarily drawn to scale. 1. Using slope and/or distance formulas, identify which of the following is the best name for the figure: π΄(β1, β4), π΅(1, β1), πΆ(2, β2). A. scalene triangle B. isosceles triangle C. equilateral triangle D. obtuse triangle 2. π(9,8) is the midpoint of Μ Μ Μ Μ π π. The coordinates of π are (10, 10). What are the coordinates of R? A. (9.5, 9) B. (18, 16) C. (11, 12) D. (8, 6) 3. In the figure, which pair of angles is supplementary? A. β 3 πππ β 7 B. β 1 πππ β 4 C. β 5 πππ β 7 D. β 4 πππ β 6 1 4 6 5 2 3 8 7 4. 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 Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 5. 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 6. 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. 7. Determine whether the conjecture is true or false. Give a counterexample for any false conjecture. 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. 8. 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. Sent on 9/27/13 Formal Geometry Semester 65ο° 1 Instructional Materials 60ο° 110ο° 120ο° 2013-2014 9. Which statement is true based on the figure? A. π β₯ π b B. π β₯ π a c C. π β₯ π D. π β₯ π 65ο° 110ο° d 60ο° 120ο° e 10. Point A is reflected over the line π΅πΆ β‘ . Which of the following is NOT true of line π΅πΆ β‘ ? A. line π΅πΆ β‘ is perpendicular to line β‘π΄π΄β² B. line π΅πΆ β‘ is perpendicular to line π΄π΅ β‘ C. line π΅πΆ β‘ bisects line segment Μ Μ Μ Μ π΄π΅ D. line π΅πΆ β‘ bisects line segment Μ Μ Μ Μ Μ π΄π΄β² 11. Given the following: β π΅ is a complement of β π΄ β πΆ is a supplement of β π΅ β π· is a supplement of β πΆ β πΈ is a complement of β π· β πΉ is a complement of β πΈ β πΊ is a supplement of β πΉ Then β πΊ β ? A. β π΅ C. β πΈ B. β πΆ D. β πΉ Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 For #12-13 use the following: Given: πΎπ bisects β π½πΎπΏ Prove: πβ 2 = πβ 3 J K 3 1 2 M L Statements Reasons πΎπ bisects β π½πΎπΏ πβ 1 = πβ 2 Given πβ 1 = πβ 3 13. πβ 2 = πβ 3 Substitution Property of Equality 12. 12. Choose one of the following to complete the proof. 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. 13. 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 angleβs measures add to 180°. D. Vertical Angle Theorem- If two angles are vertical angles, then they have equal measures. Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 For #14-15 use the following: Given: π β₯ π Prove: πβ 3 + πβ 6 = 180 t 1 4 8 2 3 5 6 7 p q Statements Reasons πβ₯π Given 14. Alternate Interior Angles Theorem πβ 5 + πβ 6 = 180 15. πβ 3 + πβ 6 = 180 Substitution Property of Equality 14. Choose one of the following to complete the proof. A. πβ 4 = πβ 5 B. πβ 2 = πβ 8 C. πβ 3 = πβ 6 D. πβ 3 = πβ 5 15. Choose one of the following to complete the proof. A. Vertical Angle Theorem- If two angles are vertical angles, then they have equal angle measures B. Congruent Supplements Theorem- If two angles are supplementary to a third angle then they are congruent C. Linear Pair Theorem- If two angles form a linear pair, then the angles are supplementary and their angle measures add to 180° D. Definition of complementary angles- If two angles are a linear pair, then the angles are complementary and their angle measures add to 90° Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 16. Complete the statement: If two lines are perpendicular, then ___________________________________________. A. the lines do not intersect B. the slopes of the two lines are equal C. the product of the slopes for the two lines is equal to negative one D. the product of the slopes for the two lines is equal to zero 17. A student proves that every right triangle is isosceles by assigning coordinates as shown and by using the distance formula to show that ππΆ = π and π΅πΆ = π. Which of the following best explains the studentβs error? A. The proof is not correct because the assigned coordinates do not result in a general right triangle. B. The proof is correct because the assigned coordinates result in a general right triangle. C. The proof is not correct because the assigned coordinates result in a rectangle. D. The proof is not correct because the assigned coordinates result in an obtuse triangle. 18. Which equation of the line passes through (15 , β 1) with a slope of π = β 2? 4 2 3 A. β8π₯ + 12π¦ = 24 C. 8π₯ + 12π¦ = β49 B. 8π₯ + 12π¦ = 24 D. 24π₯ + 36π¦ = β82 19. The equations of four lines are given. Identify which lines are parallel. I. 3π₯ + 2π¦ = 10 II. β9π₯ β 6π¦ = β8 3 III. π¦ + 1 = 2 (π₯ β 6) IV. β5π¦ = 7.5π₯ A. I, II, and IV C. III and IV B. I and II D. None of the lines are parallel Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 20. Which equation of the line passes through (29, 8) and is perpendicular to the graph of the 1 line π¦ = 13 π₯ + 17? A. B. π¦ = 385π₯ + π¦= 1 13 1 π₯ + 385 13 C. π¦ = β13π₯ + 385 D. π¦ = β13π₯ β 13 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 (β2, 9)? A. π¦ = 2π₯ β 1 B. π¦= 1 π₯+5 2 1 π₯β5 2 D. π¦ = β2π₯ + 15 C. π¦= 22. Line k is represented by the equation, π¦ = 2π₯ + 3. Which equation would you use to determine the distance between the line k and point (0, 0)? A. π¦ = 2π₯ C. 1 π¦ =β π₯+3 2 1 π₯ 2 D. 1 π¦=β π₯ 2 B. π¦= 23. 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. 24. If βπ΄π΅πΆ β βπ·πΈπΉ, which of the following is true? A. β π΄ β β π·, π΅πΆ Μ Μ Μ Μ β Μ Μ Μ Μ πΈπΉ , β πΆ β β πΉ Μ Μ Μ Μ B. β π΄ β β π·, π΄π΅ β Μ Μ Μ Μ π·πΉ β πΆ β β πΈ Μ Μ Μ Μ β π΄πΆ Μ Μ Μ Μ , β πΆ β β π· C. β π΄ β β πΉ, π΅πΆ D. β π΄ β β πΈ, Μ Μ Μ Μ π·πΉ β Μ Μ Μ Μ πΈπΉ , β πΆ β β πΉ Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 Μ Μ Μ Μ . What information is needed to prove that 25. In the figure β πΊπ΄πΈ β β πΏππ· and Μ Μ Μ Μ π΄πΈ β π·π βπ΄πΊπΈ β βππΏπ· by SAS? Μ Μ Μ Μ β πΏπ· Μ Μ Μ Μ A. πΊπΈ G B. Μ Μ Μ Μ π΄πΊ β Μ Μ Μ Μ ππΏ L C. β π΄πΊπΈ β β ππΏπ· D. β π΄πΈπΊ β β ππ·πΏ D A E O 26. In the figure β π» β β πΏ and π»π½ = π½πΏ. Which of the following statements is about congruence is true? A. βπ»πΌπ½ β βπΏπΎπ½ by ASA B. βπ»πΌπ½ β βπΎπΏπ½ by SSS C. βπ»πΌπ½ β βπΎπΏπ½ by SAS D. βπ»πΌπ½ β βπΏπΎπ½ by SAS 27. Refer To the figure to complete the congruence statement, βπ΄π΅πΆ β _________. A. βπ΄πΆπΈ B. βπΈπ·πΆ C. βπΈπ΄π· D. βπΈπ·π΄ 28. Which theorem can be used to conclude that βπΆπ΄π΅ β βπΆπΈπ·? A. SAA B. SAS C. SSS E B C D D. AAA A Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 Μ Μ Μ Μ β π·πΉ Μ Μ Μ Μ . Which theorem can be used to conclude that 29. In The figure π·πΈ = πΈπ» and πΊπ» βπ·πΈπΉ β βπ»πΈπΊ? A. SSA B. AAA C. SAS D. HL 30. Determine which postulate or theorem can be used to prove the pair of triangles congruent. A. AAS B. SAS C. ASA D. SSS 31. Given βπππ, Anna is proving πβ 1 + πβ 2 = πβ 4. Which statement should be part of her proof? A. πβ 1 = πβ 2 N B. πβ 1 = πβ 3 2 C. πβ 1 + πβ 3 = 180° D. πβ 3 + πβ 4 = 180° 1 M 32. In the figure, βπ΄π΅πΆ β βπ΄πΉπ·. What is the πβ π·? A. πβ π· = 1° B. πβ π· = 7° C. πβ π· = 42° D. πβ π· = 60° Sent on 9/27/13 3 4 P Formal Geometry Semester 1 Instructional Materials 2013-2014 For #33 use the following: Given: π is the midpoint of Μ Μ Μ Μ Μ ππ; β πππ β β πππ Prove: βπππ β βπππ Statements Reasons Μ Μ Μ Μ Μ ; β πππ β β πππ π is the midpoint of ππ Given [1] Definition of Midpoint β πππ β β πππ Μ Μ Μ Μ ππ β Μ Μ Μ Μ ππ Given βπππ β βπππ [2] Reflexive property of congruence 33. Choose one of the following to complete the proof. Μ Μ Μ Μ A. [1] Μ Μ Μ Μ Μ ππ β ππ [2] SAS Congruence Μ Μ Μ Μ B. [1] Μ Μ Μ Μ Μ ππ β ππ [2] SAS Congruence Μ Μ Μ Μ Μ β ππ Μ Μ Μ Μ C. [1] ππ [2] Linear Pair Theorem Μ Μ Μ Μ D. [1] Μ Μ Μ Μ Μ ππ β ππ [2] AAS Congruence 34. In the figure, βπππ β βπππ. What is the value of y? A. π¦ = 8 B. π¦ = 10 C. π¦ = 42 D. π¦ = 52 Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 35. In a coordinate proof, which of the following would be most useful to prove that triangles are congruent by the SSS Triangle Congruence Theorem? A. Distance formula B. Midpoint formula C. Corresponding parts of congruent triangles are congruent (CPCTC) D. Slope formula 36. For a coordinate proof concerning an isosceles triangle, which coordinates might be easiest to use? A. (π, π), (π, π), (π, π) B. (0, 0), (2π, 0), (π, π) C. (0, 0), (π, π), (2π, 3π) D. (π, π), (π, π), (π, π) 37. In the figure, πβ π΄π΅πΆ = πβ π΄πΆπ΅ = π₯. Find the πβ πΈπ΄π΅ in terms of x. A. E A 180 β 2π₯ B. 2π₯ C. π₯ B D. π₯ + 90 C 38. Suppose you wish to prove the following using indirect proof. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Which of the following would you try to contradict in an indirect proof. A. Two parallel lines are cut by a transversal. B. Alternate interior angles are congruent. C. Alternate interior angles are not congruent. D. Two parallel lines are not cut by a transversal. Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 For #39 use the following: Given: Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π΄πΆ and β 1 β β 2 β‘ β‘ Prove: π΅πΆ β₯ πΈπ· Statements Μ Μ Μ Μ β π΄πΆ Μ Μ Μ Μ π΄π΅ Reasons β 2 β β 3 39. β 1 β β 2 Given β 1 β β 3 β‘ β₯ β‘πΈπ· π΅πΆ Transitive property of congruence Given Corresponding Angles Theorem 39. 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 40. 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 Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 For #41-42 use the following: Given: Μ Μ Μ Μ πΊπΉ is a median of isosceles βπΊπΌπ½ Prove: βπ½πΊπΉ β βπΌπΊπΉ Statements Μ Μ Μ Μ πΊπΉ is a median Reasons Μ Μ Μ πΉπΌ β Μ Μ Μ πΉπ½ 41. 42. Μ Μ Μ Μ πΉπΊ β Μ Μ Μ Μ πΉπΊ Definition of isosceles triangle βπ½πΊπΉ β βπΌπΊπΉ SSS Congruence Given Reflexive property of congruence 41. 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. 42. Choose one of the following to complete the proof. Μ Μ Μ β πΊπ» Μ Μ Μ Μ A. πΊπΌ B. Μ Μ Μ πΊπΌ β Μ Μ Μ πΊπ½ C. Μ Μ Μ Μ πΎπΊ β Μ Μ Μ Μ π»πΊ Μ Μ Μ β π»π½ Μ Μ Μ Μ D. πΎπΌ 43. πΈπ΅ is the angle bisector of β π΄πΈπΆ. What is the value of x? A. π₯ = 35° B. π₯ = 51.5° C. π₯ = 70.5° D. π₯ = 142° Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 44. Which of the following terms best describe the transformation below? A. dilation B. reflection C. rotation D. translation 45. What are the coordinates for the image of βπΊπ»πΎ after a rotation 90° clockwise about the origin and a translation of (π₯, π¦) β (π₯ + 3, π¦ + 2)? A. πΊ β²β² (β3, 2), π» β²β² (β5, β1), πΎ β²β² (β1, β2) B. πΊ β²β² (0, 4), π» β²β² (β2, 1), πΎ β²β² (2, 0) C. πΊ β²β² (1, 2), π» β²β² (5, 1), πΎ β²β² (2, β1) D. πΊ β²β² (6, 0), π» β²β² (8, 3), πΎ β²β² (4, 5) K G H 46. Point Y of βπππ is (7, β8). What is the image of Y after βπππ is transformed using the translation (π₯ + 3, π¦ β 4)? A. π β² (21, β24) C. π β² (10, β12) B. π β² (21, 32) D. π β² (10, β4) 47. The point π(β2, β5) is rotated 90° counterclockwise about the origin, and then the image is reflected across the line π₯ = 3. What are the coordinates of the final image πβ²β²? A. (1, β2) C. (β2, 1) B. (11, β2) D. (2, 11) Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 48. Reflect point H across the line β‘πΉπΊ to form point Hβ, which of the following is true? A. Μ Μ Μ Μ π»πΉ β Μ Μ Μ Μ πΉπΊ H Μ Μ Μ Μ Μ Μ Μ Μ Μ β π»β²πΊ B. π»πΉ G C. Μ Μ Μ Μ πΉπΊ β Μ Μ Μ Μ Μ π»β²πΊ Μ Μ Μ Μ β Μ Μ Μ Μ Μ D. π»πΊ π»β²πΊ F 49. What is the scale factor for the dilation of βπ΄π΅πΆ to image βπ΄β²π΅β²πΆβ²? A. β2 B. 1 C. 2 D. 3 50. Apply the dilation π·: (π₯, π¦) β (4π₯, 4π¦) to the triangle given below. Which of the following is the perimeter of the image? A. 41.9 π’πππ‘π B. 20.5 π’πππ‘π C. 10.5 π’πππ‘π D. 9.2 π’πππ‘π Sent on 9/27/13 Formal Geometry Semester 1 Instructional Materials 2013-2014 1. B 11. B 21. B 31. D 41. D 2. D 12. A 22. D 32. C 42. B 3. B 13. D 23. C 33. B 43. A 4. C 14. D 24. A 34. B 44. C 5. D 15. C 25. B 35. A 45. B 6. B 16. C 26. A 36. B 46. C 7. C 17. A 27. B 37. B 47. A 8. B 18. B 28. B 38. C 48. D 9. D 19. A 29. D 39. D 49. D 10. C 20. C 30. A 40. A 50. A Sent on 9/27/13