* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Formal Geometry - Washoe County School District
Survey
Document related concepts
Rule of marteloio wikipedia , lookup
Shape of the universe wikipedia , lookup
History of trigonometry wikipedia , lookup
Cartan connection wikipedia , lookup
Algebraic geometry wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Analytic geometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Integer triangle wikipedia , lookup
Multilateration wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Geometrization conjecture wikipedia , lookup
History of geometry wikipedia , lookup
Transcript
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: ο· 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. Released 8/22/16 Formal Geometry Reference Sheet Note: You may use these formulas throughout this entire test. Linear Quadratic π¦2 β π¦1 π₯2 β π₯1 Vertex-Form π¦ = π(π₯ β h)2 + π π₯1 + π₯2 π¦1 + π¦2 π=( , ) 2 2 Standard Form π¦ = ππ₯ 2 + ππ₯ + π π = β(π₯2 β π₯1 )2 + (π¦2 β π¦1 )2 Intercept Form π¦ = π(π₯ β π)(π₯ β π) Slope π= Midpoint Distance Slope-Intercept Form π¦ = ππ₯ + π Exponential (h, k) Form Probability π¦ = ππ π₯βh + π π(π΄ πππ π΅) = π(π΄) β π(π΅) π(π΄ πππ π΅) = π(π΄) β π(π΅|π΄) π(π΄ ππ π΅) = π(π΄) + π(π΅) β π(π΄ πππ π΅) Volume and Surface Area π = ππ 2 β 4 π = ππ 3 3 ππ΄ = 2(ππ 2 ) + β(2ππ) ππ΄ = 4ππ 2 1 1 π = ππ 2 β 3 1 ππ΄ = ππ 2 + (2ππ β π) 2 Released 8/22/16 π = π΅β 3 1 ππ΄ = π΅ + (ππ) 2 Where π΅ =base area and π =base perimeter FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 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: π΄(β1, β4), π΅(1, β1), πΆ(2, β2). 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, β76.3) and the coordinates for Columbus are (39.9, β83.0). What coordinates should the pilot use for Madison? A. (β1.5, β3.3) C. (33.9, β69.6) B. (61.5, 56.6) D. (42.9, β89.7) Μ Μ Μ Μ . π is the midpoint of π΄πΆ Μ Μ Μ Μ . π is the midpoint 3. In the diagram below, π is the midpoint of π΄π΅ Μ Μ Μ Μ of π΅πΆ . Find the area of βπ ππ and π΄π΅. A. Area of βπ ππ = 4; π΄π΅ β 4β5 B. Area of βπ ππ = 8; π΄π΅ β 4β5 C. Area of βπ ππ = 4; π΄π΅ β 8β5 D. Area of βπ ππ = 8; π΄π΅ β 8β5 Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 4. 2016-2017 Μ Μ Μ Μ and ππ Μ Μ Μ Μ and determine which of the following Given the coordinates below, compare π π statements is true: π (β2, β7) π(5, β1) π(β3, 3) π(6, β1) A. The midpoints of Μ Μ Μ Μ π π and Μ Μ Μ Μ ππ have the same π₯-coordinate. B. The midpoints of Μ Μ Μ Μ π π and Μ Μ Μ Μ ππ have the same π¦-coordinate. Μ Μ Μ Μ and the length of ππ Μ Μ Μ Μ are the same. C. The length of π π D. The length of Μ Μ Μ Μ π π is longer than the length of Μ Μ Μ Μ ππ. 5. 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 which angle is congruent to β πΊ ? A. β π΅ C. β πΈ B. β πΆ D. β πΉ 6. Which diagram below shows a correct mathematical construction using only a compass and a straightedge to bisect an angle? A. C. B. D. Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS 2016-2017 Course: Formal Geometry S1 (#2215) 7. A line is constructed through point P parallel to a given line π. The following diagrams show the steps of the construction: Step 1 Step 2 Step 3 Step 4 β‘ β₯ ππ β‘ ? Which of the following justifies the statement ππ β‘ because β πππ and β πππ are congruent corresponding angles. A. β‘ππ β₯ ππ β‘ because β πππ and β πππ are congruent alternate interior angles. B. β‘ππ β₯ ππ β‘ β₯ ππ β‘ because ππ β‘ does not intersect ππ β‘ . C. ππ β‘ because a line can be drawn through point π not on ππ β‘ . D. β‘ππ β₯ ππ 8. Find the values of x and y in the diagram below. A. π₯ = 18, π¦ = 94 B. π₯ = 18, π¦ = 118 C. π₯ = 74, π¦ = 94 D. π₯ = 74, π¦ = 88 Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 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. Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 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. 65ο° 60ο° 110ο° 120ο° 14. 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 15. In the diagram below, ππ = 30, ππ = 5, ππ = ππ, and ππ = ππ. Which of the following statements is not true? A. ππ = ππ + ππ C. ππ = 3 β ππ B. ππ = ππ D. ππ = ππ Released 8/22/16 2016-2017 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 For #16-17 use the following: Given: πΎπ bisects β π½πΎπΏ Prove: πβ 2 = πβ 3 Statements Reasons πΎπ bisects β π½πΎπΏ β 1 β β 2 Given πβ 1 = πβ 2 Definition of Congruence β 1 β β 3 17. πβ 1 = πβ 3 Definition of Congruence πβ 2 = πβ 3 Substitution Property of Equality 16. 16. 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. 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 angleβs measures add to 180°. D. Vertical Angle Theorem- If two angles are vertical angles, then they have congruent angle measures. Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 18. What are the coordinates of the point π that lies along the directed segment from πΏ(β5, 7) to π(4, β8) and partitions the segment in the ratio of 1 to 4? A. (β3.2, 4) C. (1.8, β3) B. (β2.5, 3) D. (2, β5) 19. An 80 mile trip is represented on a gridded map by a directed line segment from point π(3, 2) to point π(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. 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 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 Released 8/22/16 C. π¦ = 1 π₯β5 2 D. π¦ = β2π₯ + 15 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 22. Which equation of the line passes through (29, 8) and is perpendicular to the graph of the 1 line π¦ = 13 π₯ + 17? A. π¦ = 385π₯ + B. π¦ = 23. 1 13 1 π₯ + 385 13 C. π¦ = β13π₯ + 385 D. π¦ = β13π₯ β 13 Solve for x and y so that π β₯ π β₯ π . Round your answer to the nearest tenth if necessary. A. π₯ = 17.6, π¦ = 3.1 C. π₯ = 54.3, π¦ = 8.5 B. π₯ = 17.6, π¦ = 5.5 D. π₯ = 54.3, π¦ = 26.9 Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 For #24-25 use the following: Given: π β₯ π Prove: πβ 3 + πβ 6 = 180 Statements Reasons πβ₯π Given 24. If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. πβ 3 = πβ 5 Definition of Congruence β 5 and β 6 are supplementary If two angles form a linear pair, then they are supplementary. πβ 5 + πβ 6 = 180 25. πβ 3 + πβ 6 = 180 Substitution Property of Equality 24. Choose one of the following to complete the proof. A. β 4 β β 5 B. β 2 β β 8 C. β 3 β β 6 D. β 3 β β 5 25. 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° Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 26. 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₯ B. π¦ = 1 π₯ 2 1 C. π¦ = β π₯ + 3 2 1 D. π¦ = β π₯ 2 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 π: (β2, 5) β (β2, β5). 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 Μ Μ Μ Μ π΄π΅ have coordinates π΄(1, β3) and π΅(β4, 5). After a translation π΄ is mapped on to π΄β²(β1, β7). What are the coordinates of π΅β² after the translation? A. (β6, β1) C. (β6, 1) B. (6, 1) D. (1, 6) Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 30. Figure π΄π΅πΆ is rotated 90° counterclockwise about the point (β2, β3). What are the coordinates of π΄β² after the rotation? A. π΄β²(β4, 5) B. π΄β²(β1, β6) C. π΄β²(β3, 0) D. π΄β²(4, β 5) β‘ . Which of the following is not true of line π΅πΆ β‘ ? 31. Point A is reflected over the line π΅πΆ A. line π΅πΆ β‘ is perpendicular to line β‘π΄π΄β² β‘ is perpendicular to line β‘π΄π΅ B. line π΅πΆ β‘ bisects line segment π΄π΅ Μ Μ Μ Μ C. line π΅πΆ Μ Μ Μ Μ Μ β‘ bisects line segment π΄π΄β² D. line π΅πΆ 32. A graphic designer is creating a cover for a geometry textbook by reflecting a design across line π and then reflecting the image across line π. 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 π = π D. reflection across the line π = π Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 33. 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) 34. Which composition of transformations maps βπ΄π΅πΆ into the third quadrant? A. Reflection across the line π¦ = π₯ 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 (π₯ β 5, π¦) and then a counterclockwise rotation about the origin by 90°. D. Clockwise rotation about the origin by 270° and then a translation of (π₯ + 1, π¦). 35. 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) Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 36. Describe the rigid motion(s) that would map βπ΄π΅πΆ on to βπππΆ to satisfy the SAS congruence criteria. A. Rotation B. Translation C. Rotation and Reflection D. Translation and Reflection 37. In the figure below, π·πΈ = πΈπ», Μ Μ Μ Μ πΊπ» β Μ Μ Μ Μ π·πΉ , and β πΉ β β πΊ. Is there enough information to conclude βπ·πΈπΉ β βπ»πΈπΊ? 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 βπ΄π΅πΆ β βπ·πΈπΉ, which of the following is true? Μ Μ Μ Μ β πΈπΉ Μ Μ Μ Μ , β πΆ β β πΉ A. β π΄ β β π·, π΅πΆ B. β π΄ β β π·, Μ Μ Μ Μ 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 Μ Μ Μ Μ . What information is needed to prove that 39. 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 B. Μ Μ Μ Μ π΄πΊ β Μ Μ Μ Μ ππΏ C. β π΄πΊπΈ β β ππΏπ· D. β π΄πΈπΊ β β ππ·πΏ Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 40. You are given the following information about βπΊπ»πΌ and βπΈπΉπ·. I. II. III. IV. V. β πΊ β β πΈ β π» β β πΉ β πΌ β β π· Μ Μ Μ Μ πΊπ» β Μ Μ Μ Μ πΈπΉ Μ Μ Μ Μ Μ Μ Μ πΊπΌ β πΈπ· Which combination cannot be used to prove that βπΊπ»πΌ β βπΈπΉπ·? A. V, IV, II B. II, III, V C. III, V, I D. All of the above prove βπΊπ»πΌ β βπΈπΉπ· 41. In the figure Μ Μ Μ Μ π·πΈ β Μ Μ Μ Μ πΈπ» and Μ Μ Μ Μ πΊπ» β Μ Μ Μ Μ π·πΉ . Which theorem can be used to conclude that βπ·πΈπΉ β βπ»πΈπΊ? A. SSA B. AAA C. SAS D. HL 42. In the figure, βπ΄π΅πΆ β βπ΄πΉπ·. What is the πβ π·? A. πβ π· = 57° B. πβ π· = 42° C. πβ π· = 30° D. πβ π· = 25° Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 43. Given βπππ, Anna is proving πβ 1 + πβ 2 = πβ 4. Which statement should be part of her proof? A. πβ 1 = πβ 2 B. πβ 1 = πβ 3 C. πβ 1 + πβ 3 = 180° D. πβ 3 + πβ 4 = 180° For #44 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 β ππ Μ Μ Μ Μ ππ Given βπππ β βπππ [2] Reflexive property of congruence 44. Choose one of the following to complete the proof. A. [1] Μ Μ Μ Μ Μ ππ β Μ Μ Μ Μ ππ [2] AAS Congruence B. [1] Μ Μ Μ Μ Μ ππ β Μ Μ Μ Μ ππ [2] Linear Pair Theorem C. [1] Μ Μ Μ Μ Μ ππ β Μ Μ Μ Μ ππ [2] SAS Congruence D. [[1] Μ Μ Μ Μ Μ ππ β Μ Μ Μ Μ ππ [2] SAS Congruence Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 45. In the figure, βπππ β βπππ. What is the value of y? A. π¦ = 8 B. π¦ = 10 C. π¦ = 42 D. π¦ = 52 Μ Μ Μ Μ β π΄π΅ Μ Μ Μ Μ . Find the value of y in terms of x. 46. In the figure, π΄πΆ A. π¦ = β3π₯ + 160 B. π¦ = 6π₯ β 140 C. π¦ = 6π₯ + 40 D. π¦ = 3π₯ + 20 2 Released 8/22/16 2016-2017 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 For #47 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 π΄π΅ β Μ Μ Μ Μ π΄πΆ Reasons β 2 β β 3 47. β 1 β β 2 Given β 1 β β 3 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. β‘ β₯ β‘πΈπ· π΅πΆ Given 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 Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 49. 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 50. In βπ·ππΊ, line π is drawn such that it is perpendicular to Μ Μ Μ Μ π·π at point π and Μ Μ Μ Μ π·π β Μ Μ Μ Μ ππ. Which of the following best describes line π? A. altitude C. angle bisector B. median D. perpendicular bisector 51. Reflect point H across the line β‘πΉπΊ to form point π»β², which of the following is true? A. Μ Μ Μ Μ π»πΉ β Μ Μ Μ Μ πΉπΊ B. Μ Μ Μ Μ π»πΉ β Μ Μ Μ Μ Μ π»β²πΊ C. Μ Μ Μ Μ π»πΊ β Μ Μ Μ Μ Μ π»β²πΊ Μ Μ Μ Μ β Μ Μ Μ Μ Μ D. πΉπΊ π»β²πΊ 52. Μ Μ Μ Μ is an altitude The vertices of βπ½πΎπΏ are located at π½(β5, β3), πΎ(3, 9), and πΏ(7, 2). If πΏπ of βπ½πΎπΏ, what are the coordinates of π? A. π(7, β3) C. π(β1, 3) B. π(1, 6) D. π(β2, 2) Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 53. 2016-2017 β‘ so that ππ β‘ is an angle bisector of On the graph below, β πππ is reflected over ππ β πππβ². What are the coordinates of πβ²? A. πβ²(5β, 3) B. πβ²(1, 5) C. πβ²(β9, 1) D. πβ²(1, 7) 54. A segment has endpoints π(β4, 5) and π(6, 1). Find the equation of the perpendicular bisector of Μ Μ Μ Μ ππ. 5 1 A. π₯ = 1 C. π¦ = π₯ + 2 2 2 B. π¦ = β π₯ + 4 5 Released 8/22/16 D. π¦ = 5 21 π₯β 2 2 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 For #55-56 use the following: Given: Μ Μ Μ Μ πΊπΉ is a median of isosceles βπΊπΌπ½ with Μ base πΌπ½ Prove: βπ½πΊπΉ β βπΌπΊπΉ Statements Μ Μ Μ Μ πΊπΉ is a median Reasons Μ πΉ is a midpoint of πΌπ½ Μ Μ Μ β πΉπ½ Μ Μ Μ πΉπΌ 55. 56. Μ Μ Μ Μ Μ Μ Μ Μ πΉπΊ β πΉπΊ Definition of isosceles triangle βπ½πΊπΉ β βπΌπΊπΉ SSS Congruence Given Definition of midpoint Reflexive property of 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. Μ Μ Μ πΊπΌ β Μ Μ Μ Μ πΊπ» Μ Μ Μ β πΊπ½ Μ Μ Μ B. πΊπΌ 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 Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 57. Which of the following indirect proofs is correct given the following? Given: βπ¨π©πͺ Prove: βπ¨π©πͺ has no more than one right angle Assume: βπ¨π©πͺ has more than one right angle A. Assume that β π΄ and β π΅ are both obtuse angles. So by definition of an obtuse angle, πβ π΄ = 120° and πβ π΅ = 120°. According to the Triangle Angle-Sum Theorem, πβ π΄ + πβ π΅ + πβ πΆ = 180°. By substitution, 120° + 120° + πβ πΆ = 180°. Combining like terms give the equation 240° + πβ πΆ = 180°. Subtracting 240° from both sides of the equation gives πβ πΆ = β60°. This contradicts the fact that an angle in a triangle has to be more than 0°. Therefore, the assumption βπ΄π΅πΆ has more than one right angle is false. The statement βπ΄π΅πΆ has no more than one right angle is true. B. Assume that β π΄ and β π΅ are both right angles. So by definition of a right angle, πβ π΄ = 180° and πβ π΅ = 180°. According to the Triangle Angle-Sum Theorem, πβ π΄ + πβ π΅ + πβ πΆ = 180°. By substitution, 180° + 180° + πβ πΆ = 180°. Combining like terms give the equation 360° + πβ πΆ = 180°. Subtracting 360° from both sides of the equation gives πβ πΆ = β180°. This contradicts the fact that an angle in a triangle has to be more than 0°. Therefore, the assumption βπ΄π΅πΆ has more than one right angle is false. The statement βπ΄π΅πΆ has no more than one right angle is true. C. Assume that β π΄ and β π΅ are both right angles. So by definition of a right angle, πβ π΄ = 90° and πβ π΅ = 90°. According to the Triangle Angle-Sum Theorem, πβ π΄ + πβ π΅ + πβ πΆ = 180°. By substitution, 90° + 90° + πβ πΆ = 180°. Combining like terms give the equation 180° + πβ πΆ = 180°. Subtracting 180° from both sides of the equation gives πβ πΆ = 0°. This contradicts the fact that an angle in a triangle has to be more than 0°. Therefore, the assumption βπ΄π΅πΆ has more than one right angle is false. The statement βπ΄π΅πΆ has no more than one right angle is true. D. Assume that β π΄ and β π΅ are both acute angles. So by definition of an acute angle, πβ π΄ = 60° and πβ π΅ = 60°. According to the Triangle Angle-Sum Theorem, πβ π΄ + πβ π΅ + πβ πΆ = 180°. By substitution, 60° + 60° + πβ πΆ = 180°. Combining like terms give the equation 120° + πβ πΆ = 180°. Subtracting 120° from both sides of the equation gives πβ πΆ = 60°. This contradicts the fact that an angle in a triangle has to be 90°. Therefore, the assumption βπ΄π΅πΆ has more than one right angle is true. Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 58. If a triangle has two sides with lengths of 8 ππ and 14 ππ. Which length below could not represent the length of the third side? A. 7 ππ C. 15 ππ B. 13 ππ D. 22 ππ 59. Find the range of values containing x. A. 2 < π₯ < 5 B. π₯ < 5 C. 0 < π₯ < 9 D. π₯ > 0 60. 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 πππππ B. between 43 and 107 πππππ C. between 139 and 182 πππππ D. between 150 and 214 πππππ Released 8/22/16 FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017 Formal Geometry Semester 1 Instructional Materials 2016-2017 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 Released 8/22/16