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Transcript
GEOMETRY 22 MID-TERM EXAM REVIEW January 2017 Note to student: This packet should be used as practice for the Geometry 22 midterm exam. This should not be the only tool that you use to prepare yourself for the exam. You must go through your notes, re-do homework problems, class work problems, formative assessment problems, and questions from your tests and quizzes throughout the year thus far. The sections from the book that are covered on the midterm exam are: Intro: Geometry Tool Kit (pre-requisite skills students should know coming in to Geometry) - Pythagorean theorem (8.1) - Using a protractor (1.4) - Algebra (solving equations, factoring) - Basic Area (triangle, circle (area & circumference), rectangle/square, trapezoid) - Naming Polygons - Slope - Parallel lines (3.2) - Complementary & Supplementary (1.5) - Congruence - Graphing on a coordinate plane (points and lines) (p. 893) - Types of triangles: equilateral, isosceles, scalene - Types of angles: Acute, obtuse, right, straight angle (1.4) - Solving proportions (7.1) - Perimeter Unit 1: Parallel Lines/Angles Skills needed: A. - Describe lines (using notation) (1.2) Label a diagram with given - Describe a point (using notation) (1.2) information o Collinear (1.2) Set up and solve an equation given a - Define intersection (1.2) diagram o The intersection of two lines is a point (1.2) Notation - Define postulate (1.2) - Define rays and angles (using notation) & naming angles (1.2/1.4) o Define vertex of an angle (1.4) o Define adjacent angles (1.5) - Angle addition postulate (1.4) - Definition of a Linear Pair & Linear Pair Postulate o Define supplementary & complementary - Define vertical angles and the property that vertical angles are equal (1.5) B. - Proofs (simple either two-column or flow-chart) (2.5) o o o o Substitution Property Reflexive Property of Equality Transitive Property of Equality Distributive Property (2.5) C. - Define Alternate Interior, Alternate Exterior, Same-side Interior, and Corresponding angles (3.2) Parallel lines and transversals (definitions) (3.2) Define congruence and notation Define a theorem o Alternate Interior Angles Theorem o Alternate Exterior Angles Theorem o Same-Side Interior Angles Theorem (3.2) o Corresponding Angles Postulate o Congruent Supplements/Complements Theorem (+) (2.6) o Vertical Angles Theorem (1.5) - Define conditional and converses with truth values and counterexamples (2.2) o Converse of Alternate Interior Theorem o Converse of Alternate Exterior Theorem o Converse of Same-Side Interior Theorem (3.3) o Converse of Corresponding Theorem Biconditional Statements/What is a definition (2.3) - Unit 2: Triangles A. - Define segments (using notation) (1.2) o Segment addition postulate (1.3) o Define Midpoint o Define Equidistant (5.2) B. - Define a triangle (using notation) (4.4) - Triangle inequality (5.6) - Angles in a triangle o Triangle sum theorem (3.5) o Exterior angle of a triangle theorem (3.5) o Isosceles triangle theorem & converse (4.5) o Isosceles triangle theorem corollaries (4.5) Define equilateral and equiangular (6.1) o Longest side is opposite the largest angle (8.1) - Define Bisect (1.2) - Define Perpendicular (1.6) o Perpendicular bisector (1.6) o Angle bisector (1.5) o Median (5.4) o Define Altitude (5.4) The shortest distance between a point and a line is the perpendicular distance (5.2) Use altitude to find AREA of triangles (A = ½ bh) - Points of concurrency (+) (5.3) (incenter and circumcenter) o Area and perimeter of triangles in the coordinate plane (6.7) Midpoint Formula (1.7) Triangle midsegment theorem (5.1) Unit 3: Transformations (Triangle Conguence & Similarity) A. - Rigid transformations- define reflection, rotation, translation (9.1, 9.2, 9.3) - Perform transformations in coordinate plane (ch. 9) o Define a sequence that would transform a figure onto another figure (9.5) B. - Define congruence with triangles (using correct notation) & proofs with congruent triangles (two-column or flowchart) o SSS o SAS o ASA (4.2/4.3) o AAS o HL (4.6) o CPCTC (4.4) C. - Non-rigid transformations- define dilations (9.6) Perform dilation in coordinate plane (9.6/9.7) - Define similarity with triangles (using correct notation) & proofs with similar triangles (two-column or flowchart) (setting up & solving proportions) o SSS o SAS (7.3) *Incorporate application problems throughout o AA o Side-splitting theorem (7.5) o Right triangle similarity (+) (7.4) D. Practice Problems for Midterm Exam – be sure to check your answers online!! Using the diagram to the right, give an example of each of the following: 1. Use the diagram at right to name the following using proper geometric notation. a) An obtuse angle containing point K ___________ b) An acute angle containing point K ___________ c) A right angle other than angle LQK ___________ d) Any vertex ___________ e) A ray with endpoint P ___________ f) Another way to name a ray with endpoint P ___________ g) An angle supplementary to angle NQK ___________ h) A line named two different ways ___________ ___________ i) An angle adjacent to angle NQJ that is not its complement ___________ j) the intersection of line NP and line ML _______________________ 2. Would it be accurate to name ∠𝑀𝑄𝑃 as ∠𝑄? Why or why not? _______________________________________________________________________________________ 3. Use the diagram in #1 above to fill in the blanks. ________________ + ________________ = 𝑚∠𝐽𝑄𝑃 and JQ + ______ = JK 4. Through any two points there is exactly one ___________________________. 5. Assume that B is between A and C. If AB = 6 units and BC = 13 units, what is AC? 6. Using the diagram below, if JR = 4x -12, RT = 6x +4 and JT = 8x + 10, find x. ___________ ___________ Equation: ________________________ Reason: ___________________________ 7. In the diagram below, if mDYL (5 x 11) , mLYN 27 , and mDYN (2 x 65) , solve for x and find mDYN. Equation: _____________________ x = ___________ Reason: ____________________________ mDYN. =________ 8. If A and B are complementary, and mA (2 x 15) and mB 25 , find x. 9. If 1 and 2 form a linear pair, and m1 (3 x 9) and m2 (2 x 24) , find x. 10. AB is the perpendicular bisector of PQ. If the intersection of AB and PQ is R and PR 16, find PQ. PQ = ___________ 11. AB bisects CD at point E. If CE 4 x 11 and DE 7 x 25, find the following: x ___________ DE __________ CD __________ 12. AK bisects TAN . If mTAN (8 x 20) and mKAN ( x 11) , find each of the following: x __________ mNAK __________ mTAN __________ 13. If CD = 5x – 7, find the indicated values. a) x = ____________ b) CE = ______________ 14. a) What is mDTB ? ___________ c) CD= ________________ B C D b) What is mBTC ?____________ A T 15. If mEFG 43o , what is the measure of its supplement? ___________ 16. In the figure below, mDAF 18x 3 . Find the indicated measures. a) x = _______________ b) mFAE = ______________ A 17. In the figure below, find the indicated measures. a) x = _______________ b) mKLN = _______________ 18. Solve for x in the following problems. a. b. c. 3x - 42 143° 123° (2x + 5)° x 110° 19. Refer to the statement: “All altitudes form right angles.” a) Rewrite the statement as a conditional. b) Identify the hypothesis and conclusion of the conditional. Hypothesis: Conclusion: c) Draw a Venn to illustrate the statement. d) Write the converse of the conditional. e) If the converse is false, give a counterexample: 20. Refer to the statement: “A polygon with exactly three sides is called a triangle.” a) Rewrite the statement as a conditional. b) Write the converse. c) Write the biconditional. d) Decide whether the statement is a definition. Explain your reasoning. 21. Given the following Venn diagram, state a conditional using the information. Sports teams _______________________________________________________ tennis team _______________________________________________________ 22. Explain the similarities and differences between supplementary angles and a linear pair. 23. Determine whether the following is a translation, reflection, or rotation. a) _______________________ b) _______________________ 24. Reflect ABC over the x-axis. 25. On the same graph, reflect ABC over the y-axis. 26. On the same graph, rotate ABC 180° about the origin. 27. The dashed-line figure is a dilation image of the solid-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. a. b. c. 28. Determine if the following scale factor would create an enlargement, reduction, or isometric figure. a. 3.5 b. 2/5 c. 0.6 d. 1 e. 4/3 f. -5/8 29. In the figure at right, lines r and s are parallel, m2 = 40o, and m4 = 60o; find: a) m1 ______________ b) m3 ______________ c) m5 ______________ d) m6 ______________ e) m7 ______________ f) m8 ______________ g) m9 ______________ h) m10 ______________ 14 12 13 7 9 8 11 r 10 5 i) m11 ______________ j) m12 ______________ 6 k) m13 ______________ l) m14 ______________ 1 4 s 3 2 30. If lines k and l are parallel, m4 = (3x -10)o and m5 = (x + 70)o; find: 1 3 5 7 a) m8 ______________ 2 4 k b) m6 ______________ 6 8 l c) m3 ______________ d) m7 ______________ 31. Using the diagram above (from #30), state the type of angle pair of the given angles; a. ∠2 𝑎𝑛𝑑 ∠6 _________________________ b. ∠5 𝑎𝑛𝑑 ∠6 _________________________ c. ∠2 𝑎𝑛𝑑 ∠3 _________________________ d. ∠3 𝑎𝑛𝑑 ∠6 _________________________ 32. If ∠2 ≅ ∠7 , then 𝒌 ∥ 𝒍 by what theorem? ________________________________________________ 31. If m∡HAG = (7x + 4)o, and m∡CAB = (9x – 10)o, find the following: a) x =______________ b) m∡GAH = ______________ c) m∡CAB =______________d) m∡G = ______________ 32. In the diagram below F, E, and D are midpoints. AC = ____________ If AB = 55, then FE = __________ If AB = 55 and FD = 20, find the perimeter of ABC ________ L (2x - 5)o 33. x = _________________ N mN =_________________ o (3x + 1) M 34. Solve for y, then find the value of the ∠𝑃 in the figure below. W' P' (1.5y - 12) O' 35. Can the following groups of sides be the sides of a triangle? Explain. a. 15, 10, 5 b. 20, 21, 3 c. 7, 4, 15 36. In the figure below, 𝑚∠𝑆 = 24 and 𝑚∠𝑂 = 130. Which side of ⊿𝑆𝑂𝑋 is the shortest side? Why? 37. Determine m∡ A, m∡ B, and m∡C if A is supplementary to B and complementary to C. 38. Write the conditional and converse of the statement, and determine if the converse is true. If it is not, write a counterexample. If an angle measure is 32 degrees, then it is an acute angle. 39. Find the value of x and y. a. b. c. d. 40. Solve for the given variable and find the angle measures. a. b. d. c. e. f. 41. Use the diagram to the right. a. What type of triangle is ABD ? ___________________________ b. What type of triangle is BCD ? ___________________________ c. Find mABC ___________________________ 42. Use the congruency statement to fill in the corresponding congruent parts. EFI HGI FE _____ ∠𝐸 ≅ ∠______ FI _____ ∠𝐹𝐼𝐸 ≅ ∠______ ∠𝐸𝐹𝐼 ≅ ∠______ IE _____ 43. For the following, name which triangle congruence theorem or postulate you would use to prove the triangles congruent. a. b. c. 44. Mark any additional information you can FIRST (ex: vertical angles or reflexive property). Then, label and state what ADDITIONAL information is required in order to know that the triangles are congruent for the reason given. a. b. c. extra part: ________ ________ DUT Δ________ extra part: ________ ________ LMK Δ________ extra part: ________ ________ UWV Δ _______ d. e. f. f. extra part: ________ ________ RQS Δ________ extra part: ________ ________ JIH Δ________ extra part: ________ ________ BAC Δ _______ 45. For which value(s) of x are the triangles congruent? a. x = ______________ A b. x = _______________ B m 3 = x2 W m 4 = 7x - 10 1 E 3 4 D c. x = _______________ A 2 C C x2 + 2x 7x - 4 4x + 8 R B R 46. Describe what is being constructed in the figure at right. 47. Refer to the figure at the right. ̅̅̅̅ is a ______________________ of ∆ABC a.) 𝐸𝐵 b.) _______________ is an altitude of ∆ABC. 48. Find the value of x if ̅̅̅̅ 𝐴𝐷 is an altitude of ∆ABC. 49. Solve for 𝒙. a. b. Z c. x2 + 24 S T 50. In GHI , R, S, and T are midpoints. If mG 75 and mHSR 63 . Find the mH . 51. PM = 4x + 7 and PN = 12x – 5 71. Solve for 𝑥. Find PL. 52. a) According to the diagram, what are the lengths of PQ and PS ? b.) How is PR related to SPQ? c.) Find the value of n. d.) Find mSPR and mQPR. 53. Find the value of the missing variables in the problems below. a. b. 54. Find the range of possible measures for ̅̅̅̅ 𝑿𝒀 in ΔXYZ. a. XZ = 6 and YZ = 6 b. XZ = 9 and YZ = 5 c. XZ = 11 and YZ = 6 55. Draw the angle bisectors of the triangle at right. 56. Draw the perpendicular bisectors of the triangle at right. 57. Draw the medians of the triangle at right. 58. Draw the altitudes of the triangle at right. 59. If HJ = 26, then KL = ______ 60. If HJ = 3x – 1 and KL = x + 1, then HJ = ______ 61. Solve for x. 84x 21 ̅̅̅̅, using the given coordinates of point A and point B. Then use the 62. Use the distance formula to find the length of 𝐴𝐵 ̅̅̅̅; midpoint formula to find the coordinates of the midpoint of 𝐴𝐵 a. A( -2, 4 ) , B( 6, - 8) b. A( 21, 40) , B( - 13, 6) c. A( -5, -12) , B( -3, 4) 63. Given: mSRT 88 , mQ 49 Prove: mP 43 64. Given: Q is the midpoint of PR, Prove: SQP TQR P QRT S R Q P 65. Given: RT RU , TS US Prove: TRS URS T 66. Given: 1 4 Prove: ABC is isosceles Given: VX VW 67. Y is the midpoint of WX Prove: VYX VYW A Given: AB AC , mC 80 , mB (3x 1) 68. Prove: x 27 B C 69. T Given: TRS and TRL are right angles, RTS RTL Prove: RS RL S 70. R L Given: 1 and 3 are supplementary Prove: m n 1 m 2 n 3 p