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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 NAME DATE M2 Geometry – Assignment sheet for Unit 2 Lines and Angles, Packet 3 Unit 2 includes the following sections: 1-4, 1-5, 2-8, 1-6, 6-1, 3-1 to 3-6 Due # Assignment Topics 2G p. 176 # 13-18 all, 22-30 even (In #13-18, you may assume that any objects that look parallel actually are parallel.) 2H p. 7 in this packet 2I p. 10 in this packet 3-1: Vocabulary: parallel lines, skew lines, parallel planes, transversal, consecutive interior angles, alternate interior angles, alternate exterior angles, corresponding angles Identify angle pair relationships involving lines and a transversal 3-2: Investigate angle pair relationships involving parallel lines and a transversal Use algebra to solve problems involving parallel lines and a transversal 3-5: Use angle pair relationships to determine whether lines are parallel Test on 1-4, 1-5, 2-8, 1-6, 6-1, 3-1, 3-2, 3-5 1 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 Parallel Lines and Transversals When two lines in the same plane do not intersect, they are parallel. Lines that do not intersect and are not coplanar are skew. In the figure above right, is parallel to m , or m . Arrows are drawn on the lines (not the ones at the ends) to indicate that they are parallel. You can also write PQ RS . Similarly, if two planes do not intersect, they are parallel planes. Use the figure at the right to identify each of the following. 1. a plane parallel to plane OPT 2. all segments parallel to NU 3. all segments that intersect MP A transversal is a line that intersects two or more other lines at two different points in a plane. In the figure at right, line t is a transversal. Two lines and a transversal form eight angles. Some pairs of angles have special names. The next two pages show the pairs of angles and their names. 2 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 4. Interior Angles are between the lines. Think of lines m and n as the pieces of bread in a sandwich and the transversal t as a toothpick holding the sandwich together. Interior angles are the angles inside the sandwich. List the interior angles in this diagram: ∠ ____, ∠ ____, ∠ ____, and ∠ ____ 5. Alternate interior angles are pairs of angles between two lines but on opposite sides of the transversal. In other words, they are inside the sandwich but on opposite sides of the toothpick. There are two pairs of alternate interior angles in this diagram. Name them: ∠ ____ and ∠ ____ form a pair of alternate interior angles. ∠ ____ and ∠ ____ form a pair of alternate interior angles. 6. Consecutive interior angles are between two lines and on the same side of the transversal. They are also called “same side interior”. In other words, they are inside the sandwich and on the same side of the toothpick. There are two pairs of consecutive interior angles in this diagram. Name them: ∠ ____ and ∠ ____ form a pair of consecutive interior angles. ∠ ____ and ∠ ____ form a pair of consecutive interior angles. 7. Alternate exterior Angles are outside the sandwich and on opposite sides of the toothpick. There are two pairs of alternate exterior angles in this diagram. Name them: ∠ ____ and ∠ ____ form a pair of alternate exterior angles. ∠ ____ and ∠ ____ form a pair of alternate exterior angles. 3 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 8. Corresponding angles are pairs of angles that are in the same position in relation to one of the two lines and the transversal. For example, ∠1 and ∠5 form a pair of corresponding angles because they are both above a piece of bread and on the left side of the toothpick. Name the other 3 pairs of corresponding angles: ∠ ____ and ∠ ____ form a pair of corresponding angles. ∠ ____ and ∠ ____ form a pair of corresponding angles. ∠ ____ and ∠ ____ form a pair of corresponding angles. 9. For each pair of angles: (i) Name the two lines (sandwich) and the transversal (toothpick) that form the angles. (ii) Classify as alternate interior, consecutive interior, alternate exterior, or corresponding. a. ∠10 and ∠16 2 lines: transversal: Classification: b. ∠4 and ∠12 2 lines: transversal: Classification: c. ∠12 and ∠13 2 lines: transversal: Classification: d. ∠3 and ∠9 2 lines: transversal: Classification: 4 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 ANGLES AND PARALLEL LINES 1. Open the applet at https://www.geogebra.org/m/P3nNeHjp. This link is available on the class website. 2. The diagram in this applet shows two parallel lines cut by a transversal. Name these parts below: ______ ______. The transversal is ______. 3. As usual, these 3 lines form 8 angles. How many different angle measures do you see? 4. Change the angle of the transversal by dragging point D or point E to the side. Make sure that point F stays between A and B and that point G stays between C and H. How many different angle measures do you see? 5. Suppose the angles on your screen are numbered as shown at right. a. Name the pairs of alternate interior angles in the diagram at right. What do you notice about the measures of the alternate interior angles formed by the parallel lines in the applet? b. Name the pairs of consecutive interior angles in the diagram above. What do you notice about the measures of the consecutive interior angles formed by the parallel lines in the applet? c. Name the pairs of alternate exterior angles in the diagram above. What do you notice about the measures of the alternate exterior angles formed by the parallel lines in the applet? d. Name the pairs of corresponding angles in the diagram above. What do you notice about the measures of the corresponding angles formed by the parallel lines in the applet? 5 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 e. Fill in the blanks: When two parallel lines are cut by a transversal, they form… …alternate interior angles that are _____________________, …consecutive interior angles that are _____________________, …alternate exterior angles that are ______________________, and …corresponding angles that are _________________________. 6. In the figure at right, AH EF . Fill in the blanks. a. ∠CBH ≅ ∠FCB because they are ______________________________________ angles. b. ∠DBH ≅ ∠BCG because they are ______________________________________ angles. c. ∠FCE ≅ ∠BCG because they are ____________________________________ angles. d. ∠DBA ≅ ∠GCE because they are ____________________________________ angles. e. ∠HBC and ∠GCB are supplementary because they are _____________________ angles. f. ∠ABC and ∠BCF are supplementary because they are _____________________ angles. 51° . Find the measure of each angle, 7. Suppose m∠ABC = and explain why your answer is correct. a. m∠BCG = ________ because b. m∠FCE = ________ because c. m∠BCF = ________ because d. m∠GCE = ________ because 6 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 Assignment 2H: 1. In the figure, m∠9 = 80° and m∠5 = 68° . Find the measure of each angle. a. ∠12 d. ∠3 b. ∠1 e. ∠7 c. ∠4 f. ∠16 2. Find the value of the variables in each figure. Show your work. a. b. c. 7 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 PROVING LINES PARALLEL If 2 lines are cut by a transversal and… • • • • • Then corresponding ∠ s are ≅ , alternate exterior ∠ s are ≅ , consecutive interior ∠ s are supplementary, alternate interior ∠ s are ≅ , or ⊥ to the transversal, the lines are parallel. Ex 1: If m∠1 = m∠2 , determine which lines, if any, are parallel. Explain your reasoning. Solution: ∠1 and ∠2 are corresponding angles formed by lines r and s with transversal m. Since ∠1 ≅ ∠2 , then r s . Ex 2: Find m∠ABC so that m n . Explain your reasoning, and show your work. Solution: m n if alternate interior angles are congruent. Then ∠DAB ≅ ∠ABC 3 x + 10 = 6 x − 20 10 = 3 x − 20 30 = 3 x x = 10 m∠ABC = 6 (10 ) − 20 = 40° Find x so that m . Explain your reasoning, and show your work. 1. 2. 3. 8 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 4. In the diagram at right, ∠1 ≅ ∠2 and ∠1 ≅ ∠3 . Explain why AB DC . 5. In the diagram at right, ∠1 ≅ ∠5 and ∠15 ≅ ∠5 . Explain why m and r s . 9 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 Assignment 2I: Determine which lines, if any, must be parallel if the given information is true. Explain your reasoning. Several copies of the same diagram have been provided so you can draw on them. 1. ∠1 ≅ ∠2 2. ∠2 ≅ ∠9 3. ∠5 ≅ ∠7 4. m∠3 + m∠6 =180 5. ∠3 ≅ ∠7 6. ∠4 ≅ ∠5 7. If ∠1 ≅ ∠3 and AC BD in the diagram at right, explain why AB CD . 10 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 Practice Problems for Sections 3-1, 3-2, 3-5 1. Use the figure at the right to identify each of the following. You may assume that segments appearing to be parallel actually are parallel in this diagram. a. all planes that are parallel to plane DEH b. all segments that are parallel to AB c. all segments that are skew to CD 2. Identify the two lines and transversal forming each pair of angles. Then classify the relationship between each pair of angles. a. ∠ 4 and ∠ 10 d. ∠ 2 and ∠ 12 b. ∠ 7 and ∠ 3 e. ∠ 13 and ∠ 10 c. ∠ 8 and ∠ 14 f. ∠ 6 and ∠ 14 3. In the figure at right, m∠3 = 75° and m∠10 = 110° . Find the measure of each angle. a. ∠ 2 d. ∠ 5 b. ∠ 7 e. ∠ 15 c. ∠ 14 f. ∠ 9 g. Is w x ? Explain. 11 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 4. Find the value of the variable(s) in each figure. Explain your reasoning, and show your work. a. b. 5. Determine which lines, if any, must be parallel if the given information is true. Explain your reasoning. a. ∠3 ≅ ∠7 b. ∠10 ≅ ∠12 c. ∠2 ≅ ∠16 d. m∠5 + m∠12 =180 12 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 6. Find x so that m . Explain your reasoning, and show your work. 7. If ∠1 and ∠2 are complementary and BC ⊥ CD , explain why BA CD . 13 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 Review for 1-4, 1-5, 1-6, 2-8, 3-1, 3-2, 3-5 Use a protractor to find the measure of each angle to the nearest degree. 1. 2. 3. 4. Figures on the rest of this review are not drawn to scale. Use the diagram of ∠PQS at right to solve problems 5-6. 5. m∠PQS = 4 x, m∠SQR = 2 x, m∠RQP = 24° . Find m∠SQR and m∠PQS . 6. m∠SQR = 3 x − 2, m∠SQP = 5 x, m∠PQR = 34° . Find m∠SQR and m∠PQS . 14 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 7. BD is the bisector of ∠ABC . m∠ABD= m∠ABC = ____ ( 2 y − 3) ° , and m∠DBC = ____ m∠DBC =( y + 12 ) ° . m∠ABD = ____ 10 x . Find m∠4 . 8. PD is an angle bisector of ∠BPE . m∠4 = 4 x + 12 , and m∠BPE = In problems 9-12, write an equation, and solve. 9. An angle is 40° more than its complement. What is the measure of that angle? 10. The measure of one angle is three times its complement. Find the measure of both angles. 11. The measure of an angle is 30 more than twice the measure of its supplement. Find the measure of both angles. 12. An angle is 64 ° less than its supplement. What is the measure of this angle? 15 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 In problems 13-16, find the missing values of each letter. 13. 14. 15. 16. In problems 17-20, find the values of x and y. 17. 18. 19. 20. 16 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 21. In the figure at right, TM ⊥ RS , and m∠QMS =° 58 . Find the measure of each a. m∠TMQ = ____ b. m∠RMP = ____ c. m∠SMP = ____ d. m∠PMT = ____ T M R P 3 x + 5 and m∠DEF = 2 x − 15 . 22. In the figure at right, m∠AEC = Find each measure: m∠DEF = ____ m∠DEB = ____ m∠CEB = ____ 23. Complete the proof. Given: ∠ABD ≅ ∠CBE Prove: ∠ABE ≅ ∠CBD Statements Reasons 1. 1. Given m∠CBE 2. m∠ABD = 2. ≅ ∠ s have = measures 3. m∠ABE + m∠EBD = m∠ABD _______ + _______ = m∠CBE 3. 4. m∠ABE + m∠EBD = m∠CBD + m∠EBD 4. Substitution m∠EBD 5. m∠EBD = 5. m∠CBD 6. m∠ABE = 7. ∠ABE ≅ ∠CBD 6. 7. 17 angle. Q S M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 24. Complete the proof. Given: ∠ABE ≅ ∠CBD Prove: ∠ABD ≅ ∠CBE Statements Reasons 1. ∠ABE ≅ ∠CBD 1. Given 2. 2. ≅ ∠ s have = measures 3. 3. Reflexive 4. m∠ABE + m∠EBD = m∠CBD + m∠EBD 4. Addition 5. 5. Angle Addition m∠CBE 6. m∠ABD = 6. 7. 7. 25. Complete the proof. Given: ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 Statements Reasons 1. 1. Given 2. m∠2 = m∠4 2. ≅ ∠ s have = measures 3. m∠1 + m∠2 =____ ____ + ____ = ____ 3. 4. m∠1 + m∠ = 2 ____ + ____ 4. Transitive 5. m∠1 = m∠3 5. 6. 6. ≅ ∠ s have = measures 18 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 26. Complete the proof. Given: AB ⊥ DC , ∠1 ≅ ∠4 Prove: ∠3 ≅ ∠2 Statements Reasons 1. AB ⊥ DC , ∠1 ≅ ∠4 1. Given 2. m∠1 = m∠4 2. ≅ ∠ s have = measures 3. m∠ABC = 90°, m∠ABD = 90° 3. m∠ABD 4. m∠ABC = 4. 5. ______ + ______ = ______ ______ + ______ = ______ 5. Angle Addition 6. m∠1 + m∠2 = m∠3 + m∠4 6. 7. m∠2 = m∠3 7. 8. ∠2 ≅ ∠3 8. 9. ∠3 ≅ ∠2 9. 27.Sketch a concave hexagon. 28. Sketch a convex pentagon. 29. Sketch a convex regular heptagon, and mark the sides and/or angles with tic marks and/or arcs as appropriate. 30. Sketch a concave equilateral nonagon, and mark the sides and/or angles with tic marks and/or arcs as appropriate. 19 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 31. Find the sum of the measures of the interior angles of a convex polygon with 13 sides. 32. Find the sum of the measures of the interior angles of a convex polygon with 15 sides. 33. The sum of the measures of the interior angles of a convex polygon is 7020° . Find the number of sides of the polygon. 34. The sum of the measures of the interior angles of a convex polygon is 1980° . Find the number of sides of the polygon. 35. Find the measure of each interior angle of a regular heptagon. 36. Find the measure of each interior angles of a regular nonagon. 37. Find the measure of each exterior angle of a regular decagon. 38. Find the measure of each exterior angle of a regular 18-gon. 20 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 In #39-42, find the value of x. 39. 40. 41. 42. 43. How many sides does a regular polygon have if each interior angle measures 168.75° ? 44. How many sides does a regular polygon have if each interior angle measures 150° ? 21 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 In #45-48, refer to the figure at the right to identify each of the following. 45. all planes that intersect plane STX ���� 46. all segments that intersect 𝑄𝑄𝑄𝑄 ���� 47. all segments that are parallel to 𝑋𝑋𝑋𝑋 ����� 48. all segments that are skew to 𝑉𝑉𝑉𝑉 In #49-54, classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. 49. ∠ 2 and ∠ 10 50. ∠ 7 and ∠ 13 51. ∠ 9 and ∠ 13 52. ∠ 6 and ∠ 16 53. ∠ 3 and ∠ 10 54. ∠ 8 and ∠ 14 In #55-58, name the transversal that forms each pair of angles. Then identify the special name for the angle pair. 55. ∠ 2 and ∠ 12 56. ∠ 6 and ∠ 18 57. ∠ 13 and ∠ 19 58. ∠ 11 and ∠ 7 For #59-64, m ∠ 2 = 92 ° and m ∠ 12 = 74 ° . Find the measure of each angle. 59. ∠ 10 60. ∠ 8 61. ∠ 9 62. ∠ 5 63. ∠ 11 64. ∠ 13 22 M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5 In #65-66, find the value of the variable(s) in each figure. 65. 66. In #67-70, given the following information, determine which lines, if any, must be parallel. State the reason that justifies your answer. 67. m ∠ BCG + m ∠ FGC = 180 68. ∠ CBF ≅ ∠ GFH 69. ∠ EFB ≅ ∠ FBC 70. ∠ ACD ≅ ∠ KBF In #71-72, find x so that m . 71. 72. 73. If ∠2 and ∠3 are supplementary, explain why AB CD . 23