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Honors Geometry Issue 1 Super Mathter November 10, 2004 (: 301-520-6030 Fax: 301-251-8645 Name: For class info, visit www.MathEnglish.com Direct your questions and comments to [email protected] Peter Lin Peter Lin SUPPLEMENTARY AND COMPLEMENTARY ANGLES........................................................ 2 VERTICAL ANGLES ...................................................................................................................... 3 ANGLE BISECTORS....................................................................................................................... 6 PARALLEL LINES AND ANGLES...............................................................................................10 STANDARD TEST .........................................................................................................................12 –1– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 [Linear Pair Postulate] Supplementary and Complementary Two angles of a linear pair are Angles supplementary to each other. When the two sides of an angle form a 1. What should be the value of x in the straight line, the angle is called a straight figure? angle or flat angle. Specifically, the angle is measured to be 180°. 180o x A B O Straight angle is 180°. Right angle is 90°. right angle notation, it stands for 90 o A right angle is special, it also has a special notation. Instead of writing 90° at the angle, we draw a small square near the angle vertex. Both ∠AOB and ∠BOC are right angles in the figure. Each angle of a rectangle is a right angle. B 90o A 90o O C 90o If the sum of two angles is 90°, we called them complementary angles. Namely, ∠1 + ∠2 = 90° ó ∠1 and ∠2 are complementary. If the sum of two angles is 180°, we called them supplementary angles. Namely, ∠1 + ∠2 = 180° ó ∠1 and ∠2 are supplementary. –2– This copy is for me, Peter Lin, only. 50o Honors Geometry Issue 1 Question set [4 - 7] Vertical Angles Two lines L and M intersect. Given that ∠1 = 105°. Definition: vertical angle In the figure, ∠1 and ∠2 are called vertical angles since their sides form two crossing lines and they stay on the opposite sides of the vertex. L 1 2 4 3 M O 2 1 vertical angles 4. Find the measure of ∠2 and state the reason. O 4 3 N O T vertical angles Then ∠1 = ∠2, namely, vertical angles are congruent. 2. For ∠1, name the vertical angle and adjacent angle. 5. Find the measure of ∠3 and state the reason. 3 1 4 2 6. ∠1 and ∠3 are called _ _ _ _ _ _ _ _ _ _ _ _ _, so are ∠2 and ∠4. Each pair of angles are _ _ _ _ _ _ _ _ _. 3. List all pairs of adjacent and vertical angles. 6 5 1 2 3 4 7. Find the measure of ∠4 and state the reason. –3– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 12. An angle is equal to its complementary Question set [8 - 17] angle, what is the measure of this angle? Conceptual and computational problems. 8. If two angles are complementary to the same angle, then they must be congruent. Is this statement always true? 13. What is the complementary angle for 35°? What is the supplementary for 35°? 9. If an angle has 60° as its complementary angle, what is its supplementary angle? 14. An angle is two times the complementary angle of 40°. What is the measure of this angle? 10. An angle is twice of its complementary angle, what is the measure of this angle? 15. Given: line EF bisects ∠AOB as shown in the figure, namely, ∠3 = ∠4. Prove: ∠1 = ∠2. 11. An angle is twice its supplementary angle, what is the measure of this angle? C A O E 1 3 2 4 D –4– This copy is for me, Peter Lin, only. F B Honors Geometry 16. Both ∠AOC and ∠BOD are right angles. ∠BOC = 40°. Find the measure of ∠AOD. Issue 1 B C A D 40o O 17. As the figure shows, OA bisects ∠COB, which is a right angle. Find the measures of ∠1. C A 1 D B O E 18. What is the measure of ∠AOB? A 25o O (60 - x)o (3x + 15)o B –5– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 21. What should be the value for x in the Angle Bisectors figure? Definition: angle bisector Angle bisector divides an angle equally. The ray OB divides the angle ∠AOB 60 x evenly into two congruent angles: ∠1 and ∠2, so OB is called the angle bisector of ∠AOB. o A B 1 OB bisects an angle 2 O C 22. Find the value of x. 3x+5 Question set [19 - 22] Find the value of x in each of the following. 19. What is the measure of x in the figure? 2x-5 x 40o Question set [23 - 28] Conceptual and computational problems. 20. ∠AOD is a straight angle (180°), OB and OC divide the entire angle into three congruent angles, what should be the value for x? C 23. ∠1 and ∠2 are called __ __ __ __ __ __ __ __ __ __. B 1 x D x x O A –6– This copy is for me, Peter Lin, only. 2 Honors Geometry Issue 1 24. Two angles are __________ if they add 28. Is it true that a straight angle is twice a right angle? up to be 90°. 25. Two angles are __________ if they add up to be 180°. Question set [29 - 30] ∠AOB and ∠BOC are linear pair. DO bisects ∠AOB and EO bisects ∠BOC. E B 2 C 2 O D A 29. Given that ∠AOB = 40°, what is the measure of ∠2? 26. What angles are supplementary to ∠1? 1 1 3 4 30. Prove that ∠1 + ∠2 = 90° regardless of the measure of ∠AOB. 27. If BD bisects ∠ABC, what is the measure of ∠ABD? A D o Question set [31 - 35] Computational problems. o C B –7– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 31. BD bisects ∠ABC. If ∠ABD = 30°, 34. ∠COE is bisected by OD and ∠AOC find the value for x. is bisected by OB. Find the value for x. A C D D B xo x 20o 30o E A O C B 35. P is a point on AE. BP bisects ∠APC. DP bisects ∠CPE. Find the measure of ∠DPB. 32. ∠ABC is bisected by BD in the figure. If ∠ABE = 100°, find the value for x. B C D C E B xo 100o x o E D o x A P A Name of an angle: 33. As in the following figure, is it true that x = y? A xo 1 40o O 40o B An angle is formed by one vertex and two sides connected by the vertex. As in the figure, the angle can be expressed as ∠AOB or ∠1 in short. yo Definition: congruent angle Congruent angles are equal in measure. –8– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 is congruent to 2 1 is not congruent to 3 4 When two angles ∠1 and ∠2 are measured to be the same, we called them congruent angles, or we say ∠1 is congruent to ∠2. –9– This copy is for me, Peter Lin, only. Honors Geometry Issue 1 There are two such pairs: (∠3, ∠6) and Parallel Lines and Angles (∠4, ∠5). Definition: corresponding angles Alternate exterior angles: Two lines L1 and L2 (not necessarily parallel) are cut by a transversal. Get 1 2 L familiar with the following terms. 1 1 3 5 7 2 L1 7 L2 8 4 There are two such pairs: (∠2, ∠7) and (∠1, ∠8). 6 L2 8 Consecutive exterior angles: There are four such pairs: (∠1, ∠5), (∠3, ∠7), (∠2, ∠6), (∠4, ∠8). THEOREM A [Corresponding Angles Postulate] If L1 and L2 are parallel and cut by a transversal then corresponding angles are congruent. 1 2 L1 7 L2 8 There are two such pairs: (∠1, ∠7) and (∠2, ∠8). Consecutive and alternate angles The term consecutive pair refers to both angles falling on the same side of the transversal. Consecutive interior angles: L1 6 L2 There are two such pairs: (∠3, ∠5), (∠4, ∠6). L1 co ns ec ut iv e 5 4 co ns ec ut iv e 3 L2 The term alternate pair refers to either of the angle falling at the opposite side of the transversal. Alternate interior angles: L1 6 L2 L1 al te rn at e 5 4 al te rn at e 3 – 10 – This copy is for me, Peter Lin, only. L2 Honors Geometry Interior and exterior angles The term interior pair refers to both angles falling in the interior strip formed by L1 and L2. Issue 1 L1 Interior pairs L2 The term exterior pair refers to both angles falling in the exterior strip formed by L1 and L2. L1 exterior pairs L2 36. In each of the following problems use the information to name the segments that must be parallel. If there is no such segment, write none. A 4 B 5 1 2 3 C 12 15 14 F 13 11 6 7 10 9 8G E D Given a) ∠ 2 = ∠ 8 b) ∠ 1+∠ 2=∠ 7+∠ 8 c) ∠ 3 + ∠ 13 = 180° d) ∠ 8 = ∠ 15 e) ∠ 3 = ∠ 14 f) ∠ 3+∠ 10+∠ 11=180° g) ∠ 1 + ∠ 11 = 180° h) ∠ 2 = ∠ 11 Parallel segments AB//EG Reason corr. angles – 11 – This copy is for me, Peter Lin, only. Honors Geometry Issue 1 40. A triangle has two congruent sides, and Standard Test the measure of one angle is 40°. Which 37. A square is a special case of all of the of the following types of triangles is it? following geometric figures EXCEPT a (A) isosceles (A) parallelogram (B) equilateral (B) rectangle (C) right (C) rhombus (D) scalene (D) trapezoid 41. A triangle has one 30° angle and one 60° angle. Which of the following types of triangles is it? (A) isosceles (B) equilateral (C) right (D) scalene 38. A polygon is a plane figure composed of connected lines. How many connected lines must there be to make a polygon? (A) 3 or more (B) 4 or more (C) 5 or more (D) 6 or more 42. A triangle has angles of 71° and 62°. Which of the following best describes the triangle? (A) acute scalene (B) obtuse scalene (C) acute isosceles (D) obtuse isosceles 39. Which of the following statements is true? (A) Parallel lines intersect at right angles. (B) Parallel lines never intersect. (C) Perpendicular lines never intersect. (D) Intersecting lines have two points in common. – 12 – This copy is for me, Peter Lin, only. Honors Geometry Issue 1 46. If pentagon ABCDE is similar to 43. Which of the following does NOT pentagon FGHIJ, and AB = 10, CD = have parallel two pairs of line 5, and FG = 30, what is IH? segments? (A) 3.5 (A) a rhombus (B) 5 (B) a square (C) 15 (C) a trapezoid (D) 30 (D) a rectangle 47. What is the greatest area possible enclosed by a quadrilateral with a perimeter of 24 feet? (A) 6 square feet (B) 24 square feet (C) 36 square feet (D) 48 square feet 44. In a triangle, angle A is 70° and angle B is 30°. What is the measure of angle C? (A) 90° (B) 70° (C) 80° (D) 100° 45. What is a quadrilateral with two parallel sides and an angle of 54°? (A) triangle (B) rectangle (C) square (D) parallelogram 48. What is the difference in area between a square with a base of 4 feet and a circle with a diameter of 4 feet? (A) 16 - 2π square feet (B) 16 - 4π square feet (C) 8π -16 square feet (D) 16π - 16 square feet – 13 – This copy is for me, Peter Lin, only. Honors Geometry Issue 1 49. What is the difference in perimeter 52. What is the perimeter of the regular between a square with a base of 4 feet hexagon shown below? and a circle with a diameter of 4 feet? (A) 8 - 2π feet 5 (B) 16 - 2π feet (C) 16 - 4π feet (D) 16 - 8π feet (A) 20 (B) 24 (C) 28 (D) 30 50. A rectangle’s topmost side is 3 times that of the leftmost side. If the leftmost side is A inches long, what is the area of the rectangle? (A) 3A (B) 6A (C) 3A2 (D) 6A2 53. What is the perimeter of the following figure? a 45o (A) a2 ÷ 2 (B) 2a + 2a2 (C) (2 + 2 )a (D) 4a 51. What is the perimeter of the triangle shown below? 8 10 (A) 24 (B) 28 (C) 16 (D) 14 – 14 – This copy is for me, Peter Lin, only. Honors Geometry 54. The perimeter of a rectangle is 148 feet. Its two longest sides add up to 86 feet. What is the length of each of its two shortest sides? (A) 31 feet (B) 42 feet (C) 62 feet (D) 72 feet Issue 1 55. How many feet of ribbon will a theatrical company need to tie off a performance area that is 34 feet long and 20 feet wide? (A) 54 (B) 68 (C) 88 (D) 108 56. What is the outer perimeter of the doorway shown below? 10 4 (A) 12 (B) 24 (C) 20 + 4π (D) 24 + 2π – 15 – This copy is for me, Peter Lin, only. Honors Geometry Issue 1 Answer Key 11. 120° Divide 180° into 2+1 parts. Each part gets 60°. The angle in the question should get two parts, thus it has 120°. Supplementary and Complementary Angles 1. x°+50° = 180° x = 130°. 12. 45° The complementary angle for 45° is still 45°. Vertical Angles 2. vertical: ∠4 adjacent: ∠2 13. The complementary angle of 35° is 55°. The supplementary angle of 35° is 145°. 3. Vertical angle pairs: (∠1, ∠4), (∠2, ∠5), (∠3, ∠6). Adjacent pairs (∠1, ∠6), (∠6, ∠5), (∠5, ∠4), (∠4, ∠3), (∠3, ∠2), (∠2, ∠1). 14. 100° The complementary angle of 40° is 50°. The angle is two times 50°, so it must be 100°. 4. 75° 180° - 105° = 75° since ∠1 and ∠2 are linear pair with respect to line M. 15. ∠1 = ∠4, (vertical angle) ∠4 = ∠3 (given) ∠3 = ∠2 (vertical angle) ∠1 = ∠2 (transitivity) 5. 105° 180° - 75° = 105° since ∠2 and ∠3 are linear pair with respect to line L. 16. 140° ∠AOB + 40° = 90° (Given) ∠AOB = 50° ∠AOD = ∠AOB + BOD (angle addition property) = 50° + 90° = 140° 6. vertical angles congruent 7. 75° ∠4 = ∠2 = 75° for vertical angles. 8. Yes, it is always true. 17. ∠1 = 90° + 45° = 135° since the bisector divides the right angle to two angles with 45°. 9. 150° The angle is 30° since 90-60=30. Therefore, its supplementary angle is 180°-30° = 150°. 18. 60 - x = 25 + 3x + 15 60 - x = 40 + 3x 20 = 4x x=5 ∠AOB = (60 - x)° = 55° 10. 60° The complementary angle for 60° is 30°. – 16 – This copy is for me, Peter Lin, only. Honors Geometry Issue 1 Parallel Lines and Angles Angle Bisectors 36. The answer is listed in the following table. 19. x = 50° x+40°=90° ⇒ x = 50° Given a) ∠ 2 = ∠ 8 b) ∠ 1 + ∠ 2 = ∠ 7 + ∠ 8 c) ∠ 3 + ∠ 13 = 180° d) ∠ 8 = ∠ 15 e) ∠ 3 = ∠ 14 f) ∠ 3+∠ 10+∠ 11=180° g) ∠ 1 + ∠ 11 = 180° h) ∠ 2 = ∠ 11 20. x = 60° since 180°÷3 = 60°. 21. x = 120° 22. 2x - 5 + 3x + 5 = 90° 5x = 90° x = 18° Parallel segments Reason AB//EG corr. angles BF/CD corr. angles. AE//BG consec. int. None AE//BG corr. angles BF//CD consec. int. None AB//EG alt. int. angles Standard Test 23. Linear pair 37. D 24. complementary 38. A 25. supplementary 39. B 26. ∠2 and ∠4 are supplementary to ∠1 40. A 27. 45° 41. C 28. Yes. Straight angle has 180° and a right angle is 90°. 42. A 29. ∠AOC = 180° - 40° = 140° ∠2 = 1∠AOC = 1(140°) = 70° 43. C 30. ∠AOB + ∠BOC = 180° (linear pair) ∠1 = 1∠AOB (bisector) ∠2 = 1∠BOC (bisector) ∠1 + ∠2 = 1(∠AOB + ∠BOC) = 90° 45. D 31. x = 180° - 2(30°) = 120° 48. B 32. ∠CBA = 80°, therefore, x = 1(80) = 40. 49. C 44. C 46. C 47. C 50. C 33. Yes, since x = y = 140°. 51. A 34. ∠COE = 40°. ∠AOC=140°. Thus, x = 1(140) = 70. 52. D 5×6 = 30 35. 90° 5 5 5 5 5 5 5 5 53. C – 17 – This copy is for me, Peter Lin, only. 54. A Honors Geometry 56. D Issue 1 55. D – 18 – This copy is for me, Peter Lin, only.