Chapter 1. Worked-Out Solutions. 1.5 Explorations (p. 37) BC = ( 1 2) 2 + [2 ( 2)] 2 = ( 3) = 25 = 5 AC =

Transcription

1 37. Find the lengths of the sides in terms of. B = ( 1 ) + [ ( )] = ( 3) + = 5 = 5 A = [ ( 1)] + ( ) = ( + 1) = + 1 AB = ( ) + [ ( )] = ( ) + = ( ) + 16 Triangle AB has a perimeter of 1 units. B + A + AB = ( ) + 16 = 1 ( ) + 16 = 6 ( ) + 16 = (6 ) + 0 = = = 16 = Maintaining Mathematical Proficienc = = 3 = 9 5 = 5 = 3 = = = = 5 The equation has no solution. = = = = 7 = = = = = 7 1 =. raw a ra using a straightedge and label the left endpoint of the ra. Open the compass and place the point of the compass on and the pencil point on. Without changing the setting of the compass, place the point on and mark the ra with an arc. arc X Y Label the intersection of the ra and the arc as. 1.5 Eplorations (p. 37) 1. a. m AOB = 35 and it is acute. b. m AO = 65 and it is acute. c. m BO = = 30 and it is acute. d. m BOE = = 110 and it is obtuse. e. m OE = = 80 and it is acute. f. m O = = 5 and it is acute. g. m BO = = 75 and it is acute. h. m AOE = 15 and it is obtuse.. a. heck students work. b. heck students work. c. es; for a heagon, n = 6 and 180(6 ) = 180() = 70. Each interior verte angle of the heagon is 10, so, 6(10) = 70. d. First heagon: The sum of the angle measures for each trapezoid is 360. So, the total interior angle measure is 360 = 70. Second heagon: The sum of the angle measures for each triangle is 180. The sum of the angle measures for the rectangle is 360. So, the total sum of the interior angles is = 70. Third heagon: The sum of the angle measures for each triangle is 180. So, the total interior angle measure is = no; The first two heagons have an interior polgon angle sum of 70 ; however, the third heagon includes an interior point which is the verte meeting point of the si triangles. This common verte intersection adds an additional 360 to the combined sum of interior angles. 3. Angles can be measured using a protractor. When the measure is greater than 0 and less than 90, the angle is acute. When the measure is equal to 90, the angle is right. When the measure is greater than 90 and less than 180, the angle is obtuse. When the measure is 180, the angle is straight. 1.5 Monitoring Progress (pp. 38 ) 1. Three names for the angle are PQR, RQP, and Q.. Three names for the angle are 1, Y, and XYZ (or ZYX). 3. Three names for the angle are, E, and EF (or FE). HJ lines up with 0 on the inner scale of the protractor. HM lines up with 15 on the inner scale. So, m JHM = 15. It is an obtuse angle.. HM lines up with 15 on the inner scale of the protractor. HK lines up with 55 on the inner scale. So, m JHM = = 90. It is a right angle. 5. opright Big Ideas Learning, LL Geometr 3

2 HM lines up with 15 on the inner scale of the protractor. HL lines up with 90 on the inner scale. So, m MHL = = 55. It is an acute angle AB is not congruent to FEH. AB appears to be a right angle and FEH is obtuse. 8. m KLN + m NLM = m KLM (10 5) + ( + 3) = = = 18 = 13 m KLN = (10 5) = ( ) = (130 5) = 15 m NLM = ( + 3) = ( ) = (5 + 3) = m EFH + m HFG = m EFG ( + ) + ( + 1) = = 90 3 = = 9 m EFH = ( + ) = ( 9 + ) = (58 + ) = 60 m HFG = ( + 1) = (9 + 1) = 30 M Q Eercises (pp. 3 6) N P m MNQ = 90 m QNP = 90 Vocabular and ore oncept heck 1. Two angles are congruent angles when the have the same measure.. BA does not belong because 1, BA, and AB all name the same angle. Monitoring Progress and Modeling with Mathematics 3. The names for the angle are B, AB, and BA.. The names for the angle are G, FGH, and HGF. 5. The names for the angle are K, 1, and JKL (or LKJ). 6. The names for the angle are R, 8, and QRS (or SRQ). 7. Three different angles are HMN, HMK, and KMN. 8. Three different angles are GJF, GJ, and JF. OA on 0 using the outer scale of the protractor and O lines up with 30 on the outer scale. m AO is 30 and it is an acute angle. 9. Line up OB on 0 using the inner scale of the protractor and O lines up with 65 on the inner scale. m BO = 65 and it is an acute angle. 10. Line up 11. Line up O on 150 using the inner scale of the protractor and O lines up with 65 on the inner scale. m O = = 85 and it is an acute angle. O on 60 using the inner scale of the protractor and OE lines up with 0 on the inner scale. m EO = 1. Line up 65 0 = 5 and it is an acute angle. 13. The error was the outer scale was used. The inner scale should have been used because OB passes through 0 on the inner scale; m BOG = Because neither side passes through 0, the measure of the angle should be calculated as the absolute value of the difference between the numbers on the protractor that are matched with the sides; 65 0 = Label the verte of the original angle as F A. raw a segment and label a point on the segment. raw an arc with the center at A. Label the two intersecting points as E B and. Using the same radius, draw an arc with center. Label the intersecting point as E. raw an arc with radius B and center E. Label the intersection F. raw F. 16. F E Label the verte of the original angle as A. raw a segment and label a point on the segment. raw an arc with the center at A. Label the two intersecting points as B and. Using the same radius, draw an arc with center. Label the intersecting point as E. raw an arc with radius B and center E. Label the intersection F. raw F. 17. The angles congruent to AE are EA, B, and B. 18. The angles congruent to EA are AB and B. 19. m AE = 3 0. m EA = 11 m AE = M B m EA = m AB m B = 3 m AB = m AB = m AB + m B = = 58. m LMN = m LMP + m PMN = = 108 Geometr opright Big Ideas Learning, LL

3 3. m RST = m RSV + m VST 11 = m RSV + 7 = m RSV. m GHK = m GHL + m LNK 180 = 79 + m LNK 101 = m LNK 5. m AB = m AB + m B 95 = ( + 3) + (9 5) 95 = = 11 7 = m AB = ( 7 + 3) = (1 + 3) = 37 m B = (9 7 5) = (63 5) = m XYZ = m XYW + m WYZ 117 = (6 + ) + ( ) 117 = = = m XYW = [ 6( ) + ] = ( 1 + ) = 3 m WYZ = [ 10( ) + 65 ] = (0 + 65) = m LMN = m LMP + m PMN 180 = ( ) + ( 0 + 3) 180 = = 36 = m LMP = [ 16( ) + 13 ] = (6 + 13) = 77 m PMN = [ 0( ) + 3 ] = (80 + 3) = m AB = m ABX + m XB 180 = (1 + 70) + (0 + 8) 180 = = 3 3 = m ABX = ( ) = ( + 70) = 11 m XB = ( ) = (60 + 8) = m RST = m RSQ + m QST 90 = (15 3) + (8 + 18) 90 = = 3 5 = m RSQ = (15 5 3) = (75 3) = 3 m QST = ( ) = (0 + 18) = m FE = m FEH + m HE 90 = (10 + 1) = = = m FEH = ( ) = (30 + 1) = 51 m HE = (13 3) = 39 A B Label the verte of the angle as A. Place the compass at A. raw an arc that intersects both sides of the angle. Label the intersections B and. Place the compass at and draw an arc, then place the compass point at B. Using the same radius, draw another arc. Label the intersection. Use a straightedge to draw a ra through A and. A bisects A. A B Label the verte of the angle as A. Place the compass at A. raw an arc that intersects both sides of the angle. Label the intersections B and. Place the compass at and draw an arc. Then place the compass point at B. Using the same radius, draw another arc. Label the intersection. Use a straightedge to draw a ra through A and. A bisects A. 33. m RQS = m PQS = 63 m PQR = m PQS + m RQS = = m PQS = M RQS = 71 m PQR = m PQS + m RQS = = m PQS = 1 m PQR = 1 (1 ) = 6 m RQS = 1 m PQR = 1 (1 ) = m PQS = 1 m PQR = 1 (119 ) = 59.5 m SQR = 1 m PQR = 1 (119 ) = m AB = m B (6 + 1) = (3 + 9) = 9 3 = 15 = 5 m AB = ( ) = (30 + 1) = m B = ( ) = (15 + 9) = m AB = ( + ) = 88 opright Big Ideas Learning, LL Geometr 5

4 38. m AB = m B (3 + 6) = (7 18) 6 = 18 = 6 = m AB = ( ) = (18 + 6) = m B = (7 6 18) = ( 18) = m AB = ( + ) = m AB = m B ( + 33) = ( + 81) 33 = = 6 8 = m AB = [ ( 8) + 33 ] = (3 + 33) = 65 m B = [ ( 8) + 81 ] = ( ) = 65 m AB = ( ) = m AB = m B (8 + 35) = (11 + 3) 35 = = 3 = m AB = (8 + 35) = (3 + 35) = 67 m B = (11 + 3) = ( + 3) = 67 m AB = ( ) = Use the Angle Addition Postulate to find m AB. m AB = m AB + m B, then subtract m B from m AB.. Let be the angle at which Malcom Wa intersects Park Road. 16 = = The angle at which Malcom Wa intersects Park Road is m LMN = m LMP + m PMN 76 = m LMP = m LMP. a. m AB = m EF = 11 b. BG bisects AB. m ABG = 1 m AB = 1 (11) = 56 c. m BG = 1 m AB = 1 (11) = 56 d. m EG= 1 m EF = 1 (11) = GF is a straight angle. GB bisects GF, so GE GE. Because both angles are congruent and supplementar, each angle measures Sample answer: EG is an acute angle, AB is an obtuse angle, GE is a right angle, and GF is a straight angle. 7. a. A ( + 1) B X b. m AB = m ABX + m XB 9 = ( + 1) + 9 = = 5 16 = m BX = 16 m ABX = ( ) = (6 + 1) = Sample answer: 9:00 or 3:00; The hour hand will be pointing at the 3 or 9 eactl, and the minute hand will be on the 1 eactl. 9. a. acute; When bisecting an acute angle, each angle is acute because the original angle is less than 90. b. acute; When bisecting a right angle, each angle is acute because the angles are complementar. c. acute; When bisecting an obtuse angle, each angle is acute because the original angle is less than 180 and greater than 90. d. right; When bisecting a straight angle, each angle is a right angle because the angles are supplementar. 50. a. If a ra is drawn in the interior of an acute angle, then the two angles formed are each acute. An acute angle is less than 90, so each of the smaller angles will be less than 90. b. If a ra is drawn in the interior of a right angle, each angle will have a measure less than 90 because the angles are complementar. c. If a ra is drawn in the interior of an obtuse angle, then the two angles formed could be both acute, one right and one acute, or one obtuse and one acute. An obtuse angle has a measure greater than 90 but less than 180. d. If a ra is drawn in the interior of a straight angle, the two angles formed could be both right angles or one acute and one obtuse. The sum of the angle measures is Geometr opright Big Ideas Learning, LL

5 51. a. Sample answer: (, 1) (, 0) The point (, 1) will create an acute angle. b. Sample answer: (0, ) (, 0) The point (0, ) will create a right angle. c. Sample answer: ( 1, ) (, 0) The point ( 1, ) will create an obtuse angle. d. Sample answer: (, 0) (, 0) The point (, 0) will create a straight angle. 5. no; Obtuse angles have a measure greater than 90 and less than 180. So, adding two obtuse angle measures gives a sum greater than The sum of two acute angles could ield another acute angle ( = 80 ), a right angle ( = 90 ), or an obtuse angle ( = 100 ). 5. a. m XYW + m WYV + m VYZ = = 18 The three angles appear to form a straight angle; however, the sum of the three angles is 18. b. Sample answer: hange m VYZ to P V Q R S T m RSQ = m TSQ m RSP = m PSQ = 1 m RSQ m RSV = m VSP = 1 m RSP m VSP = 17 m RSP = (17 ) = 3 m RSQ = m PSQ = (3 ) = 68 m TSQ = m RSQ = acute; It is likel that the angle with the horizontal is ver small because levels are tpicall used when something appears to be horizontal but still needs to be checked. Maintaining Mathematical Proficienc = = 90 = 113 = = = 180 = 7 = = = = 15 = = = 90 7 = 8 = = = = 105 = = = = 190 = Sample answer: You draw a segment, ra, or line in the interior of an angle so that the two angles created are congruent to each other. Angle bisectors and segment bisectors can be segments, ras, or lines, but onl a segment bisector can be a point. The two angles/segments created are congruent to each other, and their measures are each half the measure of the angle/segment. opright Big Ideas Learning, LL Geometr 7