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6 Introduction to Geometry 6 Introduction to Geometry Classwork (p. 6.13) 1. (a) 35 (b) 90 Quick Review Review Exercise 6 (p. 6.4) 1. (a) (i) s, x, y (ii) a, b (iii) c, d, e, r (c) 130 (d) 310 2. (b) s, x, y 2. (a) (a) C (b) D, E (c) B 3. (a) A (b) (b) D (c) B Classwork (p. 6.15) 1. (a) Activity Activity 6.1 (p. 6.23) 1. straight angle 2. The sum of the three interior angles of a triangle is 180. (b) EF // CD (or AF // ED // BC) Activity 6.2 (p. 6.41) 1. Polyhedron V F E V+F–E Triangular prism 6 5 9 2 Cube 8 6 12 2 Regular tetrahedron 4 4 6 2 Quadrilateral prism 8 6 12 2 Regular octahedron 6 8 12 2 2. 2. (a) (b) FA⊥EA (or FB⊥GB or GC⊥HC or HD⊥ED or EF⊥GF or FG⊥HG or GH⊥EH or HE⊥FE) Classwork (p. 6.22) V F E 2 Classified by sides Classwork Classified by angles Classwork (p. 6.6) (a) BC, CD (c) 6 cm Classwork (p. 6.7) 1. (a) (b) 120 40 50 210 180 ∴ The set of angles cannot be the interior angles of a triangle. (b) (c) (d) b= c= x= y= C A, C, D B, E A, C D, E B Classwork (p. 6.23) (a) 60 60 60 180 ∴ The set of angles can be the interior angles of a triangle. (b) BD 2. equilateral triangle isosceles triangle scalene triangle acute-angled triangle right-angled triangle obtuse-angled triangle (c) 85 15 80 180 ∴ The set of angles can be the interior angles of a triangle. ∠ABC or ∠CBA ∠BCA, ∠ACB, ∠ECA, ∠ACE or ∠C ∠DBA or ∠ABD ∠EBD or ∠DBE 125 Mathematics in Action (3nd Edition) 1A Full Solutions (d) 72 88 30 190 180 ∴ The set of angles cannot be the interior angles of a triangle. Classwork (p. 6.36) (a) (e) 24 52 84 160 180 ∴ The set of angles cannot be the interior angles of a triangle. (b) (f) Since one of the angles is 0, the set of angles cannot be the interior angles of a triangle. Classwork (p. 6.26) 1. (a) convex polygon, equilateral polygon, equiangular polygon (c) (b) convex polygon (c) convex polygon, equilateral polygon (d) (d) equilateral polygon 2. (a) (a) (b) equilateral triangle 3. Classwork (p. 6.40) (a) no (a) (b) yes (c) no Classwork (p. 6.45) 1. (a) (b) (b) Classwork (p. 6.32) (a) (c) (b) 2. (a) Classwork (p. 6.33) 1. (b) 2. 126 6 Introduction to Geometry Quick Practice (c) Quick Practice 6.1 (p. 6.9) a is an obtuse angle. b is a reflex angle. c is a right angle. d is an acute angle. Classwork (p. 6.47) Quick Practice 6.2 (p. 6.10) (a) 2 right angles 2 90 180 (b) (c) Classwork (p. 6.49) 1. 1 1 round angle 360 5 5 72 3 3 straight angle 180 4 4 135 Quick Practice 6.3 (p. 6.24) In △ABC, ∵ A B C 180 ( sum of △) ∴ A 43 37 180 A 180 43 37 100 Quick Practice 6.4 (p. 6.42) Let V, F and E be the numbers of vertices, faces and edges of the polyhedron respectively. ∵ V = 25 and F = 15 By the Euler’s formula, V FE 2 ∴ 25 15 E 2 E 38 ∴ The polyhedron has 38 edges. 2. Classwork (p. 6.50) 1. Consolidation Corner Consolidation Corner (p. 6.51) 1. (a) no. of vertices = 4, no. of edges = 6, no. of faces = 4 (b) no. of vertices = 8, no. of edges = 12, no. of faces = 6 2. 2. (a) no (b) yes (c) no 3. (a) yes (b) no (c) yes 127 Mathematics in Action (3nd Edition) 1A Full Solutions 4. (d) round angle (a) (e) obtuse angle 7. (f) straight angle (a) 2 straight angles 2 180 360 (b) (b) (c) 1 1 round angle 360 3 3 120 4 4 right angle 90 5 5 72 2 5 (d) 1 right angles 90 3 3 150 Exercise 8. Exercise 6A (p. 6.15) Level 1 1. (a) line segment: AD, AC, AB, BC, CD (or DA, CA, BA, CB, DC) marked angle: ∠BAC (or ∠CAB) (a) The marked angle 360 150 (b) The marked angle 360 2 12 60 (b) line segment: PS, PQ, PR, QR, QS (or SP, QP, RP, RQ, SQ) marked angle: ∠RQS (or ∠SQR) 9. 2. 5 12 (a) The required degree 360 (a) right angle 15 60 90 (b) round angle (b) The required degree 360 (c) straight angle 15 (d) acute angle (e) obtuse angle (f) 3. 10. (a) reflex angle (c) (b) PQS, TSQ a: obtuse angle; b: right angle; c: reflex angle; d: acute angle; e: obtuse angle 5. (a) a, h BPC APC APB 130 60 70 11. (a) a = 50°, b = 40°, c = 125°, d = 295° (b) b < a < c < d (b) b, e, f 12. (a) (c) d, j (d) c, g, i 6. APB 60 (b) APC 130 (a) QRS (or PRT), RQS, RSQ 4. 2.5 60 (a) right angle (b) (b) reflex angle (c) acute angle 128 6 Introduction to Geometry (c) 20. (a) AOB BOC COD 118 2AOB 62 118 2AOB 56 AOB 28 (d) (b) AOC AOB BOC 28 62 90 ∴ AOC is a right angle. 13. 21. (a) x 153 y 82 z 125 (b) 14. (a) DC x y z 153 82 125 360 (b) AB and DC 22. (a) Level 2 15. Angle 2 right angles 2 90 Size 180 Type of the angle straight angle 1 round 4 angle 1 360 4 90 5 right 3 angles 5 90 3 150 1 1 straight 2 angles 1 1 180 2 270 right angle obtuse angle reflex angle A 20 B 30 C 60 D 40 E 30 (b) A B C D E 20 30 60 40 30 180 1 16. Size of angle A 1 90 108 5 2 Size of angle B 360 144 5 2 Size of angle C 180 120 3 ∵ Size of angle B > size of angle C > size of angle A ∴ Angle C is not the greatest angle among the three angles. ∴ Mary’s claim is disagreed. 23. (a) (i) (ii) x 60 (b) (i) 17. ∠AOE, ∠COE, ∠COB 18. (a) AEC AEB BEC 40 25 65 (ii) (b) 65 25 Exercise 6B (p. 6.27) Level 1 1. 2.6 cm 40 19. (a) x 130 CED BED BEC BXC BXD CXD 2. (a) PQR 134 56 (b) WXYZ 78 (b) (c) ABCDE AXC AXB BXC 3. 38 78 116 (a) ABCDEF (b) AB, BC, CD, DE, EF, FA 129 Mathematics in Action (3nd Edition) 1A Full Solutions 4. (a) equilateral triangle 8. (b) scalene triangle (a) (i) (ii) (iii) (iv) D, E A, B, C A, C, E C (c) isosceles triangle (b) C 5. (a) acute-angled triangle 9. (a) convex polygon (b) right-angled triangle (b) concave polygon (c) obtuse-angled triangle (c) convex polygon 6. (a) (i) C (ii) B, C, E (iii) A, D (d) concave polygon (e) concave polygon (b) (i) C, E (ii) B, D (iii) A (f) convex polygon 10. (a) 7. (a) In △XYZ, ∵ X + Y + Z = 180 ( sum of △) ∴ 60 a 80 180 a 180 60 80 This polygon has 2 diagonals. 40 (b) (b) In △OPQ, ∵ O + P + Q = 180 ( sum of △) ∴ 90 c 65 180 c 180 90 65 25 This polygon has 5 diagonals. (c) In △ABC, ∵ A + B + C = 180 ( sum of △) ∴ w 65 65 180 (c) w 180 65 65 50 (d) In △PQR, ∵ P + Q + R = 180 ( sum of △) x 60 x 180 ∴ 2 x 180 60 This polygon has 9 diagonals. 11. The length of the sides of the regular octagon 104 8 cm 13 cm 2 x 120 x 60 Level 2 12. ∵ ∴ i.e. ∴ (e) In △XYZ, ∵ X + Y + Z = 180 ( sum of △) 90 y 3 y 180 ∴ 4 y 180 90 4 y 90 y 22.5 (f) 13. (a) In △UVW, ∵ U + V + W = 180 ( sum of △) ∴ ( z 10) (3 z 20) z 180 5 z 30 180 5 z 180 30 5 z 150 z 30 (b) 130 O is the centre of the circle. OA, OB and OC are the radii of the circle. OA = OB = OC △OAB, △OAC and △OBC are the isosceles triangles in the figure. 6 Introduction to Geometry (c) (b) ∵ The sum of an obtuse angle and the right angle is greater than 180°. ∴ An obtuse-angled triangle cannot contain a right angle. 18. (a) ∵ The hexagon ABCDEF is made up of 6 identical equilateral triangles. ∴ AB = BC = CD = DE = EF = FA Size of each interior angle of an equilateral triangle 180 3 60 ABC 2 60 120 By the similar argument, BCD = CDE = DEF = EFA = FAB = 120°. ∴ ABCDEF is a regular hexagon. (d) 14. (a) 5 (b) (i) △ABD and △ADE (ii) △BCD and △ACE (iii) △ACD (b) Length of a side of each equilateral triangle 15 cm 3 5 cm Perimeter of the hexagon 5 6 cm 15. (a) In △PRS, ∵ RPS + PRS + RSP = 180 ( sum of △) ∴ m 58 (65 15) 180 30 cm m 180 58 65 15 42 19. In △UVW, ∵ VUW + UVW + UWV = 180 ( sum of △) ∴ (a 10) 70 2a 180 3a 180 10 70 In △PQS, ∵ QPS + PQS + QSP = 180 ( sum of △) ∴ 42 n 15 180 n 180 42 15 120 a 40 123 In △UVY, ∵ VUY UVY UYV 180 ( sum of △) ∴ [(40 10) 45] 70 b 180 145 b 180 (b) In △ABD, ∵ BAD + ABD + ADB = 180 ( sum of △) ∴ (20 y ) 40 90 180 y 180 20 40 90 30 b 180 145 35 In △ACD, ∵ CAD ACD ADC 180 ( sum of △) ∴ 30 z 90 180 In △UXY, ∵ XUY UXY UYX 180 ( sum of △) ∴ 45 c (40 35) 180 120 c 180 z 180 30 90 60 c 180 120 60 16. In △ABC, ∵ BAC ABC ACB 180 ( sum of △) ∴ 40 r 80 180 Exercise 6C (p. 6.37) Level 1 1. (a), (b) r 180 40 80 60 In △BDF, ∵ DBF + BDF + BFD = 180 ( sum of △) ∴ 60 30 s 180 s 180 60 30 90 17. (a) ∵ If a triangle contains two right angles, the remaining angle is 0°. It is impossible. ∴ A triangle cannot contain two right angles. 131 Mathematics in Action (3nd Edition) 1A Full Solutions 2. 10. (a), (b) Level 2 11. 3. 12. (a) (i), (ii), (iii) 4. (a) (b) yes (b) 13. (a) (b) 5. 6. 14. 7. 15. (a) 8. (b) A = 26, B = 34, C = 120 16. (a) 9. 132 6 Introduction to Geometry (b) MN = 6.4 cm 6. (a) yes 17. (a) (b) no (c) no (b) (i) QR = 5.0 cm, RP = 5.0 cm (ii) △PQR is an equilateral triangle. (d) yes Exercise 6D (p. 6.52) Level 1 1. (a) no (b) yes (c) no 7. (a) no (d) no (b) yes (e) yes 2. (a) no. of vertices = 5, no. of edges = 8, no. of faces = 5 (b) no. of vertices = 8, no. of edges = 12, no. of faces = 6 8. (a) (c) no. of vertices = 6, no. of edges = 9, no. of faces = 5 3. 4. Let V, F and E be the numbers of vertices, faces and edges of the polyhedron respectively. ∵ F = 10 and E = 18 By the Euler’s formula, V FE 2 V 10 18 2 ∴ V 10 ∴ The polyhedron has 10 vertices. Let V, F and E be the numbers of vertices, faces and edges of the polyhedron respectively. ∵ V = 13 and F = 17 By the Euler’s formula, V F E 2 ∴ 13 17 E 2 ∴ 5. (b) E 28 The polyhedron has 28 edges. (c) (a) (b) 133 Mathematics in Action (3nd Edition) 1A Full Solutions 9. 13. (a) Isometric grid paper: (a) Oblique grid paper: (b) (b) Isometric grid paper: 10. (a) Oblique grid paper: (b) 14. (a) Isometric grid paper: Level 2 11. (a) no Oblique grid paper: (b) yes (c) no (d) yes 12. (a) (b) Isometric grid paper: (b) 134 6 Introduction to Geometry Oblique grid paper: Revision Exercise 6 (p. 6.61) Level 1 1. a: BDE (or CDE), b: BED (or MED), c: BME, d: reflex DBE (or reflex DBM or reflex CBE or reflex CBM), e: CAE (or MAE) Check Yourself (p. 6.59) 1. (a) a: 90°, b: 132°, c: 42°, d: 236°, e: 40° 2. a: acute angle, b: reflex angle, c: right angle, d: acute angle, e: reflex angle, f: obtuse angle 3. (a) 57 33 90 (b) a: right angle, b: obtuse angle, c: acute angle, d: reflex angle, e: acute angle 2. BOD BOC COD AOD AOB BOD 28 90 (a) a scalene 118 (b) an isosceles AOE AOD DOE 118 62 (c) an acute-angled 180 (d) a right-angled (b) BOD: right angle, AOD: obtuse angle, AOE: straight angle 3. Polygon A B C D 4. Equilateral polygon Equiangular polygon Convex polygon Concave polygon 4. (a) (i) concave (ii) no (b) (i) convex (ii) yes (a) (c) (i) concave (ii) yes (b) (c) 5. The angle the hour hand rotates 360 120 (d) 5. 4 12 In △ABC, ∵ CAB + ABC + ACB = 180 ( sum of △) ∴ 46 ( x 18) 88 180 152 x 180 x 28 In △ABE, ∵ BAE ABE AEB 180 ∴ ( y 46) 28 62 180 y 136 180 6. The reflex angle formed 360 10 12 300 7. ( sum of △) (a) ∵ The sum of all the interior angles of a triangle is 180. ∴ 60 45 x 180 x 180 60 45 75 y 44 (b) ∵ The sum of all the interior angles of a triangle is 180. ∴ 90 y 42 180 6. y 180 90 42 48 8. 7. (a) (a) In △ABC, ∵ A + B + C = 180 ( sum of △) ∴ 20 70 C 180 C 180 20 70 90 (b) yes 135 Mathematics in Action (3nd Edition) 1A Full Solutions (b) In △ABC, ∵ A + B + C = 180 ( sum of △) ∴ 35 B 63 180 13. (a) B 180 35 63 82 (c) In △ABC, ∵ A + B + C = 180 ( sum of △) ∴ A 18 47 180 A 180 18 47 115 (b) 9. 10. 14. (a) 11. (a) (i), (ii) (b) (b) 3 cm 12. (a) (i) no (ii) no (c) (b) (i) no (ii) no (c) (i) no (ii) yes 15. Isometric grid paper: (d) (i) yes (ii) yes Oblique grid paper: (e) (i) yes (ii) no 136 6 Introduction to Geometry Level 2 16. (a) 8 20. (a) In △PQR, ∵ P Q R 180 ( sum of △) ∴ 3x 4 x 2 x 180 (b) (i) △ABF, △BCF and △ACF (ii) △ABF, △BCF, △ACF and △DEF (iii) △BCD 9 x 180 x 20 17. There are 3 hours 15 minutes from 2:15 p.m. to 5:30 p.m. 1 15 hours ∵ 15 minutes 60 1 hours 4 0.25 hours ∴ 3 hours 15 minutes = 3.25 hours The angle the hour hand rotates 1 360 3.25 12 97.5 (b) P 3 x 3(20) 60 Q 4 x 4(20) 80 R 2 x 2( 20) 40 ∴ △PQR is an acute-angled triangle. 21. (a) In △PQR, ∵ P Q R 180 ( sum of △) ∴ 3x ( x 10) ( x 15) 180 5 x 25 180 5 x 155 x 31 18. (a) In △ABC, ∵ BAC + ABC + ACB = 180 ( sum of △) ∴ 52 32 (38 x) 180 x 180 52 32 38 58 (b) P 3x 3(31) 93 Q x 10 31 10 41 R x 15 31 15 46 ∴ △PQR is an obtuse-angled triangle. In △BCD, ∵ CBD + BCD + CDB = 180 ( sum of △) ∴ 32 58 y 180 22. (a) y 180 32 58 90 (b) In △ABD, ∵ BAD + ABD + ADB = 180 ( sum of △) ∴ 36 p 90 180 p 180 36 90 54 In △ABC, ∵ BAC + ABC + ACB = 180 ( sum of △) ∴ 36 (54 46) q 180 136 q 180 (b) 13 cm 23. (a) q 180 136 44 19. In △XYZ, ∵ YXZ + XYZ + XZY = 180 ∴ 50 x 70 180 (b) obtuse-angled triangle ( sum of △) 24. (a), (b) x 180 50 70 60 In △RST, ∵ SRT + RST + RTS = 180 ∴ 61 55 y 180 ( sum of △) y 180 61 55 64 In △MYT, ∵ YMT + MYT + MTY = 180 ( sum of △) ∴ z 60 64 180 z 180 60 64 56 137 Mathematics in Action (3nd Edition) 1A Full Solutions 25. (a) (i) (ii) 2. Answer: D ∵ A reflex A 360 ∴ The sum of A and reflex A is always equal to a round angle. ∴ The answer is D. 3. Answer: B ∵ All sides of an equilateral polygon are of equal length. ∴ The answer is B. 4. Answer: C For I: ∵ The sum of two obtuse angles is greater than 180°. ∴ It is impossible to form a triangle with two obtuse angles. ∴ There is at most one obtuse angle in any triangle. ∴ I must be true. For II: ∵ In an acute-angled triangle, all interior angles are acute. ∴ Interior angles of a triangle can be all acute. ∴ II must be true. For III: ∵ The sum of two right angles is equal to 180°. ∴ It is impossible to form a triangle with two right angles. ∴ A triangle cannot have more than one right angle. ∴ III must be false. ∴ The answer is C. 5. Answer: D In △ABC, ∵ A + B + C = 180 ( sum of △) ∴ a b 68 180 (b) Yes, the circle passes through D. △ACD is an isosceles triangle. 26. (a), (b) 27. Isometric grid paper: a b 112 6. Oblique grid paper: Answer: B With the notation in the figure, in △ADE, ∵ DAE + ADE + AED = 180 ( sum of △) ∴ DAE 73 56 180 DAE 180 73 56 DAE x 90 Multiple Choice Questions (p. 6.65) 1. Answer: C ABE is an acute angle. ∵ ABD ABC DBC 51 x 90 51 39 180 90 90 ∴ ABD is a right angle. CBE is an obtuse angle. DBE is an acute angle. ∴ The answer is C. 138 7. Answer: C ∵ Sides of cuboids can be different in length. ∴ Cuboids may not be regular polyhedra. 8. Answer: A 9. Answer: A 6 Introduction to Geometry Exam Corner Exam-type Questions (p. 6.67) 1. In △ACG, GAC + ACG + AGC = 180 ( sum of △) 54 ACG 70 180 ACG 180 54 70 56 In △BDE, EBD + EDB + BED = 180 ( sum of △) EBD 65 55 180 EBD 180 65 55 60 In △BCF, FBC + BCF + BFC = 180 ( sum of △) 60 56 x 180 x 180 60 56 64 2. 3. 4. Answer: D ∵ All sides of an equilateral polygon are of equal length. ∴ The answer is D. 5. Answer: D ∵ All interior angles of a convex polygon are less than 180°. ∴ The answer is D. 139