Worksheets for GCSE Mathematics. Geometrical Reasoning. Mr Black's Maths Resources for Teachers GCSE 1-9. Shape

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1 Worksheets for GCSE Mathematics Geometrical Reasoning Mr Black's Maths Resources for Teachers GCSE 1-9 Shape

2 Geometrical Reasoning Contents Differentiated Independent Learning Worksheets Drawing and Measuring Angles Angles about a Point Angles in a Triangle Angles in a Quadrilateral Introduction to Angles in Parallel Lines Problems with Angles in Parallel Lines Bearings Interior and Exterior Angles in Polygons Circle Theorems 1 Circle Theorems 2 - Tangents Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Solutions Drawing and Measuring Angles Angles about a Point Angles in a Triangle Angles in a Quadrilateral Introduction to Angles in Parallel Lines Problems with Angles in Parallel Lines Bearings Interior and Exterior Angles in Polygons Circle Theorems 1 Circle Theorems 2 - Tangents Page 13 Page 13 Page 13 Page 14 Page 15 Page 16 Page 17 Page 19 Page 19 Page 19

3 Constructing Angles Q1. Use a protractor to draw the following angles. a) 60 b) 55 c) 48 d) 45 e) 45 f) 135 g) 120 h) 84 i) 300 j) 200 k) 174 l) 286 Q2. Use a protractor to measure the following angle. i) Estimate the size of each angle and record your results. ii) Measure the angles to work out the difference between the actual and your estimates. 3

4 Angles about a Point Q1. Calculate the value of the unmarked angles using angles on a straight line. Q2. Calculate the value of the following using vertically opposite angles. Calculate the values of the following using angles about a point. 4

5 Q1. Calculate the value of the marked angle. Angles in Triangles Q2. Calculate the value of the marked angle. a) One angle in an isosceles triangle is 30. What could the other two angles be? b) Lee measures the three angles of a triangle as 40, 55 and 90. Could this be correct? Explain. c) Rebecca measures all the angles in a triangle to be the same. What type of triangle is it? d) Two sides of a roof are inclined at 32 to the horizontal. Calculate the angle between the sides of the roof. 5

6 Q1. Calculate the missing angle in each quadrilateral. Angles in Quadrilaterals Q2. Calculate the missing angle in each diagram. a) Two angles in a quadrilateral are 55 and 65 and the other two are equal. What size are the other two angles? b) An isosceles trapezium has an angle of 87. What are the sizes of the other three angles? c) Simon measures the angles of a quadrilateral. He says the angles are 54, 110, 64 and 134. Could he be right? Explain your answer. d) Clare measures the angles in a quadrilateral. She says two opposite angles are equal the other two are not. What type of quadrilateral was Clare measuring? 6

7 Angles in Parallel Lines Q1. Calculate the value of the unknown shaded angles. Explain your reasoning. Q2. Calculate the value of the unknown shaded angles. Explain your reasoning. 7

8 Q1. Calculate the value of the shaded angles. Problems with Angles in Parallel Lines Q2. Calculate the value of the shaded angles. a) ABCD is a parallelogram. Angle ABC = 76. Calculate the size of the other three angles. b) Prove the opposite angles of a parallelogram are equal. 8

Bearings Q1 Measure the following bearings from the star marked O. a) Central Park b) Delacorte Theatre c) Museum of Natural History d) W 85 th St and Central Park West.

9 Bearings Q1 Measure the following bearings from the star marked O. a) Central Park b) Delacorte Theatre c) Museum of Natural History d) W 85 th St and Central Park West. Q2 Use bearings to construct the following situation on a single diagram. i) A to B = 063 for 6.7 cm ii) A to C = 117 for 4.5 cm iii) A to D = 225 for 4.2 cm iv) A to E = 270 for 6 cm v) A to F = 307 for 5 cm Q3 Use you answer from Q2 to calculate the back bearings for each location from A. Q4 The bearing of Ashville from Mosely is 155. What is the bearing of Mosely from Ashville? Q5 A ship is on a bearing of 068 from a lighthouse. What is the bearing of the lighthouse from the ship? Q6 The distances from town X to town Y and Z are 180 km and 240 km respectively. The bearing of Y from X is 340 and the bearing of Z from X is 032. a) Make a scale drawing of this information. b) Find the distance between Y and Z. c) Find the bearing of Z from Y. Q7 A plane flies for 200 km on a bearing of 135. It then alters course and flies for 140 km on a bearing of 040. a) Construct a scale drawing the plane s flight. b) Describe the direct return journey. 9

10 Q1. Calculate the value of each marked angle. Interior and Exterior Angles of Polygons Q2. a) Three of the exterior angles of a quadrilateral are 105, 58 and 110. Find the size of the other exterior angle. b) A polygon has 9 sides. Eight of the angles add up to Calculate the size of the remaining angle. c) Four of the exterior angles of a hexagon are 67, 45, 65 and 17. The remaining two angles are equal. Calculate the size of the two equal angles. d) An exterior angle of a regular polygon is 18. Calculate the value an interior angle. Find the number of sides of the regular polygon with an exterior angle of a) 45 b) 24 c) d) 518 Q4. Find the number of sides of the polygon with the interior angle sum of a) 900 b) 1440 c) 2340 d) 5040 Q5. Calculate the sum of the interior angles with a) 8 sides b) 16 sides c) 25 sides d) 50 sides Q6. The diagram (not drawn accurately) shows a regular pentagon with center, O. Calculate the size of angle α. 10

11 Circle Theorems 1 Q1. Calculate the value of the angles shown. State the angle properties you used. Q2. Calculate the value of the angles shown. State the angle properties you used. Calculate the value of the angles shown. State the angle properties you used. 11

12 Circle Theorems 2 - Tangents Q1. Each diagram is constructed with a tangent. Calculate the value of the marked angle. Q2. Each diagram is constructed with a tangent. Calculate the value of the marked angle. 12

13 Q2. Constructing Angles a) 45 b) 60 c) 72 d) 100 e) 135 f) 95 g) 300 h) 190 i) 290 j) 200 k) 174 l) 286 a) 12 b) 94 c) 324 Q1. Angles about a Point a) a = 108 b) b = 34 c) c = 30 d) d = 42 e) e = 37 f) f = 52 Q2. a) g = 129, h =51 b) i = 153, j = 153, c) k = 132, l = 132 d) m = 67, n = 67, o = 113 e) p = 37, q = 143, r = 143 f) s = 162, t = 162, u = 18 a) a = 117, b =60 b) c = 59, d = 45, e = 45 c) f = 126, g = 42 Q1. Angles in Triangles a) a = 70 b) b = 63 c) c = 56 d) d = 39 e) e = 62 f) f = 112 Q2. a) g = 53 b) h = 45 c) k = 84, m = 51 d) n = 45, p = 34 e) r = 37, s = 34 f) t = 50, u = 53, v = 37 a) 30, 30, 120 or 30, 75, 75 b) Sum of angle is 185. Cannot therefore be a triangle. c) Equilateral triangle. d) 74 13

14 Q1. Angles in Quadrilaterals a) a = 104 b) b = 75 c) c = 63 d) d = 79 e) e = 54 f) f = 117, g = 63 Q2. a) h = 53 b) i = 61 c) j = 28, k = 104 d) m = 94 e) n = 67, p = 231 f) q = 73, r = 107, s = 80 a) 120 b) 87, 93 and 93 c) The angles in the quadrilateral would add to 362 not 360. d) A Kite. 14

15 Q1 Angles in Parallel Lines a) a = Alternate b) b = Corresponding c) c = 71 - Interior d) d = 59 - Alternate e) e = 68 - Corresponding f) f = 70 - Interior Q2. Q3 15

16 Q1. Problems with Angles in Parallel Lines Q2. a) 76, 104, 104 b) = Alternate angles = Corresponding angles. Therefore = 16

17 Solutions Bearings Q1 a) 042 b) 231 c) 265 d) 322 Q2 Q3 i) B to A = 243 ii) C to A = 297 iii) D to A = 135 iv) E to A = 90 v) F to A = 053 Q4 335 Q

18 Q6 Bearings Q7. 18

19 Interior and Exterior Angles of Polygons Q1. a) 63 b) 88 c) 216 d) 153 e) 135 f) 144 Q2. a) 87 b) 60 c) 83 d) 160 a) 8 sides b) 15 sides c) 11 sides d) 20 sides Q4. a) 7 sides b) 10 sides c) 15 sides d) 30 sides Q5. a) 1080 b) 2520 c) 4140 d) 8640 Q6. α = 54. Q1. Circle Theorems 1 a) a = 110 b) b = 120 c) c = 63 d) d = 146 e) e = 212 f) f = 95, g = 27 Q2. a) h = 78 b) i = 102, j = 93 c) k = 108 d) m = 67, l = 86 e) t = 78, n = 116 f) p = 84, y = 80 a) q = r = 46 b) s = t = 29 c) u = 82, v = 102 Circle Theorems 2 - Tangents Q1 a) a = 56 b) b = 53 c) 5.54 cm d) d = 125, e = 27 e) f = 104, g = 52 f) h = 30, j = 60 Q2 a) m = 44, n = 66, k = 71 b) p = 60, q = 44, r = 77 c) s = 58, t = 82, u = 58 d) v = 69, w = 53, x = 58 e) y = 67, z = 50, A = 67 f) b = 45 g) c = 95 h) e = 103, d = 58 i) f = 8, g = 53 19

Cambridge Essentials Mathematics Core 9 GM1.1 Answers. 1 a

Cambridge Essentials Mathematics Core 9 GM1.1 Answers. 1 a GM1.1 Answers 1 a b 2 Shape Name Regular Irregular Convex Concave A Decagon B Octagon C Pentagon D Quadrilateral E Heptagon F Hexagon G Quadrilateral H Triangle I Triangle J Hexagon Original Material Cambridge