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

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1 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 University Press

2 3 Check that students have drawn a regular, an irregular convex and a concave pentagon. Here are some examples: 4 a Sides must all be of equal length and the interior angles must be equal. b A regular quadrilateral is a square. c 5 a The sum of the interior angles increases by 180 each time. b Number of sides Sum of interior angles c i 160 ii 115 iii 155 iv 120 v 141 iv 97 6 a A regular pentagon has five equal angles. The sum of these angles is 540. Divide this number by 5. This gives 108 for each interior angle. b The interior angle of a regular octagon = 135 Original Material Cambridge University Press

3 c The interior angle of a dodecagon = The missing angle = = a (n 2) 180 works out the sum of the interior angles of a polygon with n sides. b i (5 2) 180 = 540 ; x = =145 and y = = 35 ii (8 2) 180 = 1080 ; x = = 143 and y = = 37 iii (4 2) 180 = 360 ; x = = 85 and y = = 95 9 The formula says that the sum of the interior angles of a quadrilateral add to 360. Alternatively, divide the shape into two triangles. As you can see, adding up the angles in each triangle is the same as adding up the four interior angles of the quadrilateral. Therefore, the sum of the interior angles of the concave quadrilateral is = 360. This is the same as the answer that the formula gave. This indicates that the formula works for convex and concave polygons. 10 a = 268, b = 139, c = a 144 b a (n 2) 180 = (7 2) 180 = = 900 b Each interior angle of a regular heptagon is =128.6 c Each side should be 4 cm and each angle Original Material Cambridge University Press

4 13 a b Exterior angles add up to 360, which is a full turn. Original Material Cambridge University Press

5 14 i a, b and c ii Exterior angles total 360 for each polygon. 15 a 40, 140 b 30, 150 c 72, Use the formula (n 2) 180 to find the sum of the interior angles of a hexagon: (n 2) 180 = (6 2) 180 = = 720 Then divide this answer by 6 to find each interior angle: Each interior angle = = 120 The exterior angle is = 60 Alternatively, use the fact that the sum of exterior angles = 360 and divide this answer by 6 to get exterior angle of regular hexagon: Each exterior angle = = 60 Interior angle = = a Equal sides, equal angles b The interior angle increases. c ii has the greatest number of sides d i 8 ii 20 iii a 24 b 30 c a Angle x is 45 because angles around a point total 360. b Angle y is 67.5 because an isosceles triangle has two equal angles: 2y = 180 x = = 135 ; y = 67.5 c z = 135 because the angle z = 2 y. 20 a True; five equal angles round a point Original Material Cambridge University Press

6 b x = 72 because angles round a point total 360 c The angles in an equilateral triangle are all 60 and x = 72 d An isosceles triangle has two equal angles: y = 1 ( ) = 54 2 e Angle z = 2 y so z = a a = 40 (corresponding angles are equal), b = 41 (angles in a triangle), c = 41 (corresponding angles are equal) b x = 140 (corresponding angles are equal), y = 140 (alternate angles or vertically opposite angles), z = 20 (angles in an isosceles triangle) c a = 55 (vertically opposite angles), c = 55 (vertically opposite angles), b = 80 (angles in a triangle) 22 a 360 b 120 c 60 d Yes e Corresponding angles 23 a v = 40 b w = 70 c x = 40 d u = 70 e z = 40 f y = 70 g They are both isosceles triangles. Original Material Cambridge University Press