<p> Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p>GM1.1 Answers</p><p>1 a</p><p> b</p><p>2</p><p>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  </p><p>Original Material © Cambridge University Press 2010 1 Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p>3 Check that students have drawn a regular, an irregular convex and a concave pentagon. </p><p>Here are some examples:</p><p>4 a Sides must all be of equal length and the interior angles must be equal.</p><p> b A regular quadrilateral is a square.</p><p> c </p><p>5 a The sum of the interior angles increases by 180° each time.</p><p> b</p><p>Number of sides 3 4 5 6 7 8 Sum of interior angles 180 360 540 720 900 1080</p><p> c i 160° ii 115° iii 155° iv 120° v 141° iv 97°</p><p>6 a A regular pentagon has five equal angles. The sum of these angles is 540°. Divide this </p><p> number</p><p> by 5. This gives 108° for each interior angle.</p><p> b The interior angle of a regular octagon = 135°</p><p> c The interior angle of a dodecagon = 150° </p><p>Original Material © Cambridge University Press 2010 2 Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p>7 The missing angle = 900° – 748° = 152°</p><p>8 a (n – 2) × 180° works out the sum of the interior angles of a polygon with n sides.</p><p> b i (5 – 2) × 180° = 540°; x = 540° – 395 =145° and y = 180° – 145 = 35°</p><p> ii (8 – 2) × 180° = 1080°; x = 1080° – 937° = 143° and y = 180° – 143 = 37°</p><p> iii (4 – 2) × 180° = 360°; x = 360° – 275° = 85° and y = 180° – 85° = 95°</p><p>9 The formula says that the sum of the interior angles of a quadrilateral add to 360°. </p><p>Alternatively, divide the shape into two triangles.</p><p>As you can see, adding up the angles in each triangle is the same as adding up the four </p><p> interior angles of the quadrilateral. Therefore, the sum of the interior angles of the concave</p><p> quadrilateral is 2 × 180° = 360°. This is the same as the answer that the formula gave. This</p><p> indicates that the formula works for convex and concave polygons.</p><p>10 a = 268°, b = 139°, c = 253°</p><p>11 a 144° b 157.5°</p><p>12 a (n – 2) × 180° = (7 – 2) × 180° = 5 × 180° = 900° </p><p> b Each interior angle of a regular heptagon is 900° ÷ 7 =128.6°</p><p> c Each side should be 4 cm and each angle 128.6.</p><p>Original Material © Cambridge University Press 2010 3 Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p>13 a</p><p> b Exterior angles add up to 360°, which is a full turn. </p><p>Original Material © Cambridge University Press 2010 4 Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p>14 i a, b and c</p><p> ii Exterior angles total 360° for each polygon.</p><p>15 a 40°, 140° b 30°, 150° c 72°, 108°</p><p>16 Use the formula (n – 2) × 180° to find the sum of the interior angles of a hexagon:</p><p>(n – 2) × 180° = (6 – 2) × 180° = 4 × 180° = 720°</p><p>Then divide this answer by 6 to find each interior angle: </p><p>Each interior angle = 720° ÷ 6 = 120°</p><p>The exterior angle is 180° – 120° = 60°</p><p>Alternatively, use the fact that the sum of exterior angles = 360° and divide this answer by </p><p>6 to get exterior angle of regular hexagon:</p><p>Each exterior angle = 360° ÷ 6 = 60°</p><p>Interior angle = 180° – 60° = 120°</p><p>17 a Equal sides, equal angles b The interior angle increases. c ii has the greatest number of sides d i 8 ii 20 iii 15</p><p>18 a 24 b 30 c 40</p><p>19 a Angle x is 45° because angles around a point total 360°.</p><p> b Angle y is 67.5° because an isosceles triangle has two equal angles: </p><p>2y = 180 – x = 180 – 45 = 135; y = 67.5</p><p> c z = 135° because the angle z = 2 × y.</p><p>20 a True; five equal angles round a point</p><p>Original Material © Cambridge University Press 2010 5 Cambridge Essentials Mathematics Core 9 GM1.1 Answers</p><p> b x = 72° because angles round a point total 360°</p><p> c The angles in an equilateral triangle are all 60° and x = 72 </p><p>1 d An isosceles triangle has two equal angles: y = (180 – 72) = 54 2</p><p> e Angle z = 2 × y so z = 108° </p><p>21 a a = 40° (corresponding angles are equal), b = 41° (angles in a triangle), </p><p> c = 41° (corresponding angles are equal)</p><p> b x = 140° (corresponding angles are equal), y = 140° (alternate angles or vertically </p><p> opposite angles), z = 20° (angles in an isosceles triangle)</p><p> c a = 55° (vertically opposite angles), c = 55° (vertically opposite angles), </p><p> b = 80° (angles in a triangle)</p><p>22 a 360° b 120 c 60° d Yes</p><p> e Corresponding angles</p><p>23 a v = 40° b w = 70° c x = 40° d u = 70°</p><p> e z = 40° f y = 70° g They are both isosceles triangles.</p><p>Original Material © Cambridge University Press 2010 6</p>