Chapter 6 Angles, triangles and polygons This chapter will show you how to 4 describe a turn, recognise and measure angles of different types 4 discover angle properties 4 use bearings to describe directions 4 investigate angle properties of triangles, quadrilaterals and other polygons 4 understand line and rotational symmetry 6.1 Angles Turning If you turn all the way round in one direction, back to your starting position, you make a full turn . The minute hand of 12 12 a clock moves a 11 1 11 1 10 2 10 2 quarter turn from 1 Minute hand makes 4 turn. 9 3 9 3 1 3 to 6. Time taken: 4 hour. 8 4 8 4 7 5 7 5 6 6 The minute hand 12 12 11 1 11 1 of a clock moves a 10 2 10 2 1 half turn from Minute hand makes 2 turn. 9 3 9 3 1 4 to 10. Time taken: 2 hour. 8 4 8 4 7 5 7 5 6 6 The hands of a clock turn clockwise. This fairground ride turns anti-clockwise . 1 2 turn anti-clockwise 142 Space and shape M06_CMC_SB_IGCSE_6850_U06.indd 142 9/6/09 16:51:44 Angles, triangles and polygons EXERCISE 6A 1 Describe the turn the minute hand of a clock makes between these times. (a) 3 am and 3.30 am (b) 6.45 pm and 7 pm Look at the clock examples on page 142. (c) 2215 and 2300 (d) 0540 and 0710 2 Here is a diagram of a compass. N NW NE You are given a starting direction W E and a description of a turn. What is the finishing direction in SW SE each case? S Starting direction Description of turn 1 (a) N 4 turn clockwise 1 (b) SE 4 turn anti-clockwise 1 (c) SW 2 turn anti-clockwise 3 (d) E 4 turn anti-clockwise 1 (e) NE 4 turn clockwise 1 (f) W 2 turn clockwise 3 (g) NW 4 turn clockwise 1 (h) S 4 turn anti-clockwise Space and shape 143 M06_CMC_SB_IGCSE_6850_U06.indd 143 9/6/09 16:51:45 Describing angles An angle is a measure of turn. Angles 11 12 1 are usually measured in degrees. 10 2 A complete circle (or full turn) is 360°. 9 3 The minute hand of a clock turns 8 4 7 5 through 360° between 1400 (2 pm) 6 and 1500 (3 pm). You will need to recognise the following types of angles. acute An angle between 0° and 90°, less than a 1 4 turn. right angle An angle of 90°, 1 a 4 turn. A right angle is usually marked with this symbol. obtuse An angle between 90° and 180°, more than 1 4 turn but less than 1 2 turn. straight line An angle of 180°, 1 a 2 turn. reflex An angle between 180° and 360°, more 1 than 2 turn but less than a full turn. You can describe angles in three different ways. AB, BC, PQ and QR are called line A • ‘Trace’ the angle using capital letters. segments. Write a ‘hat’ symbol over the middle ` letter: ABC B C • Use an angle sign or write the word P The letter on the point of the angle always goes in the middle. ‘angle’: PQR or angle PQR Q R 144 Space and shape M06_CMC_SB_IGCSE_6850_U06.indd 144 9/6/09 16:51:45 Angles, triangles and polygons • Use a single letter. x EXAMPLE 1 State whether these angles are acute, right angle, obtuse or reflex. (a) (b) (c) (d) (e) The ‘box’ symbol shows the angle is 90°. (a) right angle More than a half turn. (b) reflex (c) acute Less than 90°. (d) obtuse More than 90° but less than 180°. (e) reflex Almost a full turn. EXERCISE 6B 1 State whether these angles are acute, right angle, obtuse or reflex. (a) (b) X Z (c) LM C A B N Y F Q W (d) D (e) (f) U V RP E Space and shape 145 M06_CMC_SB_IGCSE_6850_U06.indd 145 9/6/09 16:51:46 2 Describe each of the angles in question 1 using three For example ` letters. GHK GHK angle GHK Measuring angles You use a protractor to measure angles accurately. Follow these instructions carefully. 1 Estimate the angle first, so you don’t mistake an angle of 30°, say, for an angle of 150°. 2 Put the centre point of the protractor exactly on top of the point of the angle. 3 Place one of the 0° lines of the protractor directly on top of one of the angle ‘arms’. If the line isn’t long enough, draw it longer so that it reaches beyond the edge of the protractor. 4 Measure from the 0°, following the scale round the edge of the protractor. If you are measuring from the left-hand 0°, use the outside scale. If you are measuring from the right-hand 0°, use the inside scale. 5 On the correct scale, read the size of the angle in Use your estimate to help you degrees, where the other ‘arm’ cuts the edge of the choose the correct scale. protractor. C L 136° M N 72° B Angle NML � 136° Centre point A of protractor Angle ABC � 72° Centre point of protractor Measuring from the left-hand 0°. Measuring from the right-hand 0°. 6 To measure a reflex angle (an angle that is bigger than 180°), measure the acute or obtuse angle, and subtract this value from 360°. 146 Space and shape M06_CMC_SB_IGCSE_6850_U06.indd 146 9/6/09 16:51:46 Angles, triangles and polygons EXERCISE 6C 1 Measure these angles using a protractor. (a) (b) (c) (d) (e) (f) Space and shape 147 M06_CMC_SB_IGCSE_6850_U06.indd 147 9/6/09 16:51:47 2 Draw these angles using a protractor. Label the angle in For example each case. P (a) angle PQR 5 54° (b) angle STU 5 148° 54° (c) angle MLN 5 66° Q R (d) angle ZXY 5 157° (e) angle DFE 5 42° (f) angle HIJ 5 104° Angle properties You need to know these angle facts. Angles on a straight line add up to 180°. b c a d These angles lie on a straight line, so a 1 b 1 c 1 d 5 180°. Angles around a point add up to 360°. q r p s t These angles make a full turn, so p 1 q 1 r 1 s 1 t 5 360°. Vertically opposite angles are equal. u h k v In this diagram h 5 k and u 5 v. Equal angles are shown by matching arcs. Perpendicular lines intersect at 90° and are marked with the right angle symbol. 148 Space and shape M06_CMC_SB_IGCSE_6850_U06.indd 148 9/6/09 16:51:47 Angles, triangles and polygons EXAMPLE 2 Calculate the size of the angles marked with letters. (a) (b) 100° 40° p 132° 73° q 15° (c) (d) 130° s r t 35° 122° (a) p 1 100° 1 132 5 360° (angles around a point) Solve this equation to find p. p 5 360° 2 100° 2 132° Use Chapter 5 to help you. p 5 128° (b) q 1 73° 1 40° 1 15° 5 180° (angles on a straight line) q 5 180° 2 73° 2 40° 2 15° q 5 52° (c) r 5 130° (vertically opposite) s 1 130° 5 180° (angles on a straight line) s 5 180° 2 130° 5 50° (d) t 1 122° 1 35° 1 90° 5 360° (angles around a point) t 5 360° 2 122° 2 35° 2 90° means 90°. t 5 113° EXERCISE 6D Calculate the size of the angles marked with letters. 1 2 3 300° a 150° b 60° c Space and shape 149 M06_CMC_SB_IGCSE_6850_U06.indd 149 9/6/09 16:51:49 4 d 5 e 6 95° 45° 45° f 7128 9 3 115° i a k l 64° 135° 73° 69° 65° 34° h 80° 157° 121210 11 3123 a a 64° k k l l 64°64° 73°73° 69°69° 34°34° j 80°80°157157° ° 134514 15 6 29° n 58° o 52° 66° 42° n p 57° m 33° 45° 108° 7816 17 18 9 q r 2r 3s q 2s 4s s 150 Space and shape M06_CMC_SB_IGCSE_6850_U06.indd 150 9/6/09 16:51:51 Angles, triangles and polygons Angles in parallel lines Parallel lines are the same distance apart all along their length. You can use arrows to show lines are parallel. A straight line that crosses a pair of parallel lines is called a transversal. A transversal creates pairs of equal angles. a e c g h b f d a 5 b c 5 d a and b are corresponding angles The lines make an F shape. c and d j a e c g h b f d e 5 f g 5 h e and f are alternate angles. The lines make a Z shape. g and h j In this diagram the two angles are not equal, j is obtuse and k is acute. The two angles lie on the inside of a pair of parallel lines. k They are called co-interior angles or allied angles. j Co-interior angles add up to 180°. k This angle � k j (corresponding angles) j 1 k 5 180° j � k � 180° (angles on a straight line). k Corresponding angles are equal. Alternate angles are equal. Co-interior angles add up to 180°.