6Measurement and. geometry

Transcription

1 6Measurement and geometry Geometry Geometry comes from the Greek word geometria, which means land measuring. The principles and ideas of geometry are evident everywhere in road signs, buildings, bridges and patterns for tiles and wallpaper. Many examples of geometric patterns can be seen in nature, such as the hexagonal cells on a honeycomb built by bees to store honey.

2 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Shutterstock.com/Nikolay Dimitrov ecobo n Chapter outline Proficiency strands 6-01 Triangle geometry U F R C 6-02 Quadrilateral geometry U F R C 6-03 Angle sum of a polygon U F R C 6-04 Exterior angle sum of a convex polygon U F R C n Wordbank angle sum The total of the sizes of the angles in a shape, such as a triangle bisect To cut in half convex polygon A polygon whose vertices all point outwards diagonal An interval joining two non-adjacent vertices of a shape exterior angle of a triangle An outside angle of a triangle formed after extending one of the sides of the triangle polygon A plane shape with straight sides regular polygon A polygon with all angles equal and all sides equal, such as a square

3 Chapter Geometry n In this chapter you will: classify triangles and quadrilaterals according to their side and angle properties, and solve related numerical problems using reasoning solve problems involving the angle sum of a triangle and quadrilateral and the exterior angle of a triangle solve problems involving the angle sum of a polygon and the exterior angle sum of a convex polygon SkillCheck Worksheet StartUp assignment 6 MAT09MGWK Classify each angle as acute, obtuse, reflex, right, straight or a revolution. a b c d Skillsheet Types of angles MAT09MGSS10018 Puzzle sheet Angles MAT09MGPS00050 e f g h Skillsheet Angles and parallel lines MAT09MGSS Find the value of each pronumeral, giving reasons. a b c r r 140 w c y d k e f 38 k t 110 t a a

4 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 g a 47 h 45 k i 115 c b h n j d c 76 b k b k 125 l y 58 x a n m 3 Test whether each labelled pair of lines are parallel. Give reasons for each answer. a b c A C F I E H 78 K G 75 J L B D 6-01 Triangle geometry Angle sum of a triangle Homework sheet Geometry 1 MAT09MGHS10023 Summary The angle sum of a triangle is 180. a a þ b þ c ¼ 180 b c

5 Chapter Geometry Proof: Consider any triangle PQR with angles a, b and c. Construct a line through P parallel to QR. [ \UPQ ¼ b (alternate angles, UV QR) and \VPR ¼ c (alternate angles, UV QR) But \UPQ þ \QPR þ \VPR ¼ 180 (angles on a straight line) U Q b a P c R V [ a þ b þ c ¼ 180 [ The sum of the angles of a triangle is 180. Example 1 Find the value of each pronumeral, giving reasons. a b P k Q R Solution a k þ 55 þ 42 ¼ 180 (angle sum of a triangle) k ¼ ¼ 83 b \ R ¼ \Q ¼ (npqr is isosceles) m þ m þ 52 ¼ 180 2m þ 52 ¼ 180 2m ¼ 128 m ¼ 64 (angle sum of a triangle) The exterior angle of a triangle Summary The exterior angle of a triangle is equal to the sum of the two interior opposite angles. y z ¼ x þ y x z

6 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Example 2 Find the value of k in each diagram. a b k 110 k Solution a k ¼ 42 þ 59 ðexterior angle of triangleþ ¼ 101 b k þ 50 ¼ 110 ðexterior angle of triangleþ k ¼ 60 Example 3 Find the size of the marked angle \BDF. Solution \DBC ¼ \ABE ¼ 75 ðvertically opposite anglesþ \BDF ¼ 75 þ 55 ðexterior angle of 4BCDÞ ¼ 130 A 75 E B D 55 C F Exercise 6-01 Triangle geometry 1 Find the value of each pronumeral. a b c w See Example h d k 38 e 81 f x a

7 Chapter Geometry g h i k d 55 y j k l h 15 r Worked solutions Triangle geometry MAT09MGWS10029 See Example 2 2 What is the size of each angle in an equilateral triangle? Select A, B, C or D. A 30 B 45 C 60 D 90 3 One angle of an isosceles triangle is equal to 80. Which two of the following could be the sizes of the other angles? Select two of A, B, C or D. A 80 and 20 B 40 and 60 C 40 and 40 D 50 and 50 4 Find the value of m. a b c d e f

8 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 5 What is the size of the unknown angles in this triangle? Select A, B, C or D. A 30 B 45 C 60 D 90?? 6 One angle of an obtuse-angled triangle is 50. Which of the following could be the sizes of the other angles? Select A, B, C or D. A 80 and 50 B 100 and 30 C 65 and 65 D 60 and 70 7 Find the size of \CBF in each diagram. See Example 3 a D 80 C E b D C E c 53 B A 45 B F A B F C 60 F E d A e C B f E 110 C B 120 F A 8 An isosceles triangle has a side length of 6 cm and one of its angles equal to 40. Draw all possible shapes of the triangle. 9 The diagram shows the shape of a roof truss. Find the value of each pronumeral. 50 F E D 34 c a b C F B Worked solutions Triangle geometry MAT09MGWS Copy this diagram and mark all angles equal to the angle marked d. d

9 Chapter Geometry 11 Find the value of each pronumeral. a b c 50 h 2 (2t + 1) h d e f 3g x 30 y Skillsheet Starting GeoGebra MAT09MGSS10019 Skillsheet Starting Geometer s Sketchpad Technology Sketching parallelograms and rectangles In this activity you will use GeoGebra to construct two quadrilaterals. Constructing a parallelogram 1 To construct a parallelogram, use Interval between two points and construct two sides of any length (as shown below). 2 Label the points (right-click on each point and Show label). MAT09MGSS

10 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 3 Select Parallel line from the fourth drop-down icon menu and choose side BC and point A. 4 Repeat step 2 with side AB and point C. Now select Intersect Two Objects from the second drop-down icon menu and the lines through points A and C. Label the point D. 5 Use Angle to measure the size of \DAB and \ABC. Manipulate the vertices to change the size of the parallelogram

11 Chapter Geometry Constructing a rectangle. 1 Make sure the axes and grid are removed by right-clicking and disabling them. 2 Click Enter a point and 6 (cm). Label the interval endpoints A and B. 3 Click Select point A and 6 cm interval. Repeat for point B. 4 Click New point to insert a point anywhere on the line under point B. Label it point C. 5 Construct a perpendicular line through C. 6 Click Intersect two objects and create a point of intersection at the perpendicular through A and the perpendicular through C. Label the point of intersection, D. 7 Select the Move tool and click point C. Manipulate the line through CD. 8 Click Distance or length and choose points B and C. Use the Move tool to manipulate point C until BC ¼ 3.6 cm. Click Distance or length and points A and B to show the length of AB. 9 Now draw at least two other types of quadrilaterals

12 NEW CENTURY MATHS ADVANCED for the Australian Curriculum Quadrilateral geometry A quadrilateral is any shape with four straight sides. A quadrilateral may be either convex or non-convex (concave). Convex quadrilateral Non-convex quadrilateral Worksheet Classifying quadrilaterals MAT09MGWS00066 Worksheet Properties of triangles and quadrilaterals MAT09MGWS10066 All vertices (corners) point outwards. All diagonals lie within the shape. All angles are less than 180. There are six special types of quadrilaterals. One vertex points inwards. One diagonal lies outside the shape. One angle is more than 180 (reflex angle). Worksheet Naming quadrilaterals MAT09MGWS10067 Worksheet Deductive geometry MAT09MGWS10068 Skillsheet Trapezium Parallelogram Rectangle Naming shapes MAT09MGSS10022 Homework sheet Geometry 2 One pair of parallel sides Two pairs of parallel sides Four right angles Rhombus Square Kite MAT09MGHS10024 Technology GeoGebra: Angle sum of a quadrilateral Four equal sides Four equal sides and four right angles Two pairs of equal adjacent sides Adjacent means next to each other. MAT09MGTC00010 Animated example Angles and shapes Angle sum of a quadrilateral MAT09MGAE00011 Worksheet Summary Diagonal properties of quadrilaterals The angle sum of a quadrilateral is 360. a þ b þ c þ d ¼ 360 This property is true for both convex and non-convex quadrilaterals. d a b c MAT09MGWS00067 Worksheet Shapes and angles review MAT09MGWS

13 Chapter Geometry Properties of quadrilaterals Summary Trapezium One pair of opposite sides parallel No axes of symmetry Kite Two pairs of equal adjacent sides One pair of opposite angles equal One axis of symmetry Diagonals intersect at right angles Parallelogram Opposite sides are parallel and equal Opposite angles are equal No axes of symmetry Diagonals bisect each other Rhombus All sides are equal Two axes of symmetry A special type of parallelogram Diagonals bisect each other at right angles Diagonals bisect the angles of the rhombus Rectangle All angles are 90 (right angles) Two axes of symmetry A special type of parallelogram Diagonals are equal (in length) Diagonals bisect each other Square All sides are equal All angles are 90 (right angles) Four axes of symmetry A special type of rhombus and rectangle Diagonals are equal Diagonals bisect each other at right angles Diagonals bisect the angles of the square

14 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Example 4 Find the value of each pronumeral. a b B d C Puzzle sheet Mixed angle problems MAT09MGPS00053 Puzzle sheet 78 Solution a b m þ 78 þ 125 þ 93 ¼ 360 (angle sum of a quadrilateral) A m ¼ ¼ 64 \CDA ¼ (angles on a straight line) ¼ 105 d ¼ 105 (opposite angles of a parallelogram) Can you see another method for finding d? D 75 E Parallel lines MAT09MGPS00051 Puzzle sheet Triangles and quadrilaterals MAT09MGPS00052 Example 5 Find the size of \BED. E 85 D A B C Solution \ABE ¼ 70 (equal angles in isosceles nabe) \EBC ¼ (angles on a straight line) ¼ 110 \BED ¼ (angle sum of quadrilateral BCDE) ¼ 55 Can you see another method for finding \BED?

15 Chapter Geometry Exercise 6-02 Quadrilateral geometry See Example 4 1 Find the value of m in each diagram. a b c d e f Copy and complete the table below. Property Trapezium Kite Parallelogram Rhombus Rectangle Square One pair of opposite sides parallel Opposite sides parallel Opposite sides ü equal All sides equal Two pairs of adjacent sides equal Diagonals equal ü Diagonals bisect each other Diagonals meet at right angles Diagonals bisect the angles of the shape Opposite angles equal One pair of opposite angles equal All angles 90 Axes of symmetry

16 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 3 Refer to the table you completed in question 2, and name all quadrilaterals that have: a no axes of symmetry b one pair of parallel sides c four equal sides d equal diagonals e opposite sides equal f four axes of symmetry g all adjacent sides equal h one axis of symmetry i opposite sides parallel j all angles measuring 90 k two axes of symmetry l diagonals which bisect each other m opposite angles equal n diagonals meeting at 90 4 Each triangle in this diagram is an equilateral triangle. a Name the different types of quadrilaterals you can find in the diagram. b How many of each type of quadrilateral are there in the diagram? c How many triangles are there? (It may help to copy the diagram so that you can draw on it using coloured pencils.) 5 Which statements are always true? A A rhombus is a parallelogram. B The diagonals of a parallelogram meet at right angles. C A square is a rhombus. D A parallelogram is a quadrilateral with a pair of opposite sides equal and parallel. E A square is a rectangle. F The diagonals of an isosceles trapezium bisect each other. G The opposite angles of a rhombus are equal. H The diagonals of a rhombus are equal and bisect each other at right angles. I A rectangle is a parallelogram. 6 Name each quadrilateral using the properties marked on it. a b c d e f g h i

17 Chapter Geometry 7 Find the value of each pronumeral, giving reasons. a b c k a p d e f w 35 t n g 70 h i r 110 2a 5c See Example 5 Worked solutions Quadrilateral geometry MAT09MGWS Find the size of \PQR in each diagram, giving reasons. a b c R W Q T R 74 Q T 110 P Q V W P U P 30 R d S T e f P 35 Q 55 R 105 R 110 R C D T 60 P Q P Q

18 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 9 Use geometrical instruments or geometry software such as GeoGebra to construct each quadrilateral described. a Parallelogram TUVW with sides TU ¼ 30 mm, UV ¼ 60 mm and \TUV ¼ 113. b Rhombus with side length 5 cm. c Convex quadrilateral WXYZ with WX ¼ 65 mm, WZ ¼ 40 mm, \ZWX ¼ 54, \WZY ¼ 114 and XY ¼ 24 mm. d Non-convex quadrilateral with one side 4.5 cm and one angle 200. e 6 cm D E cm G 50 F Just for the record Geometry in art Geometry has been used in artwork for centuries. Tesselations can be used to create artwork, such as the regular hexagons used in the tessellation shown below. Indian and Islamic artworks show incredible intricacy, detail and colour within the geometric images used. Shutterstock.com/Lenar Musin

19 Chapter Geometry Fractals use geometric formulas to create infinitelyoccurring images, so they are usually created using computer software. A famous fractal is the Mandelbrot set, shown here. 1 Design a tessellation or find examples of geometry in art. 2 Investigate fractals such the Mandelbrot set and the Koch snowflake, and their history. Shutterstock.com/Andrew Park Investigation: Angle sum of a polygon To find the angle sum of a pentagon ABCDE (5 sides), follow this reasoning. From one vertex of the pentagon, draw all the diagonals. The diagonals from vertex A have divided the pentagon into three triangles. ) Angle sum of a pentagon ¼ 3 3 angle sum of the 3 triangles ¼ ¼ a Draw a hexagon (6 sides) and from one vertex draw all the diagonals. b How many diagonals are there? c How many triangles did you form? d Hence find the angle sum of a hexagon. 2 Repeat the procedure to find the angle sum of: a an octagon (8 sides) b a decagon (10 sides) 3 Copy and complete this table. A E D B C Polygon Number of sides Number of triangles Angle sum triangle quadrilateral 4 2 pentagon 5 hexagon octagon decagon 4 Copy and complete this pattern: The number of triangles formed is always two than the number of of the polygon. 5 a Using your own words, describe the rule for finding the angle sum of a polygon. b What is the angle sum of a polygon with 20 sides? c For a polygon with n sides, write a formula for the sum of its angles. Discuss your result with other students

20 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 6 a Draw a non-convex octagon. b Divide the polygon into triangles as shown. c How many triangles have been formed? d Find the angle sum of this non-convex octagon. e Does your rule for the angle sum of a polygon also apply to the non-convex polygon? 6-03 Angle sum of a polygon A polygon is a general name for any shape with straight sides. The word is derived from the Greek, meaning many angles. Shapes with curved sides, such as circles, ellipses and semicircles, are not polygons. A polygon may be either convex or non-convex (concave). NSW Worksheet Angle sum of a polygon MAT09MGWK10069 Worksheet Find the unknown angle MAT09MGWK10070 Technology worksheet Angle sum of a polygon MAT09MGCT10002 Convex polygon In a convex polygon, all vertices point outwards, all diagonals lie within the shape and all angles are less than 180. In a non-convex polygon, some vertices point inwards, some diagonals lie outside the shape and some angles are more than 180 (reflex angles). A polygon s name is determined by the number of sides it has. The images below show the Pentagon building in the USA, a 50p coin from the UK, and a stop sign. Non-convex polygons Name Number of sides Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Shutterstock.com/Frontpage Shutterstock.com/Popartic Shutterstock.com/Stephen Rees

21 Chapter Geometry Summary The angle sum of a polygon with n sides is given by the formula A ¼ 180(n 2). This formula applies to both convex and non-convex polygons. Example 6 Find the angle sum of a nonagon (9 sides). Solution Angle sum ¼ 180ð9 2Þ ¼ð Þ ¼ 1260 Example 7 Find the number of sides of a polygon that has an angle sum of 720. Solution 180ðn 2Þ ¼ n 360 ¼ n ¼ 1080 n ¼ ¼ 6 [ The polygon has 6 sides (hexagon). Worksheet Equal angles MAT09MGWK00063 Regular polygons A regular polygon has all angles equal and all sides equal. For example, a regular pentagon has 5 equal sides and 5 equal angles. A square is a regular polygon but a rhombus is not. Summary The size of each angle in a regular polygon with n sides ¼ Angle sum No. of sides 180 n 2 ¼ ð n Þ

22 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Example 8 Find the size of one angle in a regular hexagon. Solution A hexagon has six sides (n ¼ 6) Size of one angle ¼ ð 6 ¼ ð Þ 6 ¼ 120 Each angle in a regular hexagon is 120. Þ Exercise 6-03 Angle sum of a polygon 1 How many sides has: a a hexagon? b a quadrilateral? c a nonagon? d a decagon? e a heptagon? f a pentagon? g a dodecagon? h an octagon? i an undecagon? 2 Name each polygon. a b c d e f g h 3 Which polygons from question 2 are: a convex? b regular?

23 Chapter Geometry 4 Which shape is a regular octagon? Select A, B, C or D. A B C D See Example 6 Worked solutions Angle sum of a polygon MAT09MGWS What is the more common name for: a a regular triangle? b a regular quadrilateral? 6 Find the angle sum of a polygon with: a 5 sides b 8 sides c 15 sides d 12 sides e 7 sides f 10 sides g 20 sides h 11 sides 7 Find the value of each pronumeral. a b c b d b d d a d d d 120 c 140 e 96 f x g 116 x c g e e e e e e 50 e e h n i y

24 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 8 Find the number of sides of a polygon that has an angle sum of: a 2160 b 1620 c 3960 d Which polygon has an angle sum of 1440? Select A, B, C or D. A pentagon B decagon C nonagon D octagon 10 Find the size of one angle in a regular a octagon b decagon c dodecagon d hexagon 11 The angle sum of a regular polygon is a How many sides does the polygon have? b Find the size of each angle. 12 How many sides does a regular polygon have if each of its angles is: a 165? b 170? c 144? See Example 7 See Example 8 Investigation: Exterior angle sum of a convex polygon 1 a Draw a triangle and extend each side to show the exterior angles of the triangle as shown. Exterior angles are x, y and z. Interior angles are a, b and c. b Use a protractor to measure angles x, y and z. c Find the sum of the exterior angles. d Looking at the diagram, what must be the value of a þ x þ b þ y þ c þ z? e But what do we know about the value of a þ b þ c? f Therefore, what must be the value of x þ y þ z? x a z c b y 2 Repeat this procedure for the exterior angles of a convex quadrilateral. What is the sum of the exterior angles of a convex quadrilateral? 3 Repeat the procedure for a convex pentagon and a convex hexagon. What do you notice about the sum of the exterior angles of those polygons? 4 a Draw any convex polygon and extend the sides. Label the vertices of your polygon A, B, C, etc. b Start at A and move around the polygon, turning in the direction indicated at each vertex. c Continue until you return to A, facing the same way you started. What must be the sum of the turns in any round trip of a convex polygon? d Test whether this rule works for a non-convex polygon. B A C E D

25 Chapter Geometry NSW 6-04 Exterior angle sum of a convex polygon Summary The sum of the exterior angles of a convex polygon is 360. C B D A E Example 9 For a regular hexagon, find the size of: a each exterior angle b each (interior) angle. Solution a Sum of exterior angles ¼ 360 One exterior angle ¼ ¼ 60 b Each angle ¼ ðangles on a straight lineþ ¼ OR: Each angle ¼ ð Þ 6 ¼

26 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Example 10 Find the number of sides of a regular polygon if: a each exterior angle is 12 b each (interior) angle is 160. Solution a Sum of exterior angles ¼ 360 Number of exterior angles ¼ ¼ 30 [ The regular polygon has 30 sides. b Exterior angle ¼ ðangles on a straight lineþ ¼ 20 Sum of exterior angles ¼ 360 Number of exterior angles ¼ ¼ 18 [ The regular polygon has 18 sides. OR: 180ðn 2Þ ¼ 160 n 180ðn 2Þ ¼160n 180n 360 ¼ 160n 20n 360 ¼ 0 20n ¼ 360 n ¼ ¼ 18 [ The regular polygon has 18 sides. Exercise 6-04 Exterior angle sum of a convex polygon 1 Find the size of each exterior angle of a regular: a octagon b decagon c 15-sided polygon d nonagon 2 Find the size of each angle of a regular: a pentagon b dodecagon c nonagon d 16-sided polygon 3 Find the number of sides of a regular polygon if each exterior angle is: a 24 b 36 c 40 d 10 e 18 f 60 4 Find the number of sides of a regular polygon if each angle is: a 150 b 175 c 162 d 140 e 108 f 176 See Example 9 See Example

27 Chapter Geometry Mental skills 6 Maths without calculators Dividing decimals To divide one decimal by another, first move the decimal points in both decimals the same number of places to the right so that the second decimal is a whole number. 1 Study each example. a = 24 6 = 4 c = = b = = 0.9 d = = 30 e = 16 4 = 4 f = = 80 2 Now evaluate each quotient. a b c d e f g h i j k l Study each example. Given that ¼ 8, evaluate each expression. a = = = = = 8 10 = 80 Estimate: = 112 b = = = = = 8 10 = 0.8 Estimate: = 1 c = = = = = 800 d = = = = 0.08 Estimate: = Estimate: = Now given that ¼ 16, evaluate each quotient. a b c d e f g h i j k l

28 NEW CENTURY MATHS ADVANCED for the Australian Curriculum9 Power plus 1 How many diagonals has: a a quadrilateral? b an octagon? c a dodecagon? 2 a By drawing polygons with 3 to 10 sides and counting diagonals, copy and complete this table. No. of sides, n No. of diagonals, d b Find a formula for the number of diagonals, d, in a polygon with n sides. 3 Name all quadrilaterals whose diagonals: a bisect each other at right angles b bisect each other c intersect at right angles d have equal length e bisect the angles of the quadrilateral f are equal and bisect each other 4 Find the value of each pronumeral, giving reasons. a b c 83 h k p w 113 e d e f 40 y f x g n 34 v h 25 i a c t 117 d

29 Chapter 6 review n Language of maths Puzzle sheet Geometry crossword MAT09MGPS10071 Quiz Shapes and angles MAT09MGQZ00011 adjacent angle sum bisect co-interior convex corresponding equilateral exterior angle interior isosceles kite parallel parallelogram polygon quadrilateral rectangle regular polygon rhombus right angle square supplementary trapezium vertex vertically opposite 1 What shape has two pairs of equal adjacent sides? 2 Name one property of a convex quadrilateral. 3 What type of angle is created when one side of a triangle is extended? 4 What is the sum of the exterior angles of any polygon? 5 What is a regular polygon? Are all regular polygons also convex? 6 Copy and complete: The exterior angle of a triangle is equal to the of the angles. n Topic overview Worksheet Geometry summary poster MAT09MGWK10072 Worksheet Mind map: Geometry (Advanced) Write three ideas from this topic that were new to you. Summarise what you know about the angle sum of a triangle, quadrilateral and polygon. Name the three types of triangles, choose one and list all of its properties. Name the six special quadrilaterals, choose one and list all of its properties. Copy (or print) and complete this mind map of the topic, adding detail to its branches and using pictures, symbols and colour where needed. Ask your teacher to check your work. MAT09MGWK10074 Triangle geometry GEOMETRY Polygons Quadrilateral geometry

30 Chapter 6 revision 1 Find the value of each pronumeral, giving reasons. See Exercise 6-01 a 42 b c x 72 k e 127 d e f 3 35 b 2 5 t g h i y 87 y y What are the sizes of the angles in: a an equilateral triangle? b a right-angled isosceles triangle? 3 Find the value of each pronumeral, giving reasons. a x b c y 82 d e 56 f 2e 35 g See Exercise 6-01 See Exercise 6-02 x y 110 3e 4 Name all quadrilaterals that have: a both pairs of opposite sides parallel b two equal diagonals c all sides equal d diagonals that bisect each other. 5 Draw each pentagon. a a regular pentagon b a non-regular pentagon c a non-convex pentagon 6 a Show that the angle sum of a decagon is b Find the size of one angle in a regular nonagon. 7 a Find the size of each exterior angle in a regular hexagon. b If each angle in a regular polygon is 150, how many sides does it have? See Exercise 6-02 See Exercise 6-03 See Exercise 6-03 See Exercise