Angles, Triangles and Polygons

Chapter 6

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 .

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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

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• Use a single letter.

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

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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°.

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EXERCISE 6C

1 Measure these angles using a protractor.

(a) (b)

(c)

(d)

(e)

(f)

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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.

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EXAMPLE 2

Calculate the size of the angles marked with letters. (a) (b)

100° 40° p 132° 73° q 15°

(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°

(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

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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

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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°.

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M06_CMC_SB_IGCSE_6850_U06.indd 151 9/6/09 16:51:52 EXAMPLE 3

Calculate the size of the angles marked with letters.

(a) (b) y w x 65°

u 126° 100° v

(a) Method A You could use method A or u 5 65° (alternate) method B. v 5 65° (vertically opposite) Method B v 5 65° (corresponding) u 5 65° (vertically opposite) (b) w = 100° (corresponding) x 1 126° 5 180° (co-interior angles) x 5 180° 2 126° x 5 54° y 5 54° (vertically opposite)

EXERCISE 6E

Calculate the size of the angles marked with letters. 1 2 3 62° c b

a 115° 109°

4 5 6 e 75° 63°

d h g 110° f

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7 8 9 i r q 74° p m l j n 127° 105° k 82°

6.2 Three-figure bearings You can use compass points to describe a direction, but in Mathematics we use three-figurebearings .

A three-figure bearing gives a direction in degrees. It is an angle between 0° and 360°. It is always measured N from the north in a clockwise direction. N The diagram shows the cities of Marseille and Rome. 110° The bearing of Rome from Marseille is 110°. Marseille The bearing of Marseille from Rome is 290°. Rome A bearing must always be written with three figures. 290°

A bearing of 073° You write 073° because the bearing N must have three figures.

73°

A bearing of 254°

254°

You need to be able to measure bearings accurately using a Your answer needs to be within 1° protractor. of the correct value.

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M06_CMC_SB_IGCSE_6850_U06.indd 153 9/6/09 16:51:54 EXAMPLE 4

(a) Write down the bearing of N Q from P. (b) Work out the bearing of P from Q. 80° Q P

If you stand at P and face north, (a) 080° you need to turn through 80° to face Q. (b) N N Draw in the north line at Q.

This angle is the bearing of P from x Q. 80° Q P The two north lines are parallel, so 80° 1 x 5 180° (co-interior angles) 80° 2 80° 1 x 5 180° 2 80° x 5 100° Bearing of P from Q 1 x 5 360° (angles around a point) Bearing of P from Q 1 100° 5 360° Bearing of P from Q 5 260°.

EXERCISE 6F

1 For each diagram, write down the bearing of Q from P. (a) N (b) N (c) N Q

44° 63° Q P P P

(a) N(a) N (a) N (d)(b) N (b) N (b) N (c) (c)(e) N(c) N (d) N Q(d) QN(d)(f) QN N

Q Q Q

65° 65° 65° 115° 115° 115° P P P P P P P P P P P P 225° 225° 225° 325° 325° 325°

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2 For each diagram work out the bearing of B from A. (a) N (b) N (c) N (d) N B B 212° A A 135° A

A 28° 314° B B

N N B N N (e) (f) (g) (h) B

A 63° B A 49° 48° A A B 200°

3 Draw accurate diagrams to show these three-figure bearings. (a) 036° (b) 145° (c) 230° (d) 308° (e) 074° (f) 256° (g) 348° (h) 115° 4 The diagram shows the peaks, P, L and M, of three mountains in the Himalayas.

P Use your protractor to find these bearings. (a) L from P (b) M from P (c) M from L

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M06_CMC_SB_IGCSE_6850_U06.indd 155 9/6/09 16:51:58 5 In each diagram work out the bearing of X from Y.

N N N (a) N (b) (c)

N N

136° X Y X 254° 76° X Y Y (d) N (e) N (f) N

N 34° N N Y X X 72° X 149° Y Y 6 In each case • Sketch a diagram to show the bearing of D from C • Work out the bearing of C from D Bearing of D from C (a) 037° (b) 205° (c) 167° (d) 296°

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6.3 Triangles Types of triangle There are four types of triangle. They can be described by their properties. Triangle Picture Properties scalene The three sides are different lengths. The three angles are different sizes. isosceles A Two equal sides. AB 5 AC Two equal angles The marks across the sides of the (‘base angles’) triangles show which sides are equal and which sides are different. BCangle ABC 5 angle ACB. equilateral All three sides are 60° equal in length. All angles 60°. 60° 60°

right-angled X One of the angles is a right angle (90°). Y Angle XZY 5 90°. Z

Angles in a triangle 1 Draw a triangle on a piece of paper. 2 Mark each angle with a different letter or shade them different colours. 3 Tear off each corner and place them next to each other on a straight line.

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M06_CMC_SB_IGCSE_6850_U06.indd 157 9/6/09 16:52:01 You will see that the three angles fit exactly onto the You are not proving this result, you straight line. are simply showing that it is true So the three angles add up to 180°. for the triangle you drew. You can prove this for all triangles using facts about See page 151. alternate and corresponding angles.

B b D

x acy A E C

For the triangle ABC: 1 Extend side AC to point E. 2 From C draw a line CD parallel to AB. Let BCD 5 x and DCE 5 y. x 5 b (alternate angles) y 5 a (corresponding angles) c 1 x 1 y 5 180° (angles on a straight line at point C ) which means that c 1 b 1 a 5 180°.

The sum of the angles of a triangle is 180°.

EXAMPLE 5

Calculate the size of the angles marked with letters. (a) (b) (c) f 63° d

145° 48° a b c 70° e

n (a) a 5 180° 2 63° 2 48° means ‘triangle’ 63° (angle sum of n) a 5 69° 48° a b b 5 180° 2 69° (angles on a straight line) b 5 111°

Continued...

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(b) c 5 180° 2 145° d (angles on a straight line) c 5 35° 145° c d 5 180° 2 90° 2 35° (angle sum of n) d 5 55°

EXERCISE 6G

Calculate the size of the angles marked with letters.

1 2 3 j i b c 60° 53° 30° 37° d 42° 76° a 34° 66° q 50° e p

4 5 6 53° 60° f g 30° a 50° e 52° 134° h p 7 8 9 j i 65° m c 70°

37° d 42° 76° k l q

10 11 j i 12 c 60° 30°53° 30° 37° a d o 42°nn76° 50° e q p

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M06_CMC_SB_IGCSE_6850_U06.indd 159 9/6/09 16:52:04 13 14 15 u u b b t t w w 62° 62° r s s 36° 36° a 46°a 46° 75° 75° v v c c

16 17 18 68° u b t x w 62° s y 36° z a 46° 75° v c

Interior and exterior angles in a triangle The angles inside a triangle are called interior angles. An exterior angle is formed by extending one of the sides of the triangle. Angle BCE in this diagram is an exterior angle. B Let BCE 5 d b

then c 1 d 5 180° (angles on a straight line) acd E but c 1 b 1 a 5 180° (angle sum of nABC) A C which means that d 5 a 1 b

In a triangle, the exterior angle is equal to the sum of the two opposite interior angles.

Sometimes you need to calculate the size of ‘missing’ angles This helps make your explanations before you can calculate the ones you want. clear. You can label the extra angles with letters.

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EXAMPLE 6

Calculate angles g and h. Let the third angle of the triangle 5 x.

24° g h

g 1 x 1 24° 5 180° (angle sum of n) x g 1 x 5 180° 2 24° g 1 x 5 156° g 5 x (base angles isosceles n) 24° g h }1} so both g and x 5 2 of 156° (or 156° 4 2) g 5 78° h 5 180° 2 78° (angles on a straight line) 156 4 2 5 78.

EXAMPLE 7

Calculate angle i. i

42°

x 5 42° x i (base angles isosceles n 1) Label the unknown angles x, y and z. y 5 42° 1 42° 12 (exterior angle of n 1) Label the two triangles 1 and 2. 42° y z y 5 84° z 5 84° (base angles isosceles n 2) i 5 180° 2 84° 2 84° (angle sum n 2) i 5 12°

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M06_CMC_SB_IGCSE_6850_U06.indd 161 9/6/09 16:52:07 EXAMPLE 8

Calculate angle j.

58° j

Label the ‘missing’ angles x, y and z. x 5 180° 2 58° (co-interior angles, parallel lines) x 5 122° 58° x y 5 122° (vertically opposite) y j z 1 j 5 180° 2 122° (angle sum of n) z z 1 j 5 58° z 5 j (base angles isosceles n) }1} so both z and j 5 2 of 58° (or 58° 4 2) 58 4 2 5 29. j 5 29°

EXERCISE 6H Copy the diagram and label any ‘missing’ angles. Calculate the size of the angles marked with letters. 1122 3 c b e 80° d 44° a 53° f

70° 32° 96°

454 h 28° 5 i 6 41° 30° 24° g 66° 37° j k

78787 8 n m n m

86° 86°

p p

l l 62° 62° 72° 72°

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6.4 Quadrilaterals and other polygons Quadrilaterals

A quadrilateral is a 2-dimensional 4-sided shape. This quadrilateral can be divided into 2 triangles:

f b e You can divide any quadrilateral into 2 triangles.

c d

In each triangle, the angles add up to 180°, so a 1 b 1 c 5 180° and d 1 e 1 f 5 180°

The 6 angles from the 2 triangles add up to 360°. a, (b 1 f), e, c 1 d are the angles a 1 b 1 c 1 d 1 e 1 f 5 360° of the quadrilateral.

In any quadrilateral the sum of the interior angles is 360°.

Some quadrilaterals have special names and properties. Quadrilateral Picture Properties square Four equal sides. All angles 90°. Bisect means ‘cut in half’. Diagonals bisect each other at 90°. rectangle Two pairs of equal sides. All angles 90°. Diagonals bisect each other.

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M06_CMC_SB_IGCSE_6850_U06.indd 163 9/6/09 16:52:10 Quadrilateral Picture Properties parallelogram Two pairs of equal and parallel sides. Opposite angles equal. Diagonals bisect each other.

rhombus Four equal sides. Lines with equal arrows are parallel Two pairs of parallel to each other. sides. Opposite angles equal. Diagonals bisect each other at 90°. trapezium One pair of parallel sides.

kite Two pairs of adjacent sides equal. One pair of opposite Adjacent means ‘next to’. angles equal. One diagonal bisects the other at 90°.

EXERCISE 6I

1 Write down the name of each of these quadrilaterals. (a) (b) (c)

(d) (e) (f)

2 Write down the names of all the quadrilaterals with these properties: (a) diagonals which cross at 90° (b) all sides are equal in length (c) only one pair of parallel sides

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(d) two pairs of equal angles but not all angles equal (e) all angles are equal (f) two pairs of opposite sides are parallel (g) only one diagonal bisected by the other diagonal (h) two pairs of equal sides but not all sides equal (i) the diagonals bisect each other (j) at least one pair of opposite sides are parallel (k) diagonals equal in length (l) at least two pairs of adjacent sides equal.

Polygons A polygon is a 2-dimensional shape with many sides and angles. Here are some of the most common ones.

Pentagon (5 sides) Hexagon (6 sides) Octagon (8 sides)

Polygon means ‘many angled’.

A polygon with all of its sides the same length and all of its angles equal is called regular.

Regular pentagon Regular hexagon Regular octagon

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M06_CMC_SB_IGCSE_6850_U06.indd 165 9/6/09 16:52:12 The sum of the exterior angles of any polygon is 360°. You can explain this result like this. Exterior angle If you ‘walk round’ the sides of this hexagon until you get back to where you started, you complete a full turn or 360°. The exterior angles represent your ‘turn’ at each corner, so they must add up to 360°. On a regular hexagon, all 6 exterior angles are the same size. 360° 360° 6 5 number of sides For a regular hexagon: 360° exterior angle 5 6 5 60° For a regular octagon 360° exterior angle 5 A regular octagon has 8 equal 8 exterior angles. 5 45°

For a regular polygon, exterior angle 5 360° number of sides

You can use this equation to work out the number of sides in a regular polygon.

For a regular polygon, number of sides 5 360° exterior angle

In a regular polygon with exterior angles of 18°: 360° number of sides 5 5 20 18° Once you know the exterior angle, you can calculate the

interior angle. Interior angle At each vertex, the exterior and interior angles lie next to each other (adjacent) on a straight line. Exterior angle In a polygon, each pair of interior and exterior angles adds up to 180°.

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So, for any polygon In a regular polygon all the exterior angles are equal. So all the interior interior angle 5 180° 2 exterior angle angles are equal. For example, • regular hexagon interior angle 5 180° 2 60° 5 120° • regular octagon interior angle 5 180° 2 45° 5 135° When a polygon is not regular the interior angles could all be different sizes. You can find the sum of the interior angles. These diagrams show how you can divide any polygon into triangles.

Draw dotted lines from the vertices. 4 sides 6 sides 2 triangles 5 sides 4 triangles 3 triangles You can see that the number of triangles is always 2 less than the number of sides of the polygon.

The sum of the interior angles for each polygon is given by the formula You should learn this formula. (n 2 2) 3 180° where n is the number of sides of the polygon.

Name of Number Number of Sum of interior polygon of sides triangles angles Triangle 3 1 1 3 180° 5 180° Quadrilateral 4 2 2 3 180° 5 360° Pentagon 5 3 3 3 180° 5 540° Hexagon 6 4 4 3 180° 5 720° Heptagon 7 5 5 3 180° 5 900° Octagon 8 6 6 3 180° 5 1080° Nonagon 9 7 7 3 180° 5 1260° Decagon 10 8 8 3 180° 5 1440°

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M06_CMC_SB_IGCSE_6850_U06.indd 167 9/6/09 16:52:13 EXAMPLE 9

Calculate the size of the angles marked with letters in each of You could use the fact that the sum these diagrams. of the angles of a quadrilateral is 360° to calculate q.

(a) q p 5 118° (opposite angles of 118° parallelogram are equal p q 5 62° (p and q are co-interior angles, so p 1 q 5 180°) 5 2 Sum of the interior angles (b) 110° 124° r 540° (the sum of the other of a pentagon (5 sides) 5 4 angles) 3 3 180° 5 540°. r 5 540 2 (95° 1 110° 1 124° 1 73°) 95° 73° r 5 138° r Sum of interior angles of (c) x 1 x 1 133° 1 79° 5 360° quadrilateral 5 360°. 79° 133° 2x 5 360° 2 133° 2 79° x 2x 5 148° x 5 74° x

EXAMPLE 10

Exterior angle of (a) Calculate the size of the exterior and interior angles of a 360° regular polygon 5 number of sides regular polygon with 20 sides. (b) How many sides has a regular polygon with interior angle 168°?

Exterior angle (a) Interior angle 360° Exterior angle 5 }} 5 18° 20 Interior angle 5 180° 2 exterior angle 5 180° 2 18° 5 162° (b) Exterior angle 5 180° 2 interior angle 5 180° 2 168° 5 12° 360 Number of sides 5 }} 5 30 12°

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EXERCISE 6J

Calculate the size of the angles marked with letters. 1 2 3 1284° 55° 3 111°

115°

d 69° a bc

454 5 160° 66 g 119° 69° 132° g ee f 83° 114° 155°

7 8899 h 112° 85° l j i m

k 72° 50° 116° 80° 143°

10 2p 11 r 1212 142° 80° 118° 94° 120° h

q 116° t 103° p

13 1414 x 55° y 58°

104° 52° 68° u w

15 Calculate the interior angle of a regular polygon with 18 sides. 16 Can a regular polygon have an interior angle of 130°? Explain your answer.

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M06_CMC_SB_IGCSE_6850_U06.indd 169 9/6/09 16:52:18 6.6 Symmetry Line symmetry

A kite is symmetrical.

If you fold it along the dashed line, one half fits exactly The dashed line is called a line of onto the other. symmetry.

A line of symmetry divides a shape into two halves. One half is the mirror image of the other.

Some shapes have more than one line of symmetry. Rectangle … 2 lines of symmetry You can draw lines of symmetry with solid lines.

Equilateral triangle … 3 lines of symmetry

Regular hexagon … 6 lines of symmetry

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Some shapes have no lines of symmetry. A parallelogram has no lines of symmetry.

The dashed line is not a line of symmetry. If you reflect the parallelogram in it you get the red parallelogram.

These shapes have no lines of symmetry.

EXERCISE 6K

Copy these shapes and draw in all the lines of symmetry (if any). 1122 3

454 5 6

7 Copy and complete these grids so that they are symmetrical about the dashed line.

(a) (b) (c)

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M06_CMC_SB_IGCSE_6850_U06.indd 171 9/6/09 16:52:22 Rotational symmetry Look again at the shapes before Exercise 6K. They have no lines of symmetry but they have rotational symmetry. You can turn them and they will fit exactly into their original shape again. The order of rotational symmetry is the number of times a shape looks the same during one full turn.

A J P M Q

C B K L If A turns to If J turns to J, K, If P turns to positions any of the L or M the shape P or Q the shape will positions A, will look exactly look exactly the same. B or C the the same. Rotational symmetry shape will look Rotational of order 2. exactly the symmetry of same. order 4. Rotational symmetry of order 3.

Some shapes have line symmetry and rotational symmetry.

2 lines of symmetry. 3 lines of symmetry. Rotational symmetry of Rotational symmetry of order 2. order 3.

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EXERCISE 6L

1 Which of these shapes has rotational symmetry. ABCD EF GH

2 Write down the letters of the shapes that have rotational symmetry of order (a) 2 (b) 3 (c) 4 (d) 5

A BCDEF

GH IJKL

2 Shade one more square in each grid to make shapes with rotational symmetry of order 2 (a) (b) (c)

3 Copy and complete this table

Name of shape Order of rotational symmetry 3 Square Pentagon 5

4 Eight squares have been shaded in this grid to make a shape with rotational symmetry of order 4. Find other ways of shading eight squares to make shapes with rotational symmetry of order 4.

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M06_CMC_SB_IGCSE_6850_U06.indd 173 9/6/09 16:52:24 EXAMINATION QUESTIONS

P Q

(a) Write down the number of lines of symmetry of shape (i) p, [1] (ii) q. [1] (b) Write down the order of rotational symmetry of shape q. [1]

(CIE, Paper 1, Jun 2000)

Q S 2 u° 119° 84° t°

P v° R In the diagram the lines PQ and RS are parallel. Calculate the values of t, u and v. [3]

(CIE Paper 1, Jun 2000)

3 MATHEMATICS (a) In the word above (i) Which letters have two lines of symmetry? [2] (ii) Which letter has rotational symmetry but no line symmetry? [1]

(CIE Paper 1, Nov 2000)

174 Space and shape

M06_CMC_SB_IGCSE_6850_U06.indd 174 9/6/09 16:52:27 Angles, triangles and polygons

Ch6 EQ4

Insert drawing Nov 2000) Paper1 Q13 – find in folder and renumber as question 4 ????

(CIE Paper 1, Nov 2000)

Ch6 EQ5

Insert drawing Paper 3 Q6 Jun 2001)– find in folder and renumber as question 5 ???

(CIE Paper 3, Jun 2001)

6 DC

(Copy and) draw accurately the lines of symmetry of the rectangle ABCD. [1]

(CIE Paper 3, Nov 20001)

7 North Samira (S) and Tamara (T) walk towards each other. Samira walks on a bearing of 140°. S 140° North Find the bearing on which Tamara walks. [2]

(CIE Paper 1, Nov 2001)

Space and shape 175

M06_CMC_SB_IGCSE_6850_U06.indd 175 9/6/09 16:52:29 8 (a) Write down the name of the special quadrilateral which has rotational symmetry of order 2 but no line symmetry. [1] (b) Draw, on a grid, a quadrilateral which has exactly one line of symmetry but no rotational symmetry. Draw the line of symmetry on your diagram. [2]

(CIE Paper 1, Jun 2002)

X 9 (a) Y p° D 140°

A q°

150° r° B C

In the diagram, AX and CY are straight lines which intersect at d. BA and CY are parallel. Angle CDX = 40° and angle ABC = 150°. (i) Find p, q and r. [3] (ii) What is the name of the special quadrilateral ABCD? [1] (b) (i) A nonagon is a polygon with nine sides. Calculate the size of an interior angle of a regular nonagon. [3] (ii) Each angle of another regular polygon is 150°. Calculate the number of sides of this polygon. [3]

(CIE Paper 3, Jun 2002)

10 A

128°

B C

In the triangle ABC, AB = BC. (a) What is the special name of this triangle? [1] (b) Angle BAC = 128°. Work out angle ABC [2]

(CIE Paper 1, Nov 2002)

11 Find the size of one of the ten interior angles of a regular decagon. [3]

(CIE Paper 1, Nov 2003)

176 Space and shape

M06_CMC_SB_IGCSE_6850_U06.indd 176 9/6/09 16:52:31 Angles, triangles and polygons

For the shape shown, write down (a) the number of lines of symmetry, [1] (b) the order of rotational symmetry. [1]

(CIE Paper 1, Jun 2004)

13 (a) (i) What is the special name given to a fi ve-sided polygon? [1] (ii) Calculate the total sum of the interior angles of a regular fi ve-sided polygon. [2] (iii) Calculate the size of one interior angle of a regular fi ve-sided polygon. [1]

(CIE Paper 3, Jun 2004)

14 Reﬂ ex Right Acute Obtuse Use one of the above terms to describe each of the angles give. (a) 100°, (b) 200°.

(CIE Paper 1, Nov 2004)

15 Write down the order of rotational symmetry of each of the following shapes. (a) (b)

Equilateral triangle Rhombus

(CIE Paper 1, Nov 2004)

Space and shape 177

M06_CMC_SB_IGCSE_6850_U06.indd 177 9/6/09 16:52:34