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Transcript

2 Plane figure geometry cyan magenta yellow 95 100 50 75 Turning Measuring angles Classifying and naming angles Complementary and supplementary angles Angles in a revolution Bisecting angles Constructing 90° angles to a line Plane shapes Points and lines 25 0 5 95 A B C D E F G H I 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 Contents: black Y:\HAESE\SA_08-6ed\SA08-6_02\033SA08-6_02.cdr Friday, 27 October 2006 3:01:51 PM PETERDELL SA_08-6 34 PLANE FIGURE GEOMETRY (Chapter 2) If we look carefully we notice angles in many objects and situations. For example, we see angles in building frameworks, pitches in roof structures, steepness of ramps, and the positioning of boats and aeroplanes from their home bases. Angle measurement dates back more than 2500 years and is still very important today in architecture, building, surveying, engineering, navigation, space research, etc. DEGREE MEASURE To measure the size of angles accurately we use a small unit of measure. This unit is called a degree. There are 360 degrees, 360o , in one complete turn or revolution. Degrees have been used from ancient Babylonian times when all the numbers were written and calculated in base 60. 360o is a useful measure because it is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, etc., and this makes it easy to find fractions of 360o such as halves, thirds, quarters, and so on. OPENING PROBLEM There are four posts at the corners A, B, C and D of a paddock with a raised mound as shown. The distances between the posts are easily measured and are shown on the figure. 125 m B A 97 m 113 m Your task is to draw an accurate scale diagram of the situation. Because of the mound you cannot accurately measure the distances AC or BD. C 84 m rough figure D cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 Things to discuss, think about and do: ² Is knowing the lengths of the four sides enough to accurately redraw the figure? Try doing this using millimetres instead of metres. ² What other facts are necessary to accurately redraw the figure? ² How would you accurately measure the four angles at A, B, C and D in the real world? ² How would you accurately measure the four angles of the figure on paper? ² If the angle at A is known to be 100o , could you accurately peg out on the school oval 1 th scale model of paddock ABCD? You may only use four pegs, a large measuring a 10 tape (as used on athletics days) and an angle measuring device (such as a theodolite or modern laser equipment). black Y:\HAESE\SA_08-6ed\SA08-6_02\034SA08-6_02.CDR Friday, 27 October 2006 3:03:13 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY A (Chapter 2) 35 TURNING Many machines are operated with a dial showing different settings. The dial is rotated in order for the machine to perform a particular operation. Example 1 Self Tutor Here is the dial of a ceiling fan. There are three settings as well as an ‘off’ position, as shown. i 1 4 3 4 c clockwise What setting will be beside the arrow for each of the following rotations? Through how many degrees has the dial turned in each case? ii a anticlockwise turn clockwise b 1 2 turn anticlockwise d a full revolution anticlockwise turn clockwise a i low ii 1 4 of 360o = 90o b i medium ii c i low ii 3 4 of 360o = 270o d i off ii 1 2 of 360o = 180o 360o EXERCISE 2A 1 Using the fact that a full rotation or turn is equal to 360o , what is the size of: a a half rotation b a quarter turn c a three-quarter turn d three full turns e one and a half rotations f one-eighth rotation? 2 For the ceiling fan described in the example above, copy each diagram and fill in the missing settings: a b cyan magenta HIGH LOW yellow 95 100 50 75 25 0 5 95 100 50 MED 75 25 0 high to off 5 iii 95 off to very high 100 50 ii 75 25 0 5 95 100 50 75 25 0 5 3 An industrial fan is used to remove chemical fumes from a factory. It has five settings as shown, and can only be turned clockwise. OFF clockwise a There are six equal divisions on the dial. How many degrees is each division worth? VERY VERY HIGH LOW b How many degrees does each of the following rotations require? i off to medium black Y:\HAESE\SA_08-6ed\SA08-6_02\035SA08-6_02.CDR Thursday, 26 October 2006 2:06:51 PM PETERDELL SA_08-6 36 PLANE FIGURE GEOMETRY (Chapter 2) B MEASURING ANGLES An angle is made up of two arms which meet at a point called the vertex. arm The size of the angle is measured by using the amount of turning (rotation) from one arm to the other. vertex arm angle MEASURING DEVICES We use a protractor (or geoliner) which has tiny 1o markings on it. 170 180 160 0 150 20 10 30 40 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 0 5 12 50 0 13 centre To use a protractor to measure angles we: ² place it so its centre is at the angle’s vertex and 0o lies exactly on one arm ² start at 0o and follow the direction the angle turns through to reach the other arm. base line The small ° symbol means degrees. Example 2 Self Tutor Measure these angles: a yellow 0 10 20 3 180 170 1 60 1 0 50 40 14 0 100 95 50 25 0 5 95 100 50 75 25 0 5 95 50 75 25 100 magenta b This angle has size 123o : 75 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 5 95 100 50 75 25 0 5 cyan 170 180 160 0 150 20 10 30 a This angle has size 47o . 40 170 180 160 0 150 20 10 30 40 A 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 50 12 50 0 13 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 50 12 50 0 13 A B b B black Y:\HAESE\SA_08-6ed\SA08-6_02\036SA08-6_02.CDR Friday, 27 October 2006 3:04:32 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY (Chapter 2) 37 EXERCISE 2B 1 What is the size of the angle being measured? a b 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 95 100 50 75 25 0 5 95 100 50 75 25 5 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 0 10 20 3 180 170 1 60 1 0 50 40 14 0 95 100 50 75 25 0 5 95 170 180 160 0 150 20 10 30 40 100 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 50 120 50 0 13 0 14 50 170 180 160 0 150 20 10 30 Y:\HAESE\SA_08-6ed\SA08-6_02\037SA08-6_02.CDR Thursday, 9 November 2006 5:14:05 PM DAVID3 black 40 170 180 160 0 150 20 10 30 40 75 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 0 5 12 50 0 13 0 14 0 14 yellow 170 180 160 0 150 20 10 30 170 180 160 0 150 20 10 30 40 j 40 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 50 120 50 0 13 magenta h 0 14 170 180 160 0 150 20 10 30 40 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 0 5 12 50 0 13 i 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 50 120 50 0 13 0 14 g 25 170 180 160 0 150 20 10 30 40 170 180 160 0 150 20 10 30 40 f 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 50 120 50 0 13 0 0 14 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 0 2 5 1 50 0 13 e 5 170 180 160 0 150 20 10 30 d 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 0 2 5 1 50 0 13 cyan 40 170 180 160 0 150 20 10 30 40 c 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 50 12 50 0 13 0 14 80 90 100 11 0 1 70 80 7 60 110 100 0 6 20 1 30 0 0 50 12 50 0 13 SA_08-6 38 PLANE FIGURE GEOMETRY (Chapter 2) 2 Look at the following angles and estimate their sizes. Record your estimates. Now measure each angle accurately with a protractor and record your answers. Copy and complete a table like the one given. How close was your estimate to the measured angle? Estimate Measure a b .. . c b a d e C CLASSIFYING AND NAMING ANGLES Angles are classified according to their size. Revolution Straight Angle Right Angle 1 2 1 4 One complete turn. One revolution = 360o : turn. 1 straight angle = 180o : turn. 1 right angle = 90o : Acute Angle Obtuse Angle Reflex Angle Less than a 14 turn. An acute angle has size between 0o and 90o : Between 14 turn and 12 turn. An obtuse angle has size between 90o and 180o : Between 12 turn and 1 turn. A reflex angle has size between 180o and 360o : NAMING ANGLES To avoid confusion in complicated figures, three point angle notation is used. The shaded angle is named ]ADB or ]BDA where the vertex must be the middle letter. A B cyan magenta yellow 95 100 50 75 0 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 C 25 Calling the shaded angle ]D, would be unsatisfactory. Why? 5 D black Y:\HAESE\SA_08-6ed\SA08-6_02\038SA08-6_02.CDR Thursday, 26 October 2006 2:07:29 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY 39 (Chapter 2) EXERCISE 2C 1 Match the names to the correct angles: a ]ABC b ]CAB A B C C B d D A ]CBD B B A B ]BCA c C C C D A 2 Draw and label a diagram for each of the following angles: a ]DEF b ]ZXY c ]XYZ d ]PQR 3 Using only a ruler and pencil, draw angles you estimate to be: a 90o b 45o c 30o d 60o Check your estimations using a protractor. 4 Find the size of each of the following angles without your protractor: a ]ABC b ]DBC c ]ABD d ]ABE B e ]RPQ e 135o A 35° 35° 24° C D E 5 Use your ruler and protractor to draw angles with the following sizes: a 35o b 131o c 258o Get someone else to check the accuracy of your angles. Z 6 Which is the larger angle, ]ABC or ]XYZ? C B Y A X 7 Draw a free-hand sketch of: a acute angle BPQ b d obtuse angle CPT e c f right angle NXZ reflex angle DSM straight angle QDT revolution ]E 8 Use a protractor to measure the named angles: a i ]PMN ii ]OPL b i ]VTU ii reflex ]VST iii ]PON iii reflex ]TVU S Q P M L T cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 V O 100 50 75 25 0 5 95 100 50 75 25 0 5 N black Y:\HAESE\SA_08-6ed\SA08-6_02\039SA08-6_02.CDR Thursday, 26 October 2006 2:07:40 PM PETERDELL U SA_08-6 40 PLANE FIGURE GEOMETRY (Chapter 2) 9 Kim hits the billiard ball so that it follows the path shown. What acute angle will it make with the edge of the table? 10 A golfer completing his swing holds the golfclub behind his body. What is the size of the reflex angle between his body and the club? 0 4 1 3 2 7 5 11 A fan switch has seven operating speeds apart from the off position (0). a If the knob is turned clockwise to 5, through how many degrees has it rotated? b How many less degrees would it have to turn if it rotated anticlockwise? 6 12 a How many obtuse angles can be drawn joining any three dots? How many right angles can be drawn? b D COMPLEMENTARY AND SUPPLEMENTARY ANGLES Two angles are complementary if their sizes add to 90o . Two angles are supplementary if their sizes add to 180o . For example, O U M P magenta yellow V 95 100 50 ]STU and ]UTV are supplementary because 79o + 101o = 180o . 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 ]MNO and ]MNP are complementary because 52o + 38o = 90o . cyan 79° 101° T S 75 N 52° 38° black Y:\HAESE\SA_08-6ed\SA08-6_02\040SA08-6_02.CDR Friday, 27 October 2006 3:05:33 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY (Chapter 2) 41 Note: Two angles are equal if they have the same size (degree measure). Example 3 Self Tutor a Are angles with sizes 37o and 53o complementary? b What angle size is the supplement of 48o ? a 37o + 53o = 90o . So, the angles are complementary. b The angle size is 180o ¡ 48o = 132o . EXERCISE 2D 1 Add the following pairs of angles and state whether they are complementary, supplementary, or neither: b 30o , 150o c 110o , 40o a 20o , 70o d 47o , 43o e 107o , 63o f 35o , 55o 2 Find the size of the angle complementary to: a 30o b 5o c 85o 3 Find the size of the angle supplementary to: a 100o b 5o c 90o 4 Classify the following angle pairs as complementary, supplementary or neither: a ]BOC and ]COD B C D b ]AOC and ]COE The symbol c ]COD and ]DOE indicates a right angle d ]AOB and ]BOE A E O 5 Copy and complete: a the size of the complement of xo is .......... b the size of the supplement of y o is .......... Example 4 Self Tutor Find the value of the pronumeral in: a b x° x° 58° 72° cyan magenta yellow 95 100 50 75 25 0 b The three angles add up to 180o . ) x = 180 ¡ 72 ¡ 78 ) x = 30 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a The angles 58o and xo are . complementary ) x = 90 ¡ 58 ) x = 32 78° black Y:\HAESE\SA_08-6ed\SA08-6_02\041SA08-6_02.CDR Thursday, 26 October 2006 2:08:03 PM PETERDELL SA_08-6 42 PLANE FIGURE GEOMETRY (Chapter 2) 6 Find the value of the pronumeral in: a b c a° 25° b° 147° d c° c° e 62° f 12° f° f° d° e° 30° 33° g 38° h i 32° g° g° 48° h° E h° x° x° 74° x° ANGLES IN A REVOLUTION A The angles around a point total 360o . D So, ]ABC + ]CBD + ]DBA = 360o . B C Example 5 Self Tutor The sum of the five angles is 360o . Find a in: ) the three equal angles add to 360o ¡ 90o ¡ 120o = 150o a° a° 120° a° So, each must be 150o ¥ 3 = 50o ) a = 50: EXERCISE 2E 1 Find the size of the angles marked with a pronumeral: a b c 266° z° y° 27° cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 x° black Y:\HAESE\SA_08-6ed\SA08-6_02\042SA08-6_02.CDR Thursday, 26 October 2006 2:08:14 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY 2 Find the value of the pronumerals in: a b c 35° 109° 121° p° r° 60° 60° r° 60° 113° q° d 43 (Chapter 2) e f 138° t° 41° 63° t° 58° 52° s° F u° 74° u° w° BISECTING ANGLES When we bisect an angle with a straight line we divide it into two angles of equal size. The following construction shows angle bisection with a compass and ruler only. Example 6 Self Tutor A Bisect angle ABC. B Step 1: C A With centre B, draw an arc of convenient radius which cuts BA and BC at P and Q respectively. P B C Q A Step 2: With Q as centre, draw an arc within the angle ABC. P B C Q A Step 3: Keeping the same radius and with centre P draw another arc to intersect the previous one at M. P M B C Q A Step 4: Join B to M. BM bisects angle ABC, so ]ABM = ]CBM. P M cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 B black Y:\HAESE\SA_08-6ed\SA08-6_02\043SA08-6_02.CDR Thursday, 26 October 2006 2:08:22 PM PETERDELL Q C SA_08-6 44 PLANE FIGURE GEOMETRY (Chapter 2) EXERCISE 2F 1 Use your protractor to draw an angle ABC of size 80o : a Bisect angle ABC, using a compass and ruler only. b Check with your protractor the size of each of the two angles you constructed. 2 Draw an acute angle XYZ of your own choice. a Bisect the angle without using a protractor. b Check your construction using your protractor. 3 Draw an obtuse angle ABC of your own choice. a Bisect the angle using a compass only. b Check your construction using your protractor. a Using a ruler, draw any triangle with sides greater than 5 cm. b Bisect each angle using a compass and straight edge only. c What do you notice about the three angle bisectors? 4 G CONSTRUCTING 90° ANGLES TO A LINE A right angle or 90o angle can be constructed without a protractor or set square. Consider the following example: Example 7 Self Tutor Construct an angle of 90o at P on the line segment XY. Step 1: Step 2: X On a line segment XY, draw a semicircle with centre P and convenient radius which cuts XY at M and N. X With centre M and convenient radius larger than MP, draw an arc above P. X Y P M P N M P N Y Y W cyan magenta yellow 95 100 50 X 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 With centre N and same radius draw an arc to cut the first one at W. 25 0 5 95 100 50 75 25 0 5 Step 3: black Y:\HAESE\SA_08-6ed\SA08-6_02\044SA08-6_02.CDR Friday, 27 October 2006 3:05:49 PM PETERDELL M P N Y SA_08-6 PLANE FIGURE GEOMETRY Step 4: Draw the line from P through W. Angles WPY and WPX are 90o . 45 (Chapter 2) W X M N P Y EXERCISE 2G 1 a Draw a horizontal line segment AB which is 5 cm long. b Use your ruler to find C on AB where AC = 2:5 cm. c Construct a 90o angle at C using your compass and ruler. 2 a Draw a line segment AB of length 3 cm. b Construct an angle of 90o at A. (You will need to extend the line segment AB on the other side of A.) c Draw AC of length 4 cm, such that ]BAC is 90o . d Join BC to form triangle ABC and measure the length of BC. 3 a b c d e f Draw a line segment XY of length 8 cm. At X construct an angle of 90o . Draw XZ of length 5 cm. Join ZY. Use your compass to bisect angle ZXY. What is the size of angle YXP? If ZY and XP meet at Q, measure QY to the nearest mm. Horizontal lines are drawn across the page like a horizon. P Z Q 5 cm X H 8 cm Y PLANE SHAPES A shape that is drawn on a flat surface or plane is called a plane figure. If the shape has no beginning or end it is said to be closed. We can use properties like these to classify objects. POLYGONS Polygons are closed plane figures with straight line sides which do not cross over themselves. Simple examples are: cyan magenta yellow 95 100 50 pentagons 75 25 0 5 95 100 50 75 25 0 5 95 quadrilaterals 100 50 75 25 0 5 95 100 50 75 25 0 5 triangles black Y:\HAESE\SA_08-6ed\SA08-6_02\045SA08-6_02.CDR Thursday, 26 October 2006 2:08:58 PM PETERDELL SA_08-6 46 PLANE FIGURE GEOMETRY (Chapter 2) Example 8 Self Tutor Classify as a polygon or not, giving reasons: a b c a It is closed and has straight line sides, so it is a polygon. b It does not have all straight line sides, so it is not a polygon. c It is not closed and so it is not a polygon. Polygons are named according to the number of sides they have. For example, a 9 sided polygon can be called a 9-gon. However, many polygons are known by other more familiar names. Here are the first few: Number of Sides 3 4 5 6 7 8 9 10 Polygon Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon CIRCLES Suppose we make a loop from a length of light rope and place it over a fixed spike in the ground. The rope is made taut and a stick is placed at the opposite end to the fixed spike. By keeping the rope taut and moving the stick around the spike, a circle is produced. The fixed spike is the circle’s centre. This method of drawing a circle was known and used by builders in ancient Egypt. Parts of a circle this is a circle this is the circle’s centre (and is often not drawn) this is a radius which joins the circle’s centre to any point on the circle AB is a diameter of the circle as it passes through the circle’s centre and its end points lie on the circle A cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 B black Y:\HAESE\SA_08-6ed\SA08-6_02\046SA08-6_02.CDR Thursday, 26 October 2006 2:09:17 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY (Chapter 2) 47 this is a semi-circle (half of a circle) this is an arc of the circle (part of the circle) this is a chord of the circle this is a segment of the circle this is a sector of the circle EXERCISE 2H 1 Classify as polygons or not, giving reasons: a b c d convex (no reflex angles) e f g h non-convex (has a reflex angle) 2 Draw a free-hand sketch of: a a convex 4-sided polygon c a convex 5-sided polygon ACTIVITY a non-convex 4-sided polygon a non-convex 6-sided polygon b d 3 What special name is given to a polygon with: a three sides b four sides d six sides e eight sides c f five sides ten sides cyan magenta yellow A B C D E F G H I J 95 100 50 75 25 0 5 95 100 50 75 25 0 j 5 i 95 h 100 g 50 f 75 e 25 0 5 95 100 50 75 25 0 5 4 Match the part of the figure indicated to the phrase which best describes it: a b c d black Y:\HAESE\SA_08-6ed\SA08-6_02\047SA08-6_02.CDR Friday, 27 October 2006 3:06:07 PM PETERDELL a a a a a a a a a a semi-circle radius minor arc major arc diameter chord minor segment major segment major sector minor sector SA_08-6 48 PLANE FIGURE GEOMETRY (Chapter 2) GROUP ACTIVITY GEOMETRICAL IDEAS What to do: 1 In the classroom find two examples of each of the following: ² a point ² a line ² an angle ² a flat surface ² a curved surface 2 Discuss the following: ² What is meant by a point? I ² How small can a point be? POINTS AND LINES POINTS IN GEOMETRY Examples of a point in the classroom are: ² the intersection of two adjacent walls and the floor ² a speck of dust in the room at a particular instant in time. In geometry, a point could be represented by a small dot. To help identify it we could name it with or assign a capital letter. Consider: The letters A, B and C are useful for identifying points to which we are referring. B We can make statements like: “the distance from A to B is .....” or “the angle at B measures .....”, etc. A C a point marks a position and does not have any size. To a mathematician, FIGURES AND VERTICES A Any special collection of points in a plane (flat surface) is called a figure. The figure alongside contains four points which have been labelled A, B, C and D. These corner points are also known as vertices. B C D magenta yellow 95 100 50 75 25 0 5 95 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 cyan 100 Vertices is the plural of vertex. (Note: Point B is a vertex of the figure.) black Y:\HAESE\SA_08-6ed\SA08-6_02\048SA08-6_02.CDR Thursday, 26 October 2006 2:09:34 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY (Chapter 2) GROUP DISCUSSION 49 LINES What to do: Discuss the following questions: 1 How many different straight lines could be drawn through a single point? 2 Suppose A and B are two different points. How many straight lines can be drawn such that each line passes through both A and B? STRAIGHT LINES A straight line (usually called a line) is a continuous infinite collection of points which lie in a particular direction. B This line which passes through points A and B is called “line AB” or “line BA”. A A This part line which joins points A and B is called “line segment AB” or “line segment BA”. B Note: ² When three or more points lie on a line we say that the points are collinear. The points A, B, C and D shown are collinear. ² If three or more lines meet (intersect) at the same point we say that the lines are concurrent. The lines shown are concurrent at point B. B A C D B PARALLEL AND INTERSECTING LINES Imagine a plane, like a table top, which goes on indefinitely in all directions, i.e., it has no boundaries. In such a plane, two straight lines are either parallel or intersecting. Arrow heads are used to show parallel lines. point of intersection parallel Parallel lines are lines which are always a fixed distance apart and so never meet. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 a point of intersection concurrent lines 100 c f 50 2 Describe with a sketch the meaning of: a a vertex b an angle d parallel lines e collinear points 75 a flat surface 25 c 0 1 Give two examples in the classroom of: a a point b a line 5 95 100 50 75 25 0 5 EXERCISE 2I black Y:\HAESE\SA_08-6ed\SA08-6_02\049SA08-6_02.CDR Thursday, 26 October 2006 2:09:46 PM PETERDELL SA_08-6 50 PLANE FIGURE GEOMETRY (Chapter 2) 3 Give all ways of naming the following lines: a b C B Q P (6 answers) A 4 Name the point of intersection of: a line 1 and line 2 b line 2 and line 3 c line AB and line MN M A line 3 line 2 line 1 B C N 5 ABCD is a quadrilateral and line segment BD is called a diagonal. a Name the four sides of the quadrilateral. A b Name the two diagonals of the quadrilateral. c At what point do the diagonals meet? d How many line segments meet at A? e What can be said about points A, X and C? f What can be said about the line segments AB, D DB and CB? B X C 6 How many different lines do you think you can draw through: a two points A and B b all three collinear points A, B and C c one point A d all three non-collinear points A, B, C? 7 Draw a different diagram to fit each statement: a C is a point on line AB. b Lines AB and CD meet at point X. c Point A does not lie on line BC. d X, Y and Z are collinear. e Line segments AB, CD and EF are concurrent at G. a Name line AC in two other ways. b Name two different lines containing point B. c What can be said about i points A, B and D ii lines BF and AD iii lines FE, CD and AB? 8 A B C D F E 9 When drawing lines through three different points there are two possible cases: 3 different lines 1 line, as the points are collinear a How many different cases can we have for four different points? Illustrate each case. b Now draw the cases for five different points. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 c You may also wish to experiment with a higher number of points. black Y:\HAESE\SA_08-6ed\SA08-6_02\050SA08-6_02.CDR Thursday, 26 October 2006 2:09:53 PM PETERDELL SA_08-6 PLANE FIGURE GEOMETRY 51 (Chapter 2) REVIEW SET 2A 1 a State the complement of 41o . b What is the size of one revolution? c Determine the measure of angle ABC. C A B 2 For the given figure: a find the angle number corresponding to: i ]BDA ii ]DCB iii ]BAC b classify the following angles as acute, obtuse or reflex: i 3 ii 1 iii 4 A 1 119° 2 6 5 D 4 C 3 Find the value of the pronumeral in: a b n° 3 B c x° x° 68° y° 100° 140° 4 Name the following polygons: a b a 5 Define the following terms: an arc of a circle b a sector of a circle 6 If the radius of a circle is 5 cm in length, what is the length of a diameter? 7 How many points are needed to determine the position of a line? 8 Draw a diagram to illustrate the following statement: “Line segments AB and CD intersect at P.” 9 a b R Q S U P Name line RS in two other ways. Name two different lines containing point P. What can be said about i points P, Q and R ii lines PQ and RS? c T cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 10 Use your protractor to draw an angle of 56o . Bisect this angle using your compass. Check that the two angles produced are 28o each. black Y:\HAESE\SA_08-6ed\SA08-6_02\051SA08-6_02.CDR Thursday, 26 October 2006 2:10:12 PM PETERDELL SA_08-6 52 PLANE FIGURE GEOMETRY (Chapter 2) REVIEW SET 2B 1 a Draw a diagram to illustrate an obtuse angle. b If angle ABC and angle DEF are equal angles and the measure of angle ABC is 72o , find the measure of angle DEF. c Determine the measure of the reflex angle PQR. P Q R A 2 Match the angle name to the angle number: a ]ADC b ]BAC 1 6 ]ABD c 2 B 3 4 C 3 Match each angle description to one of the angles shown: a revolution b right angle c d reflex angle e acute angle f D C E F a What is the complement of 63o ? 4 D straight angle obtuse angle B A 5 b What is the supplement of 70o ? 5 Find, giving a reason, the value of y in each of the following: a b c y° 131° y° 123° y° y° y° 6 Name the following polygons: a b 7 Draw a circle with radius 2 cm. Mark and label: a a radius b a sector c a segment d a minor arc 8 Copy and complete: “Points are collinear if ......”. 9 State the total number of points of intersection of the lines: cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 10 Draw AB of length 4 cm. Construct an angle of 90o at B using a compass and ruler only, and draw BC of length 3 cm and perpendicular to AB. Draw in and measure the length of AC. black Y:\HAESE\SA_08-6ed\SA08-6_02\052SA08-6_02.CDR Monday, 30 October 2006 9:08:43 AM PETERDELL SA_08-6