Calculate Angles on Straight Lines, at Points, in s & involving Parallel Lines iss1

Transcription

1 alculate ngles on Straight Lines, at Points, in s & involving Parallel Lines iss1 ngles on a Straight Line dd to 180 x 44 x = = 136 ngles at a Point dd to 360 ngles in a Triangle dd to z 30 z = =135 Isosceles Triangles In an isosceles the two base angles, opposite the equal sides, are equal. 30 These have been labelled m. The two angles, m, together make 150 (180 30). So one of them is = 75 So m = 75. m m 160 y 50 y = = /F/Z-ngles between Parallel Lines Notice the two ends of the and Z are made from parallel lines! Which parts of the F are made from parallel lines? a F-angles are Equal c 55 b The shown is isosceles and is a straight line. a) alculate the missing angles. upside down F backwards F upside down backwards F Z-angles are Equal -angles sum to 180 backwards stretchy Z What sort of angle do the following pairs make? b) b/50 c) c/50 d) d/50 e) e/55 f) Explain why angles c and e are not Z-angles. g) alculate the angles b, c, d, e and f, justifying your results. Hint: To justify use your answers from b-e. 50 b c d e 55 f RPID ID TEST lank out the page above before answering these! 60 d a alculate the angles a, b, c, d and e, justifying your results. b c e

2 alculate ngles in Polygons iss7 Irregular Pentagons/Hexagons and Regular Polygons of any number of sides Interior ngles The angle sum of a triangle is 180. ut what about for quadrilaterals (4-sided shapes), pentagons (5-sided shapes) and hexagons (6-sided shapes)? The angle sum for these shapes, called the sum of interior angles, can be worked out by breaking the shape into triangles. y breaking the following shape into triangles write down the angle sum of this pentagon. The 5-side shape has been drawn as 3 triangles. See how the 9 angles in the triangles make up the 5 angles in the pentagon. See how this obtuse angle in the pentagon is made from 3 of the angles from the triangles. Reminder: Obtuse means greater than 90, less than 180. Know This: The number of triangles is always 2 less than the number of sides. 5-sided shape has, 5 2 = 3 s inside! 6-sided shape has, 6 2 = 4 s inside! There are 3 triangles in the 5-sided shape. The angles of these 3 triangles make up the angles inside the 5-sided shape. The angle sum of a 5 sided shape is therefore = 540. For a 6-sided shape there will be 4 triangles and the angles sum will be, = 720. There are always 2 fewer triangles than sides. In general terms, for an n-sided shape, there are (n 2) triangles and the angle sum is: ( n 2 ) 180. Exterior ngles n exterior angle is part of the angle on the outside of the shape. If you were to walk around the shape it is the angle you would turn through to start walking down a new side. For a 5-sided shape there are 5 exterior angles. y the time you have walked around the whole shape you will be facing the same way and will have turned 1 revolution, 360! Exterior ngle Remember the Formulae: Sum of the Interior ngles = ( n 2 ) 180 Sum of the Exterior ngles = 360 (One full turn). In your examination you need to know the interior angle sums for quadrilaterals, pentagons and hexagons. The above formulae can help you work out that the angle sums are 360, 540 and 720 respectively! 80 a) alculate the missing angle i) ii) Regular Polygons regular polygon has all its sides of equal length ND all its angles are equal!! In a regular 5-sided polygon there are 5 exterior angles. The 5 angles add together to make 360 and one of them is 360 therefore or 5 which is 72. n exterior angle and an interior angle add to make 180, because they form a straight line, so an interior angle of a regular pentagon will be = 108. In general terms you have the following additional formulae to remember for Regular Polygons: ( n 2 ) 180 Exterior ngle = Interior ngle = 180 OR = n n n b) alculate the i) exterior angle & ii) interior angle of both a regular hexagon & a regular 10-sided polygon RPID ID TEST lank out the page above before answering these! 1. alculate the missing angles i) ii) alculate: a) the exterior angle and b) the interior angle of a regular 9-sided shape.

3 onstruct Triangles and Regular Polygons (from inscribed circles) iss8 onstruct a Triangle with Given Lengths onstruct a triangle with sides 2, 4 and 5cm. Draw the longest With compasses centred at one end, With compasses centred at the other end, side as the base. draw an arc with radius. draw a crossing arc with radius 4cm. 4cm 5 cm a) onstruct a triangle with sides 3, 5 and 7cm. Join the ends of the base line to the point where the arcs cross. 4cm Regular Polygons 0 Draw a regular hexagon, using a circle. Hexagons are 6 sided. Divide a circle into 6 equal parts so that every part is = 60. On a 360 protractor this means making marks at; 0, 60, 120, 180, 240 & 300. Draw a circle. Mark the points 0, 60, 120, 180, 240 & 300. Join the points together b) Draw a regular pentagon, using a circle. 180 RPID ID TEST lank out the page above before answering these! 1. onstruct a triangle with sides 3, 4 and 6cm. 2. Draw a regular 10-sided polygon.

4 Draw, and Elevation Views of 3-D Shapes iss9 This is all about drawing 3-D shapes in 2-D. One way is to draw out the net of the shape, this is where each side is drawn out flat; more of this later. nother way is to consider what can be seen from different directions. The plan means the view from above. One way to learn about this is to take some 3-D shapes and/or construct some simple shapes with cubes and view the shape at a distance from different directions. onsider this cuboid. From above, the plan view, all that can be seen is a by 3cm rectangle. From the side and front all that can be seen are also rectangles we will come back to this after looking at an example. 3cm Elevation Draw the accurate front, side and plan elevations of: a) a square based pyramid with a base of sides 1.5cm and a perpendicular height of. 1.5cm Optionally show the centre of the pyramid and/or the diagonals to show height change Elevation b) a cylinder with a diameter of 1.5cm and a height of. 1.5cm 1.5cm Square ylinder ased Pyramid a) Draw accurately the plan, side and front view of the cuboid shown on the top right of this page. Remember: n immediate change in height or depth is shown as a solid line Drawn half size assuming cubes have sides of b) Similarly draw 3 accurate views and assume the side of each cube is. Hint: The front view needs one dotted line and the side view one solid line to show height change. Remember: hidden change in height (one that cannot be seen from a given view) is shown as a dotted line TIP: Draw the outside of your view first! Only then add in the change of height lines! RPID ID TEST lank out the page above before answering these! 1. a) ccurately draw the front, right side & plan elevations of the given shape. ssume the side of each cube is. b) Draw the view from the left side. Left Right 2. The front elevation of a small house 1m is as shown. The house is 3.5m 1m deep and the shape of the house does not change from front to back. Draw the side & plan views. Use 2m a similar scale of to 1m as used in this front elevation. 4m Hint: Draw height difference lines on your views. You need one for your side view and one for your plan view. These show the height difference of the slanted roof.

5 Draw Nets of 3-D Shapes iss10 The net of a 3-D shape are the faces of the shape laid out flat. It is therefore important to consider the number of faces the shape has and the size of each face. How many faces does a a) uboid have? b) ylinder (closed at both ends) have? c) Square based Pyramid have? The curved surface of the cylinder counts as one face Sketch the net of a by by 3cm cuboid. Show the correct dimensions of the net. The front and back faces are both by 3cm. The two side faces are both by. The top and bottom faces are both by 3cm. 3cm 2 cm d) Sketch a copy of the net of the by by 3cm cuboid and mark the front and back faces, the side faces and the top and bottom faces. 2 cm 2 cm e) Sketch the net of a by by cuboid. Show the correct dimensions of the net. 3 cm f) The net of the square based pyramid - see c) above - includes 1 square face and 4 other identical faces. Sketch roughly the shape of the net of the square based pyramid given in c) above. Draw the net of a cylinder with a radius of 0.75cm & a height of. Use π = 3.14 in your calculation and show the correct dimensions of the net. Practical Take a piece of paper in the shape of a rectangle and form an open cylinder by joining two sides of the rectangle, let go and see how the curved surface is a rectangle. Remember Laid out flat the curved surface of a cylinder is a rectangle. The cylinder is made of two identical faces that are circles and a curved surface. The curved surface laid out flat is a rectangle. The length of the rectangle is the length of the curved surface. The length of the curved surface is given by the circumference, where circumference = 2πr, see iss5. The height of the rectangle is the same as the height of the cylinder. g) Sketch the net of a cylinder, closed at both ends, 3cm tall and with a diameter of. Show the correct dimensions of the net. Use π = 3.14 in your calculation. 0.75cm ircumference = 2πr = cm RPID ID TEST lank out the page above before answering these! 1. Sketch the net of a cylinder, closed at both ends, 4cm tall and with a diameter of 4cm. Show the correct dimensions of the net. Use π = 3.14 in your calculation. 2. Sketch the net of a cube with sides of. Show the correct dimensions of the net.

6 Explain asic Geometric Facts iss11 You should be able to explain the angle sum of a triangle, angle sum of a quadrilateral and that an exterior angle of a triangle is equal to the sum of the two interior opposite angles asic Facts 1. The ngle Sum of a Triangle is The ngle Sum of a Quadrilateral is n Exterior angle of a triangle is equal to the sum of the two interior opposite angles. Explanations 1. The ngle Sum of a Triangle is 180. Draw a line parallel to the base of the triangle as shown In simple terms the two angles are equal because they make a Z-angle; similarly the two angles are equal because they make a Z-angle. The three angles, and sum to 180 because angles on a straight line sum to 180. The angles in the triangle are also, and and so they must also sum to For a reminder of Z-angles see iss 1 lternative explanation using 3 letter angle notation  = ĈX (Z-angle) ˆ = ĈY (Z-angle) ĈX + ĈY + Ĉ = 180 (ngles on a straight line)  + ˆ + Ĉ = 180 Therefore, the 3 angles in the triangle sum to 180. X dd the letters X and Y as shown Y a) Sketch a copy of above and without looking at the explanation given, explain in simple terms why the angle sum of is The ngle Sum of a Quadrilateral is 360. ny quadrilateral can be divided into 2 triangles, such that the angles in the two triangles make up the 4 angles in the quadrilateral. The quadrilateral can be divided as 2 triangles as shown. The 6 angles in the 2 triangles make up the 4 angles in the quadrilateral. ngles in a triangle sum to 180. Therefore angles in a quadrilateral sum to = 360 as required. 3. n exterior angle of a triangle is equal to the sum of the two interior opposite angles. In simple terms we need to explain why + = Z. We can start by saying that angles, and added will make 180 because angles in a triangle make 180. Z In 3 letter angle notation we need to explain why  + Ĉ = ˆ X. b) Now finish off this explanation. c) Explain why  + Ĉ = ˆ X using 3 letter angle notation. X RPID ID TEST lank out the page above before answering these! 1. With reference to this diagram, explain in simple terms why the angle sum must be 180. D E 2. Explain why the angle sum of a quadrilateral (4-sided closed shape) has an angle sum of 360.