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Transcript
2.2 Angles and proof Identify interior and exterior angles in triangles and quadrilaterals Calculate interior and exterior angles of triangles and quadrilaterals Understand the idea of proof Recognise the difference between conventions, definitions and derived properties Why learn this? Angles can be crucia lly important in some extreme spor ts. interior angle Interior and exterior angles add up to 180°. Level 5 exterior angle The interior angles in a triangle sum to 180°. Level 5 The interior angles in a quadrilateral sum to 360°. Level 6 b The exterior angle of a triangle is equal to the sum of the two interior opposite angles. For example, a ⴙ b ⴝ e. Level 6 a e Did you know? A convention is an accepted mathematical way to show some information. Level 7 A definition is a precise description. For example, the definition of a square is: a shape with exactly four equal sides and four equal angles. Level 7 The word ‘angle’ comes from the Latin word A derived property is information that can be worked out from a definition. For example, each angle of a square is 90° because they sum to 360° and are all equal. Level 7 ‘angulus’, which means ‘a corner’. Level 5 Work out the size of angle q. q is an interior angle. q and 52° lie on a straight line, so they sum to 180°. q = 180 – 52 = 128° 80° I can use interior and exterior angles to calculate angles p q 52° Work out the size of angle p. a Copy and complete these sentences to identify the interior and exterior angles. E A D U Z T Tip S 117° C 105° B EBC is X B 98° 84° C Y W V W X an interior angle. i YZW is . ii SWX is iii XTU is . A The marked angle is ABC or BCA. . b Calculate the missing angles marked on the diagrams. Calculate the size of the lettered angles, stating any angle facts that you use. a b c d 94° d 107° p 70° 68° 22 Get in line x y s t 110° 58° u convention q 105° definition 64° 98° e derived property Level 6 I can calculate interior and exterior angles of triangles and quadrilaterals exterior angle Sketch this diagram. x b Then copy and complete these sentences. a Angle x is equal to angle a because they are b Angle y is equal to angle xⴙbⴙyⴝ because they are because they lie on a c Since x ⴝ a and y ⴝ , aⴙbⴙcⴝ ⴙbⴙ angles. a y c angles. . . Angles in a triangle sum to 180°. Angles in a quadrilateral sum to 360°. . Sketch this diagram. The quadrilateral has been split into two triangles. a ⴙ b ⴙ c ⴝ 180° Continue the proof to show that angles in a quadrilateral sum to 360°. b c e a d b Then copy and complete this proof. triangle sum to b cⴙxⴝ c x . because they lie on a c So a ⴙ b ⴙ c ⴝ c ⴙ So a ⴙ b ⴝ x. a I can follow a proof that the sum of angles in a quadrilateral is 360° f Sketch this diagram. ⴝ 180° because angles in a I can follow a proof that the sum of angles in a triangle is 180° Learn this This proves that angles in a triangle sum to a aⴙbⴙ Level 6 . I can follow a proof that the exterior angle of a triangle is equal to the sum of the two interior opposite angles . Decide whether each statement is a definition, a convention or a derived property. Angles on a straight line sum to 180°. An interior angle is an angle inside a shape. Derived property Definition a The exterior angle of a triangle is equal to the sum of the two interior opposite angles. Level 7 I can recognise the difference between conventions, definitions and derived properties b The dashes on opposite sides of a rectangle show that the sides are the same length. c A triangle has three sides and three interior angles. d Parallel lines are marked with arrows pointing in the same direction. A Angle problems Work with a partner. Each draw a triangle with the interior and exterior angles marked. Tell your partner two of the interior angles from your triangle. Challenge them to work out the other interior angle and the exterior angles. Check their answers to see if they are correct. B Triangle properties Use a dynamic geometry program to construct a triangle with a line going through one vertex that is parallel to the opposite side. Drag any of the vertices to explore what happens. interior angle proof quadrilateral triangle 2.2 Angles and proof 23 2.3 Constructing triangles Draw an angle accurately using a protractor Construct a triangle using a protractor and a ruler Construct a triangle using compasses and a ruler Draw a right-angled triangle using compasses and a ruler Why learn this? Triangles are a stron g shape used in the constructi on of many bridges. You can construct a triangle using a ruler and a protractor if you know either two sides and the included angle (SAS) or two angles and the included side (ASA). Level 5 You can construct a triangle using a ruler and compasses if you know the length of all three sides (SSS). Level 6 hypotenuse The hypotenuse of a right-angled triangle is the longest side and is opposite the right angle. Level 6 Did you know? The word ‘triangle’ is made up of ‘tri’, which means ‘three’ and ‘angle’. A triangle has three angles. Lines that meet at right angles are perpendicular. Perpendicular lines can be constructed using compasses. Level 6 You can construct a right-angled triangle using a ruler and compasses if you know the lengths of the hypotenuse and one of the shorter sides (RHS). Level 7 Level 5 Draw these angles accurately using a ruler and protractor. Label each angle as reflex or obtuse. a 138° b 294° Construct these triangles using a ruler and protractor. c 197° a I can use a protractor to draw obtuse and reflex angles to the nearest degree d 176° b B 4 cm X I can construct a triangle given two sides and the included angle (SAS) 6 cm 72° A Make an accurate drawing of these triangles. a C 6 cm 4 cm Z b B 45° Y 85° 5 cm 73° I can construct a triangle given two angles and the included side (ASA) 60° 3 cm C A An architect is calculating the length of wood required to make trussels for a roof. The width of the roof is 5 m and the two angles to the horizontal are 88° and 65°. a Using a scale of 1 cm represents 1 m, draw an accurate scale drawing of the roof. b Measure the length of each sloping beam to find how much wood is needed for one truss. 24 Get in line acute 88° 65° 5m angle compasses These triangles are all right-angled triangles. Which letter marks the hypotenuse of each triangle? c a b c e a b d h d g Level 6 l I can identify the hypotenuse in a right-angled triangle k i j f I can construct a triangle given three sides Use compasses and a ruler to construct a triangle with sides AB ⴝ 7 cm, AC ⴝ 6 cm and BC ⴝ 5 cm. Construct a triangle with sides of length 9 cm, 7 cm and 8 cm using compasses and a ruler. Using compasses and a ruler, draw the perpendicular to the line at point A. I can use a ruler and compasses to construct the perpendicular from a point on a line segment A 4 cm 5 cm Level 7 A motor cycle stunt man is building a ramp so he can jump over four cars. Here is the side-view of his ramp. I can construct a right-angled triangle if I know the lengths of the hypotenuse and another side (RHS) 10 m 8m a Draw an accurate scale drawing of the ramp using a ruler and compasses. b What is the height of the top of the ramp? A 4 m ladder leans against a wall with its base 1 m from the wall. Watch out! 4m a Draw an accurate scale drawing of the ladder against the wall. b Use your drawing to find how far the ladder reaches up the wall. Don’t rub out your construc tion lines as they show that you have used the compasses correctly. 1m Mark wants to construct a triangle with sides of length 5 cm, 3 cm and 9 cm. Explain why Mark’s triangle is impossible to construct. Hint: Try to construct the triangle first. A B Drawing triangles 1 Drawing triangles 2 1 Draw a triangle and label the vertices A, B and C. 1 Draw a right-angled triangle using compasses and a ruler. 2 Measure the sides AB and AC. 3 Measure the angle BAC. 2 Measure the hypotenuse and one of the other sides. 4 Describe the triangle to your partner by telling them the information about the two sides and the angle. Your partner draws the triangle you have described. 3 Describe the triangle to your partner by telling them the information about the two sides and the angle. Your partner draws the triangle you have described. 5 Check your partner’s triangle with the original. 4 Check your partner’s triangle with the original. construction hypotenuse perpendicular obtuse 2.3 Constructing triangles 25 2.4 Special quadrilaterals Know the properties of quadrilaterals Solve geometrical problems involving quadrilaterals and explain the reasons Why learn this? Many buildings are made of rectangles and othe r quadrilaterals. How many different shapes can you see in this photo? Quadrilateral properties: rectangle square parallelogram rhombus isosceles trapezium kite arrowhead Level 6 When solving problems using the properties of shapes it is important to explain your reasoning. Level 6 & Level 7 Level 6 I can identify angle, side and symmetry properties of simple quadrilaterals Nathaniel said ‘A square is a rectangle’. Is this true? Explain your answer. Which of these statements are always true for a rectangle? A All its sides are equal. B It has four lines of symmetry. C It has four right angles. Copy this table. Complete it by writing each shape name in the correct position. Number of pairs of parallel sides Tip Number of lines of symmetry 0 1 2 Some cells may contain more than one shape. 4 0 1 2 Did you know? a rectangle b square c parallelogram d rhombus e kite f arrowhead g isosceles trapezium The prefix ‘quadri-’ comes from the Latin word for four. Can you think of any other words that begin with ‘quad’? Draw a rectangle and cut it out. I can identify and begin to use angle, side and symmetry properties of quadrilaterals a Cut along one of the diagonals. Rearrange the pieces to make another quadrilateral. b Write the name of the new quadrilateral that you have made. c Write one geometrical fact about this shape. 26 Get in line arrowhead isosceles trapezium kite parallelogram Level 6 Sketch an equilateral triangle in one of its sides. a Write the name of the quadrilateral that is formed. b Which of these statements are always true for this special quadrilateral? A The diagonals bisect at right angles. B The angles are all equal. C It has two pairs of parallel sides. D It has four lines of symmetry. Look at this rectangle. One of the diagonals is drawn. Work out the sizes of angles angles a, b and c. b c 30° In a rhombus, one of the angles is 40°. Work out the sizes of the other angles. x 40° The square and rhombus are quadrilaterals with equal length sides. T Look at this arrowhead. TSV ⴝ 45°, STV ⴝ 30° I can solve geometrical problems using properties of triangles and quadrilaterals 30° b TVU c SVU 45° V S In this rectangle, calculate angle EBD. Show your steps for solving this problem and explain your reasoning. U A B 72° b ABE D A Work out the sizes of these angles. Explain your reasoning. B b c a c CBE B Special quadrailaterals A game for two players. Each secretly draw a special quadrilateral. Take turns to tell each other one property of your shape. Try to guess each other’s shape. Score 1 point if you guess correctly from one property, 2 points from two properties, and so on. The player with the lowest score wins. quadrilateral rectangle rhombus C Level 7 C d I can use reasoning to solve more complex geometrical problems d BCD F A I can use reasoning to solve geometrical problems 56° E a FAB I can solve simple geometrical problems using properties of quadrilaterals Learn this y z Calculate a TUV a I can identify and begin to use angle, side and symmetry properties of quadrilaterals 70° E 65° D Parallelograms Draw a parallelogram like this. Label three angles with their sizes. Challenge your partner to work out the missing angles and explain their reasons. Use what you know about the properties of parallelograms to check their answers. square symmetry 2.4 Special quadrilaterals 27 2.5 More constructions Know the names of parts of a circle Use a straight edge and compasses to construct the perpendicular bisector of a line and an angle, and the perpendicular to a line Use a straight edge and compasses to investigate the properties of overlapping circles Why learn this? Understanding perpendicular lines can help you appreciate their use in Lines that meet at right angles are perpendicular. Perpendicular buildings and on ro ads. lines can be constructed using compasses. Level 6 The angle bisector cuts the angle in half. The perpendicular bisector cuts the line in half at right angles. Both can be constructed using compasses. Level 6 The perpendicular from a point to a line segment is the shortest distance to the line. Level 6 When the points of intersection of two identical overlapping circles are joined to the centres, a rhombus is formed. Level 7 A right-angled triangle can be constructed using a ruler and compasses if you know the length of the hypotenuse and one of the shorter sides. Level 7 Level 6 I can construct the mid-point and perpendicular bisector of a line segment Using only a ruler and compasses, draw the perpendicular bisectors of these line segments. Mark the mid-point of each line segment. a a straight line segment AB of length 6 cm b a straight line segment BD of length 8 cm A construction company is building two houses, 10 m apart. The architect’s plans look like this. 10 m a Copy the plan, using a scale of 1 cm to represent 1 m. b Construct the perpendicular bisector of the 10 m line: c A fence will be built on the perpendicular bisector. What can you say about the position of the fence? I can name the parts of a circle Copy this circle with radius 4 cm. Add these labels. a radius b diameter c chord d arc e tangent f circumference Use compasses and a ruler to draw the bisector of these angles. Learn this a an acute angle of your choice ‘Bisect’ means to cut something into two equal parts. b an angle of 90° drawn with a protractor c an obtuse angle of your choice 28 Get in line I can construct the bisector of an angle arc bisector (bisect) chord circle compasses I can construct the perpendicular from a point to a line segment Tip Copy the diagram. Using compasses and a ruler, draw the perpendicular at X. Check after you have drawn a perpendicular line to see if it looks to be at a right angle. X 3 cm Level 6 A Make a copy of this diagram. Construct the perpendicular from point A to the line. 5 cm A construction company is building a bridge across a river. Copy the diagram and draw the perpendicular from point S across the river to show where the bridge should be built. S Level 7 a Using compasses, draw two circles of radius 4 cm that overlap. b Join the centres of the circles with a straight line and draw the chord that is common to both circles. c Join the centres of the circles the points where the circles intersect. What do you notice about the quadrilateral that is formed? A I can explain how standard constructions using a ruler and compasses relate to the properties of two intersecting circles with equal radii Triangles in circles 1 Draw a circle, using compasses or dynamic geometry software. 2 Mark three points on the circumference of the circle. 3 Join up these points to make a triangle. 4 Construct the perpendicular bisector of each side of your triangle. 5 What do you notice? 6 What happens when the vertices of the triangle are moved to different points on the circumference? B Polygons in circles 1 Draw a circle, using compasses or dynamic geometry software. 2 Mark four points on the circumference of the circle. 3 Join up these points to make a quadrilateral. 4 Construct the perpendicular bisector of each side of your quadrilateral. 5 What do you notice? 6 Investigate other polygons inside a circle. diameter perpendicular radius right angle tangent 2.5 More constructions 29 2.6 Angles in polygons Find the sum of the interior and exterior angles of polygons Find an interior and exterior angle of a regular polygon Use the interior and exterior angles of regular and irregular Why learn this? polygons to solve problems Polygons are found in many places in nature. Whe n lava cools it can form colum ns in the shape of polygon s. An interior angle and its corresponding exterior angle sum to 180°. Level 5 The sum of the interior angles in an n-sided polygon is (n ⴚ 2) ⴛ 180°. Level 6 The sum of the exterior angles in any polygon is always 360°. Level 6 A regular polygon has all sides of equal length and all angles equal. Level 6 sum of interior angles The interior angle of a regular polygon ⴝ _________________________. Level 6 number of sides You can use interior and exterior angles in polygons to solve problems. Level 7 Level 6 Explain how you calculate the sum of the interior angles in I can explain how to find the interior angle sum of a polygon a a quadrilateral b a pentagon. I can explain how to calculate the interior and exterior angles of regular polygons a Explain how you find the size of an interior angle in a regular pentagon. b Explain how you find the size of an exterior angle in a regular pentagon. I can calculate the sums of the interior and exterior angles of irregular polygons a What is the sum of the interior angles in i a quadrilateral ii a pentagon iii a hexagon? b Calculate the sum of the interior angles in a 10-sided polygon. Look at this quadrilateral. At each vertex the sum of the interior and exterior angles is 180°. I E I ⴙ E ⴝ 180° Explain why this is true. a Draw a quadrilateral with the exterior angles marked, like the one in Q4. b Use a protractor to measure each exterior angle. Find the sum of the exterior angles. c Repeat parts a and b for a pentagon and a hexagon. d What do you notice about the sum of the exterior angles of a polygon? 30 Get in line exterior angle hexagon interior angle irregular polygon Level 6 a Calculate the exterior angle of a regular hexagon. I can calculate the interior and exterior angles of regular polygons b Calculate the size of each interior angle in a regular hexagon. Copy and complete this table. Regular polygon Number of sides equilateral triangle 3 square 4 regular pentagon 5 Sum of interior Size of each Sum of exterior Size of each angles interior angle angles exterior angle 180° 360° regular hexagon regular octagon a How do you find the sum of the interior angles in an n-sided polygon? b Calculate the size of the interior and exterior angles in a regular 16-sided shape. a The exterior angle of a regular polygon is 18°. ii How many sides does the polygon have? ii Calculate the size of each interior angle. Level 7 Learn this b The interior angle of a regular polygon is 156°. How many sides does the polygon have? The exterior angles of a polygon always add up to 360°. It is possible to draw a polygon that has interior angles that sum to 1300°? Explain your reasoning The diagram shows a regular octagon. The line BC is parallel to the line AD. Calculate the size of a BCD b CDA B C A H D I can solve harder problems using properties of angles, parallel and intersecting lines, and triangles and other polygons E 67.5° G c ADH I can use the interior and exterior angles of regular polygons to solve problems F Did you know? A Polygon poster Make a poster of all the facts you know about the interior and exterior angles of polygons. B Polygons are used to create complex-shaped computer graphics. Next time you play a computer game, see how many polygons you can spot. Tessellating polygons Investigate which regular polygons tessellate. Look at the interior angles. How can you tell by looking at the interior angles whether a shape will tessellate? Why will a regular hexagon and a square tessellate? octagon pentagon quadrilateral regular polygon triangle 2.6 Angles in polygons 31