Upcoming SlideShare
×

# Ks3 mathematics

• 2,194 views

linne an andles …

linne an andles

• Comment goes here.
Are you sure you want to
Be the first to comment

Total Views
2,194
On Slideshare
0
From Embeds
0
Number of Embeds
0

Shares
0
0
Likes
2

No embeds

### Report content

No notes for slide

### Transcript

• 1. © Boardworks Ltd 20041 of 69 KS3 Mathematics S1 Lines and Angles
• 2. © Boardworks Ltd 20042 of 69 A A SS1.4 A Contents S1 Lines and angles S1.1 Labelling lines and angles S1.3 Calculating angles S1.2 Parallel and perpendicular lines S1.4 Calculating angles in triangles andquadrilaterals
• 3. © Boardworks Ltd 20043 of 69 Lines In Mathematics, a straight line is defined as having infinite length and no width. Is this possible in real life?
• 4. © Boardworks Ltd 20044 of 69 Labelling line segments When a line has end points we say that it has finite length. It is called a line segment. We usually label the end points with capital letters. For example, this line segment A B has end points A and B. We can call this line ‘line segment AB’.
• 5. © Boardworks Ltd 20045 of 69 Labelling angles When two lines meet at a point an angle is formed. An angle is a measure of the rotation of one of the line segments relative to the other. We label points using capital letters. A B C The angle can then be described as ABC or ABC or B. Sometimes instead an angle is labelled with a lower case letter.
• 6. © Boardworks Ltd 20046 of 69 Contents A A A S1.2 Parallel and perpendicular lines S1.1 Labelling lines and angles S1.3 Calculating angles S1 Lines and angles S1.4 Calculating angles in triangles andquadrilaterals
• 7. © Boardworks Ltd 20047 of 69 Lines in a plane What can you say about these pairs of lines? These lines cross, or intersect. These lines do not intersect. They are parallel.
• 8. © Boardworks Ltd 20048 of 69 Lines in a plane A flat two-dimensional surface is called a plane. Any two straight lines in a plane either intersect once … This is called the point of intersection.
• 9. © Boardworks Ltd 20049 of 69 Lines in a plane … or they are parallel. We use arrow heads to show that lines are parallel. Parallel lines will never meet. They stay an equal distance apart. Where do you see parallel lines in everyday life? We can say that parallel lines are always equidistant.
• 10. © Boardworks Ltd 200410 of 69 Perpendicular lines What is special about the angles at the point of intersection here? a = b = c = d Lines that intersect at right angles are called perpendicular lines. a b c d Each angle is 90 . We show this with a small square in each corner.
• 11. © Boardworks Ltd 200411 of 69 Parallel or perpendicular?
• 12. © Boardworks Ltd 200412 of 69 Contents A A AS1.3 Calculating angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1 Lines and angles S1.4 Calculating angles in triangles andquadrilaterals
• 13. © Boardworks Ltd 200413 of 69 Angles Angles are measured in degrees. A quarter turn measures 90 . It is called a right angle. We label a right angle with a small square. 90
• 14. © Boardworks Ltd 200414 of 69 Angles Angles are measured in degrees. A half turn measures 180 . This is a straight line.180
• 15. © Boardworks Ltd 200415 of 69 Angles Angles are measured in degrees. A three-quarter turn measures 270 . 270
• 16. © Boardworks Ltd 200416 of 69 Angles Angles are measured in degrees. A full turn measures 360 .360
• 17. © Boardworks Ltd 200417 of 69 Getting to know angles Use SMILE programs Angle 90 and Angle 360 To get to know angles.
• 18. © Boardworks Ltd 200418 of 69 You must learn facts about angles. So you can calculate their size without drawing or measuring. • Learn facts about • Angles between intersecting lines • Angles on a straight line • Angles around a point
• 19. © Boardworks Ltd 200419 of 69 Intersecting lines
• 20. © Boardworks Ltd 200420 of 69 Vertically opposite angles When two lines intersect, two pairs of vertically opposite angles are formed. a b c d a = c and b = d Vertically opposite angles are equal.
• 21. © Boardworks Ltd 200421 of 69 Angles on a straight line
• 22. © Boardworks Ltd 200422 of 69 Angles on a straight line Angles on a line add up to 180 . a + b = 180 a b because there are 180 in a half turn.
• 23. © Boardworks Ltd 200423 of 69 Angles around a point
• 24. © Boardworks Ltd 200424 of 69 Angles around a point Angles around a point add up to 360 . a + b + c + d = 360 a b c d because there are 360 in a full turn.
• 25. © Boardworks Ltd 200425 of 69 b c d 43 43 68 Calculating angles around a point Use geometrical reasoning to find the size of the labelled angles. 103 a167 137 69
• 26. © Boardworks Ltd 200426 of 69 You can use the facts you have learnt to calculate angles. Work out the answers to the following ten ticks questions.
• 27. © Boardworks Ltd 200427 of 69 Complementary angles When two angles add up to 90 they are called complementary angles. a b a + b = 90 Angle a and angle b are complementary angles.
• 28. © Boardworks Ltd 200428 of 69 Supplementary angles When two angles add up to 180 they are called supplementary angles. a b a + b = 180 Angle a and angle b are supplementary angles.
• 29. © Boardworks Ltd 200429 of 69 Angles made with parallel lines When a straight line crosses two parallel lines eight angles are formed. Which angles are equal to each other? a b c d e f g h
• 30. © Boardworks Ltd 200430 of 69 Angles made with parallel lines
• 31. © Boardworks Ltd 200431 of 69 dd hh a b c e f g Corresponding angles There are four pairs of corresponding angles, or F-angles. a b c e f g d = h because Corresponding angles are equal
• 32. © Boardworks Ltd 200432 of 69 ee aa b c d f g h Corresponding angles There are four pairs of corresponding angles, or F-angles. b c d f g h a = e because Corresponding angles are equal
• 33. © Boardworks Ltd 200433 of 69 gg cc Corresponding angles There are four pairs of corresponding angles, or F-angles. c = g because a b d e f h Corresponding angles are equal
• 34. © Boardworks Ltd 200434 of 69 ff Corresponding angles There are four pairs of corresponding angles, or F-angles. b = f because a b c d e g h b Corresponding angles are equal
• 35. © Boardworks Ltd 200435 of 69 ff dd Alternate angles There are two pairs of alternate angles, or Z-angles. d = f because Alternate angles are equal a b c e g h
• 36. © Boardworks Ltd 200436 of 69 cc ee Alternate angles There are two pairs of alternate angles, or Z-angles. c = e because a b g h d f Alternate angles are equal
• 37. © Boardworks Ltd 200437 of 69 Calculating angles Calculate the size of angle a. a 29º 46º Hint: Add another line. a = 29º + 46º = 75º
• 38. © Boardworks Ltd 200438 of 69 Calculating angles involving parallel lines. Calculate these angles from this ten ticks worksheet.
• 39. © Boardworks Ltd 200439 of 69 Contents A A A A S1.4 Angles in triangles and quadrilaterals S1.1 Labelling lines and angles S1.3 Calculating angles S1.2 Parallel and perpendicular lines S1 Lines and angles
• 40. © Boardworks Ltd 200440 of 69 Angles in a triangle
• 41. © Boardworks Ltd 200441 of 69 Angles in a triangle For any triangle, a b c a + b + c = 180 The angles in a triangle add up to 180 .
• 42. © Boardworks Ltd 200442 of 69 Angles in a triangle We can prove that the sum of the angles in a triangle is 180 by drawing a line parallel to one of the sides through the opposite vertex. These angles are equal because they are alternate angles. a a b b Call this angle c. c a + b + c = 180 because they lie on a straight line. The angles a, b and c in the triangle also add up to 180 .
• 43. © Boardworks Ltd 200443 of 69 Calculating angles in a triangle Calculate the size of the missing angles in each of the following triangles. 233 8231 116 326 43 49 28 a b c d 33 64 88 25
• 44. © Boardworks Ltd 200444 of 69 Calculating angles in a triangle. Calculate the angles shown on this ten ticks worksheet.
• 45. © Boardworks Ltd 200445 of 69 Angles in an isosceles triangle In an isosceles triangle, two of the sides are equal. We indicate the equal sides by drawing dashes on them. The two angles at the bottom of the equal sides are called base angles. The two base angles are also equal. If we are told one angle in an isosceles triangle we can work out the other two.
• 46. © Boardworks Ltd 200446 of 69 Angles in an isosceles triangle For example, Find the sizes of the other two angles. The two unknown angles are equal so call them both a. We can use the fact that the angles in a triangle add up to 180 to write an equation. 88 + a + a = 180 88 a a 88 + 2a = 180 2a = 92 a = 46 46 46
• 47. © Boardworks Ltd 200447 of 69 Calculating angles in special triangles. Calculate the angles on this ten ticks worksheet.
• 48. © Boardworks Ltd 200448 of 69 Interior angles in triangles c a b The angles inside a triangle are called interior angles. The sum of the interior angles of a triangle is 180 .
• 49. © Boardworks Ltd 200449 of 69 Exterior angles in triangles f d e When we extend the sides of a polygon outside the shape exterior angles are formed.
• 50. © Boardworks Ltd 200450 of 69 Interior and exterior angles in a triangle a b c Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. a = b + c We can prove this by constructing a line parallel to this side. These alternate angles are equal. These corresponding angles are equal. b c
• 51. © Boardworks Ltd 200451 of 69 Interior and exterior angles in a triangle
• 52. © Boardworks Ltd 200452 of 69 Calculating angles Calculate the size of the lettered angles in each of the following triangles. 823164 34 a b 33 116 152 d25 127 131 c 272 43
• 53. © Boardworks Ltd 200453 of 69 Calculating angles Calculate the size of the lettered angles in this diagram. 56 a 73 b86 69 104 Base angles in the isosceles triangle = (180º – 104º) 2 = 76º 2 = 38º 38º 38º Angle a = 180º – 56º – 38º = 86º Angle b = 180º – 73º – 38º = 69º
• 54. © Boardworks Ltd 200454 of 69 Sum of the interior angles in a quadrilateral c a b What is the sum of the interior angles in a quadrilateral? We can work this out by dividing the quadrilateral into two triangles. d f e a + b + c = 180 and d + e + f = 180 So, (a + b + c) + (d + e + f ) = 360 The sum of the interior angles in a quadrilateral is 360 .
• 55. © Boardworks Ltd 200455 of 69 Sum of interior angles in a polygon We already know that the sum of the interior angles in any triangle is 180 . a + b + c = 180 Do you know the sum of the interior angles for any other polygons? a b c We have just shown that the sum of the interior angles in any quadrilateral is 360 . a b c d a + b + c + d = 360
• 56. © Boardworks Ltd 200456 of 69 Interior and exterior angles in an equilateral triangle In an equilateral triangle, 60 60 Every interior angle measures 60 . Every exterior angle measures 120 . 120 120 60 120 The sum of the interior angles is 3 60 = 180 . The sum of the exterior angles is 3 120 = 360 .
• 57. © Boardworks Ltd 200457 of 69 Interior and exterior angles in a square In a square, Every interior angle measures 90 . Every exterior angle measures 90 . The sum of the interior angles is 4 90 = 360 . The sum of the exterior angles is 4 90 = 360 . 90 90 90 90 90 90 90 90