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KS3 Mathematics

KS3 Mathematics

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KS3 Mathematics

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  1. KS3 Mathematics S1 Lines and Angles

  2. S1 Lines and angles Contents S1.2 Parallel and perpendicular lines S1.1 Labelling lines and angles S1.3 Calculating angles S1.4 Angles in polygons

  3. Lines In Mathematics, a straight line is defined as having infinite length and no width. Is this possible in real life?

  4. Labelling line segments A B 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 has end points A and B. We can call this line, line segment AB.

  5. Labelling angles or ABC or B. This angle can be described as ABC When two lines meet at a point an angle is formed. A B C An angle is a measure of the rotation of one of the line segments to the other. We label angles using capital letters.

  6. Conventions, definitions and derived properties A convention is an agreed way of describing a situation. For example, we use dashes on lines to show that they are the same length. A definition is a minimum set of conditions needed to describe something. 60° For example, an equilateral triangle has three equal sides and three equal angles. 60° 60° A derived property follows from a definition. For example, the angles in an equilateral triangle are each 60°.

  7. Convention, definition or derived property?

  8. S1 Lines and angles Contents S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons

  9. Lines in a plane What can you say about these pairs of lines? These lines do not intersect. These lines cross, or intersect. They are parallel.

  10. 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.

  11. 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. This means that they are always equidistant. Where do you see parallel lines in everyday life?

  12. Perpendicular lines What is special about the angles at the point of intersection here? a a = b = c = d b d Each angle is 90. We show this with a small square in each corner. c Lines that intersect at right angles are called perpendicularlines.

  13. Parallel or perpendicular?

  14. The distance from a point to a line What is the shortest distance from a point to a line? O The shortest distance from a point to a line is always the perpendicular distance.

  15. Drawing perpendicular lines with a set square We can draw perpendicular lines using a ruler and a set square. Draw a straight line using a ruler. Place the set square on the ruler and use the right angle to draw a line perpendicular to this line.

  16. Drawing parallel lines with a set square We can also draw parallel lines using a ruler and a set square. Place the set square on the ruler and use it to draw a straight line perpendicular to the ruler’s edge. Slide the set square along the ruler to draw a line parallel to the first.

  17. S1 Lines and angles Contents S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons

  18. Angles Angles are measured in degrees. A quarter turn measures 90°. 90° It is called a right angle. We label a right angle with a small square.

  19. Angles Angles are measured in degrees. A half turn measures 180°. This is a straight line. 180°

  20. Angles Angles are measured in degrees. A three-quarter turn measures 270°. 270°

  21. Angles Angles are measured in degrees. A full turn measures 360°. 360°

  22. Intersecting lines

  23. Vertically opposite angles a d b c When two lines intersect, two pairs of vertically opposite angles are formed. and a = c b = d Vertically opposite angles are equal.

  24. Angles on a straight line

  25. Angles on a straight line Angles on a line add up to 180. a b a + b = 180° because there are 180° in a half turn.

  26. Angles around a point

  27. Angles around a point Angles around a point add up to 360. b a c d a + b + c + d = 360 because there are 360 in a full turn.

  28. Calculating angles around a point Use geometrical reasoning to find the size of the labelled angles. 69° 68° d 167° a 43° c 43° b 103° 137°

  29. 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.

  30. Supplementary angles When two angles add up to 180° they are called supplementary angles. b a a + b = 180° Angle a and angle b are supplementary angles.

  31. Angles made with parallel lines When a straight line crosses two parallel lines eight angles are formed. a b d c e f h g Which angles are equal to each other?

  32. Angles made with parallel lines

  33. Corresponding angles There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h g g d = h because Corresponding angles are equal

  34. Corresponding angles There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h g g a = e because Corresponding angles are equal

  35. Corresponding angles There are four pairs of corresponding angles, or F-angles. a b d c c e f h g g c = g because Corresponding angles are equal

  36. Corresponding angles There are four pairs of corresponding angles, or F-angles. a b b d c e f f h g b = f because Corresponding angles are equal

  37. Alternate angles There are two pairs of alternate angles, or Z-angles. a b d d c e f f h g d = f because Alternate angles are equal

  38. Alternate angles There are two pairs of alternate angles, or Z-angles. a b d c c e e f h g c = e because Alternate angles are equal

  39. Calculating angles Calculate the size of angle a. 29º Hint: Add another line. a 46º 75º a = 29º + 46º =

  40. S1 Lines and angles Contents S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.4 Angles in polygons S1.3 Calculating angles

  41. Angles in a triangle

  42. Angles in a triangle c a b For any triangle, a + b + c = 180° The angles in a triangle add up to 180°.

  43. 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. a b c a b These angles are equal because they are alternate angles. Call this angle 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°.

  44. Calculating angles in a triangle Calculate the size of the missing angles in each of the following triangles. 64° b 116° 33° a 326° 31° 82° 49° 43° 25° d 88° c 28° 233°

  45. 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 on 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. Angles in an isosceles triangle 46° 46° For example, 88° a a Find the size 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° + 2a = 180° 2a = 92° a = 46°

  47. Polygons A polygon is a 2-D shape made when line segments enclose a region. A The end points are called vertices. One of these is called a vertex. B The line segments are called sides. E C D 2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no height.

  48. Naming polygons Polygons are named according to the number of sides they have. Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon

  49. Interior angles in polygons b c a The angles inside a polygon are called interior angles. The sum of the interior angles of a triangle is 180°.

  50. Exterior angles in polygons When we extend the sides of a polygon outside the shape exterior angles are formed. e d f