This document provides an overview of key concepts related to lines, angles, and shapes in geometry: 1. It defines lines, line segments, and angles, and explains how they are labeled. 2. It describes parallel and perpendicular lines, and explores properties like corresponding angles. 3. It covers calculating and classifying angles, such as complementary, supplementary, and vertically opposite angles. 4. It examines angles in triangles and quadrilaterals, noting the sums of interior and exterior angles.

1. KS3 Mathematics S1 Lines and Angles

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 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’. A B

5. 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 Sometimes instead an angle is labelled with a lower case letter. The angle can then be described as ABC or ABC or B.

7. 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. 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. 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. 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. Parallel or perpendicular?

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

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

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

17. Getting to know angles Use SMILE programs Angle 90 and Angle 360 To get to know angles.

19. Intersecting lines

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

21. Angles on a straight line

22. 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. Angles around a point

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

26. You can use the facts you have learnt to calculate angles. Work out the answers to the following ten ticks questions.

27. 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. 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. 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. Angles made with parallel lines

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

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

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

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

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

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

38. Calculating angles involving parallel lines. Calculate these angles from this ten ticks worksheet.

40. Angles in a triangle

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

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

44. Calculating angles in a triangle. Calculate the angles shown on this ten ticks worksheet.

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 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. 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° + 2 a = 180° 2 a = 92° a = 46° 46° 46°

47. Calculating angles in special triangles. Calculate the angles on this ten ticks worksheet.

48. Interior angles in triangles The angles inside a triangle are called interior angles . The sum of the interior angles of a triangle is 180°. c a b

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

50. 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. Interior and exterior angles in a triangle

52. Calculating angles Calculate the size of the lettered angles in each of the following triangles. 82° 31° 64° 34° a b 33° 116° 152° d 25° 127° 131° c 272° 43°

53. Calculating angles Calculate the size of the lettered angles in this diagram. 56° a 73° b 86° 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. 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. 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 a + b + c + d = 360 ° We have just shown that the sum of the interior angles in any quadrilateral is 360°. a b c d

56. Interior and exterior angles in an equilateral triangle In an equilateral triangle, Every interior angle measures 60°. Every exterior angle measures 120°. The sum of the interior angles is 3 × 60° = 180°. The sum of the exterior angles is 3 × 120° = 360°. 60° 60° 120° 120° 60° 120°

57. 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°