1 KS3 Mathematics S1 Lines and Angles
2 Contents S1 Lines and angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines 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 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 Labelling angles 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. This angle can be described as ABC or ABC or B.
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. For example, an equilateral triangle has three equal sides and three equal angles. A derived property follows from a definition. 60 60 60 For example, the angles in an equilateral triangle are each 60.
7 Convention, definition or derived property?
8 Contents S1 Lines and angles 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 cross, or intersect. These lines do not 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? d a c b a = b = c = d Each angle is 90. We show this with a small square in each corner. Lines that intersect at right angles are called perpendicular lines.
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 Contents S1 Lines and angles 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. 90 A quarter turn measures 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. 180 This is a straight line.
20 Angles Angles are measured in degrees. 270 A three-quarter turn measures 270.
21 Angles Angles are measured in degrees. 360 A full turn measures 360.
22 Intersecting lines
23 Vertically opposite angles When two lines intersect, two pairs of vertically opposite angles are formed. a d b c a = c and 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. a b d c a + b + c + d = 360 because there are 360 in a full turn.
Calculating angles around a point Use geometrical reasoning to find the size of the labelled angles. a 167 103 69 68 d 43 b 43 c 137 28
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. a b 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. b c a d f g e h 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. b a c d f e g h d = h because Corresponding angles are equal
34 Corresponding angles There are four pairs of corresponding angles, or F-angles. b c a d f g e h a = e because Corresponding angles are equal
35 Corresponding angles There are four pairs of corresponding angles, or F-angles. b c a d f g e h c = g because Corresponding angles are equal
36 Corresponding angles There are four pairs of corresponding angles, or F-angles. b c a d f g e h b = f because Corresponding angles are equal
37 Alternate angles There are two pairs of alternate angles, or Z-angles. b c a d f g e h d = f because Alternate angles are equal
38 Alternate angles There are two pairs of alternate angles, or Z-angles. b c a d f g e h c = e because Alternate angles are equal
39 Calculating angles Calculate the size of angle a. 29º a 46º Hint: Add another line. a = 29º + 46º = 75º
40 Contents S1 Lines and angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons
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 c b 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. 116 33 a 64 b 31 82 326 43 88 c 49 25 d 233 28
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 For example, 88 46 a 46 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. The line segments are called sides. B A E The end points are called vertices. One of these is called a vertex. 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. Number of sides 3 4 5 6 7 8 9 10 Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon
49 Interior angles in polygons The angles inside a polygon are called interior angles. b c a 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
51 Interior and exterior angles in a triangle Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. c b ca b 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.
52 Interior and exterior angles in a triangle
53 Calculating angles Calculate the size of the lettered angles in each of the following triangles. 33 a 116 b 64 31 82 34 25 d 127 152 43 c 131 272
54 Calculating angles Calculate the size of the lettered angles in this diagram. 56 38º 38º 86 a 69 b 104 73 Base angles in the isosceles triangle = (180º 104º) 2 = 76º 2 = 38º Angle a = 180º 56º 38º = 86º Angle b = 180º 73º 38º = 69º
55 Sum of the interior angles in a quadrilateral What is the sum of the interior angles in a quadrilateral? c d f a b e We can work this out by dividing the quadrilateral into two triangles. 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.
56 Sum of interior angles in a polygon We already know that the sum of the interior angles in any triangle is 180. c a + b + c = 180 a b a b d c We have just shown that the sum of the interior angles in any quadrilateral is 360. a + b + c + d = 360 Do you know the sum of the interior angles for any other polygons?
57 Sum of the interior angles in a pentagon What is the sum of the interior angles in a pentagon? a b c e d h f g i We can work this out by using lines from one vertex to divide the pentagon into three triangles. a + b + c = 180 and d + e + f = 180 So, (a + b + c) + (d + e + f ) + (g + h + i) = 560 and g + h + i = 180 The sum of the interior angles in a pentagon is 560.
58 Sum of the interior angles in a polygon We ve seen that a quadrilateral can be divided into two triangles and a pentagon can be divided into three triangles. A How hexagon many triangles can be divided can a hexagon into four be divided triangles. into?
59 Sum of the interior angles in a polygon The number of triangles that a polygon can be divided into is always two less than the number of sides. We can say that: A polygon with n sides can be divided into (n 2) triangles. The sum of the interior angles in a triangle is 180. So, The sum of the interior angles in an n-sided polygon is (n 2) 180.
60 Interior angles in regular polygons A regular polygon has equal sides and equal angles. We can work out the size of the interior angles in a regular polygon as follows: Name of regular polygon Sum of the interior angles Size of each interior angle Equilateral triangle 180 180 3 = 60 Square 2 180 = 360 360 4 = 90 Regular pentagon 3 180 = 540 540 5 = 108 Regular hexagon 4 180 = 720 720 6 = 120
61 Interior and exterior angles in an equilateral triangle In an equilateral triangle, 120 60 Every interior angle measures 60. Every exterior angle measures 120. 120 60 60 120 The sum of the interior angles is 3 60 = 180. The sum of the exterior angles is 3 120 = 360.
62 Interior and exterior angles in a square In a square, 90 90 90 90 Every interior angle measures 90. Every exterior angle measures 90. 90 90 90 90 The sum of the interior angles is 4 90 = 360. The sum of the exterior angles is 4 90 = 360.
63 Interior and exterior angles in a regular pentagon In a regular pentagon, 72 72 108 108 108 72 108 108 72 72 Every interior angle measures 108. Every exterior angle measures 72. The sum of the interior angles is 5 108 = 540. The sum of the exterior angles is 5 72 = 360.
64 Interior and exterior angles in a regular hexagon In a regular hexagon, 60 60 120 120 120 120 60 120 120 60 60 60 Every interior angle measures 120. Every exterior angle measures 60. The sum of the interior angles is 6 120 = 720. The sum of the exterior angles is 6 60 = 360.
65 The sum of exterior angles in a polygon For any polygon, the sum of the interior and exterior angles at each vertex is 180. For n vertices, the sum of n interior and n exterior angles is n 180 or 180n. The sum of the interior angles is (n 2) 180. We can write this algebraically as 180(n 2) = 180n 360.
66 The sum of exterior angles in a polygon If the sum of both the interior and the exterior angles is 180n and the sum of the interior angles is 180n 360, the sum of the exterior angles is the difference between these two. The sum of the exterior angles = 180n (180n 360 ) = 180n 180n + 360 = 360 The sum of the exterior angles in a polygon is 360.
67 Take Turtle for a walk
68 Find the number of sides
69 Calculate the missing angles 50º This pattern has been made with three different shaped tiles. The length of each side is the same. What shape are the tiles? Calculate the sizes of each angle in the pattern and use this to show that the red tiles must be squares. = 50º = 40º = 130º = 140º = 140º = 150º