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1 1 of 69
2 2 of 69 Intersecting lines
3 3 of 69 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.
4 4 of 69 Angles on a straight line
5 5 of 69 Angles around a point
6 6 of 69 Angles made with parallel lines
7 7 of 69 Angles in a triangle
8 8 of 69 Angles in a triangle c a b For any triangle, a + b + c = 180 The angles in a triangle add up to 180.
9 9 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. 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.
10 10 of 69 Calculating angles in a triangle Calculate the size of the missing angles in each of the following triangles a 64 b c d
11 11 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 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.
12 12 of 69 Angles in an isosceles triangle For example, 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 a + a = a = 180 2a = 92 a = 46
13 13 of 69 Polygons A polygon is a 2-D shape made when line segments enclose a region. The line segments are called sides. B C A D E Each end point is called a vertex. We call more than one vertices. 2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no thickness.
14 14 of 69 Naming polygons Polygons are named according to the number of sides they have. Number of sides Name of polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon
15 15 of 69 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.
16 16 of 69 Exterior angles in polygons When we extend the sides of a polygon outside the shape exterior angles are formed. e f d
17 17 of 69 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 c a 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.
18 18 of 69 Interior and exterior angles in a triangle
19 19 of 69 Calculating angles Calculate the size of the lettered angles in each of the following triangles. 33 a 116 b d c
20 20 of 69 Calculating angles Calculate the size of the lettered angles in this diagram º 38º 86 a 69 b Base angles in the isosceles triangle = (180º 104º) 2 = 76º 2 = 38º Angle a = 180º 56º 38º = 86º Angle b = 180º 73º 38º = 69º
21 21 of 69 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.
22 22 of 69 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?
23 23 of 69 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) = 540 and g + h + i = 180 The sum of the interior angles in a pentagon is 540.
24 24 of 69 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 into hexagon four triangles. be divided into?
25 25 of 69 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.
26 26 of 69 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 = 60 Square = = 90 Regular pentagon = = 108 Regular hexagon = = 120
27 27 of 69 Interior and exterior angles in an equilateral triangle In an equilateral triangle, Every interior angle measures 60. Every exterior angle measures The sum of the interior angles is 3 60 = 180. The sum of the exterior angles is = 360.
28 28 of 69 Interior and exterior angles in a square In a square, Every interior angle measures 90. Every exterior angle measures The sum of the interior angles is 4 90 = 360. The sum of the exterior angles is 4 90 = 360.
29 29 of 69 Interior and exterior angles in a regular pentagon In a regular pentagon, Every interior angle measures 108. Every exterior angle measures 72. The sum of the interior angles is = 540. The sum of the exterior angles is 5 72 = 360.
30 30 of 69 Interior and exterior angles in a regular hexagon In a regular hexagon, Every interior angle measures 120. Every exterior angle measures 60. The sum of the interior angles is = 720. The sum of the exterior angles is 6 60 = 360.
31 31 of 69 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.
32 32 of 69 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 The sum of the exterior angles in a polygon is 360.
33 33 of 69 Take Turtle for a walk
34 34 of 69 Find the number of sides
35 35 of 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
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