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°.

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°

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.

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°

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.

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

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°.

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

Interior and exterior angles in a triangle c b Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. 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.

Interior and exterior angles in a triangle

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

Calculating angles Calculate the size of the lettered angles in this diagram. 38º 38º 56° 73° 86° a b 69° 104° Base angles in the isosceles triangle = (180º – 104º) ÷ 2 = 76º ÷ 2 = 38º = 86º Angle a = 180º – 56º – 38º = 69º Angle b = 180º – 73º – 38º

Sum of the interior angles in a quadrilateral What is the sum of the interior angles in a quadrilateral? d c f a e b 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°.

Sum of interior angles in a polygon We have just shown that the sum of the interior angles in any quadrilateral is 360°. d a c b We already know that the sum of the interior angles in any triangle is 180°. c a + b + c = 180 ° a b a + b + c + d = 360 ° Do you know the sum of the interior angles for any other polygons?

Sum of the interior angles in a pentagon What is the sum of the interior angles in a pentagon? c d a f g b e i h 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° and g + h + i = 180° So, (a + b + c)+ (d + e + f ) + (g + h + i) = 540° The sum of the interior angles in a pentagon is 540°.

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 hexagon can be divided into four triangles. How many triangles can a hexagon be divided into?

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°.

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: 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°

Interior and exterior angles in an equilateral triangle 120° 60° 120° 60° 60° 120° 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°.

Interior and exterior angles in a square 90° 90° 90° 90° 90° 90° 90° 90° 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°.

Interior and exterior angles in a regular pentagon 72° 72° 108° 108° 108° 72° 108° 108° 72° 72° In a regular pentagon, 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°.

Interior and exterior angles in a regular hexagon 60° 60° 60° 120° 120° 120° 120° 60° 120° 120° 60° 60° In a regular hexagon, 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°.

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°. 180n° – 360°. We can write this algebraically as 180(n – 2)° =

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°.

Take Turtle for a walk

Find the number of sides

Calculate the missing angles = 50º = 40º = 130º = 140º = 140º = 150º This pattern has been made with three different shaped tiles. The length of each side is the same. 50º 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.

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