# S1 Lines and angles Contents S1.1 Labelling lines and angles

How to calculate the sum of interior angles of a hexagon
How to calculate the sum of interior angles of a hexagon

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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

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Angles in a triangle Change the triangle by moving the vertex. Pressing the play button will divide the triangle into three pieces. Pressing play again will rearrange the pieces so that the three vertices come together to form a straight line. Conclude that the angles in a triangle always add up to 180º. Pupil can replicate this result by taking a triangle cut out of a piece of paper, tearing off each of the corners and rearranging them to make a straight line.

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

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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. Discuss this proof that angles in a triangle have a sum of 180º. 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°.

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Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles. 64° b 116° 33° a 31° 326° 82° Ask pupils to calculate the size of the missing angles before revealing them. 49° 43° 25° d 88° c 28° 233°

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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.

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Angles in an isosceles triangle
For example, 88° a 46° 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. As an alternative to using algebra we could use the following argument. The three angles add up to 180º, so the two unknown angles must add up to 180º – 88º, that’s 92º. The two angles are the same size, so each must measure half of 92º or 46º. 88° + a + a = 180° 88° + 2a = 180° 2a = 92° a = 46°

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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 These two dimensions are length and width. A polygon has no height. 2-D stands for two-dimensional.

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Naming polygons Polygons are named according to the number of sides they have. Number of sides Name of polygon 3 4 5 6 7 8 9 10 Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon

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

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

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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 c a b 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.

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Interior and exterior angles in a triangle
Drag the vertices of the triangle to show that the exterior angle is equal to the sum of the opposite interior angles. Hide angles by clicking on them and ask pupils to calculate their sizes.

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

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

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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°.

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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 We have just shown that the sum of the interior angles in any quadrilateral is 360°. a b c d Pupils should be able to understand a proof that the that the exterior angle is equal to the sum of the two interior opposite angles. Framework reference p183 a + b + c + d = 360 ° Do you know the sum of the interior angles for any other polygons?

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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 h 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° 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°.

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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?

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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°.

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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° Ask pupils to complete the table for regular polygons with up to 10 sides. Regular pentagon 3 × 180° = 540° 540° ÷ 5 = 108° Regular hexagon 4 × 180° = 720° 720° ÷ 6 = 120°

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

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Interior and exterior angles in a square
Every interior angle measures 90°. 90° 90° 90° 90° Every exterior angle measures 90°. The sum of the interior angles is 4 × 90° = 360°. 90° 90° 90° The sum of the exterior angles is 4 × 90° = 360°. 90°

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Interior and exterior angles in a regular pentagon
Every interior angle measures 108°. 108° 72° 72° Every exterior angle measures 72°. 108° 108° 72° The sum of the interior angles is 5 × 108° = 540°. 108° 108° 72° 72° The sum of the exterior angles is 5 × 72° = 360°.

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Interior and exterior angles in a regular hexagon
Every interior angle measures 120°. 60° 120° 120° 60° Every exterior angle measures 60°. 60° 120° 120° 60° The sum of the interior angles is 6 × 120° = 720°. 120° 120° 60° 60° The sum of the exterior angles is 6 × 60° = 360°.

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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°.

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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° Discuss this algebraic proof that the sum of the exterior angles in a polygon is always 360°. = 360° The sum of the exterior angles in a polygon is 360°.

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Take Turtle for a walk Use this activity to demonstrate that the sum of the exterior angles in a convex polygon is always 360º. Select the polygon required by choosing the number of sides and drag the vertices to make a convex polygon. Hitting the turtle button will make Turtle walk around the outside of the shape. As Turtle walks around the outside of the shape ask pupils to estimate the size of the next exterior angle. This activity is ideal for getting pupils to think about the size of exterior angles and would make a good introduction to drawing polygons using Logo.

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Find the number of sides
Challenge pupils to find the number of sides in a regular polygon given the size of one of its interior or exterior angles. Establish that if we are given the size of the exterior angle we have to divide this number into 360° to find the number of sides. This is because the sum of the exterior angles in a polygon is always 360° and each exterior angle is equal. Establish that if we are given the size of the interior angle we have to divide 360° by (180° – the size of the interior angle) to find the number of sides. This is because the interior angles in a regular polygon can be found by subtracting 360° divided by the number of sides from 180°.

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Calculate the missing angles
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. Start by establishing that all of the tiles are rhombuses. Establish that all of the dark purple angles must be equal and so equal 50º. The dark green angles must all be equal because the green tiles are also rhombuses. We can calculate the size of the dark green angle as follows. The angles around the point at the centre of the pattern equal 360º. Four of these angle measure 50º so the four dark green angles must together measure 360º – (4 × 50º) = 160º. One dark green angle therefore measures 160º ÷ 4 = 40º. The light purple angle can be found by considering the sum of the angles in a quadrilateral or by deducing that this angle plus the dark purple angle must add up to 180º because the opposite sides of a rhombus are parallel. The light purple angles therefore equal 130º. Using similar reasoning the light green angles measure 140º. Using the fact that angles around a point equal 360º we can deduce that the angle inside the red shape is a right angle. If one angle inside a rhombus is a right angle, all of the interior angles must be right angles and so the rhombus must be square. Using the fact that angles around a point equal 360º we can deduce that the dark yellow angle is 140º and the light yellow angle is 150º. Ask pupils how we could show that the red tiles are squares without finding any of the angles. Link: S2 2-D shapes – quadrilaterals = 50º = 40º = 130º = 140º = 140º = 150º

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