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**KS3 Mathematics S1 Lines and Angles**

The aim of this unit is to teach pupils to: Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions and derived properties Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp S1 Lines and Angles

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**S1.1 Labelling lines and angles**

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

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**Is this possible in real life?**

Lines In Mathematics, a straight line is defined as having infinite length and no width. A line is the shortest distance between two points. Mathematically, a line only has one dimension, length and no width. We cannot draw a line like this in real life because it would be invisible. The two arrows at either end indicate that the line is infinite. We could not draw an infinitely long line in reality. Is this possible in real life?

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**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.

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**Labelling angles When two lines meet at a point an angle is formed. A**

C An angle is a measure of the rotation of one of the line segments to the other. Pupils often find the naming of angles difficult particularly when there is more than one angle at a point. At key stage 3 this confusion is often avoided by using single lower case letters to name angles. We label angles using capital letters. This angle can be described as ABC or ABC or B.

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**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. 60° For example, an equilateral triangle has three equal sides and three equal angles. Discuss the difference between a convention, a definition and a derived property. 60° 60° A derived property follows from a definition. For example, the angles in an equilateral triangle are each 60°.

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**Convention, definition or derived property?**

Decide whether the information given is a convention, a definition or a derived property.

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**S1.2 Parallel and perpendicular lines**

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

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**Lines in a plane What can you say about these pairs of lines?**

When we discuss lines in geometry, they are assumed to be infinitely long. That means that two lines in the same plane (that is in the same flat two-dimensional surface) will either intersect at some point or be parallel. These lines cross, or intersect. These lines do not intersect. They are parallel.

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**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. Ask pupils how we could draw two infinitely long lines that will never meet. The answer would be to draw them in different planes. We can imagine, for example, one plane made by one wall in the room and another plane made by the opposite wall. If we drew a line on one wall and a line on the other, they would never meet, even if the walls extended to infinity.

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**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. Pupils should be able to identify parallel and perpendicular lines in 2-D and 3-D shapes and in the environment. For example: rail tracks, double yellow lines, door frame or ruled lines on a page. This means that they are always equidistant. Where do you see parallel lines in everyday life?

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**What is special about the angles at the point of intersection here?**

Perpendicular lines What is special about the angles at the point of intersection here? a a = b = c = d b d Each angle is 90. We show this with a small square in each corner. c Pupils should be able to explain that perpendicular lines intersect at right angles. Lines that intersect at right angles are called perpendicular lines.

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**Parallel or perpendicular?**

Use this activity the identify whether the pairs of lines given are parallel or perpendicular. This activity will also practice the labeling of lines using their end points.

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**The distance from a point to a line**

What is the shortest distance from a point to a line? O Ask pupils to point out which line they think is the shortest and ask them what they notice about it. Ask pupils if they think that the shortest line from a point to another line will always be at right angles. Reveal the rule. The shortest distance from a point to a line is always the perpendicular distance.

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**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.

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**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.

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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 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.

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**Angles Angles are measured in degrees. A half turn measures 180°.**

This is a straight line. 180°

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**Angles Angles are measured in degrees.**

A three-quarter turn measures 270°. 270°

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

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Intersecting lines Use this activity to demonstrate that vertically opposite angles are always equal.

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**Vertically opposite angles**

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

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**Angles on a straight line**

Use this activity to demonstrate that the angles on a straight line always add up to 180°. Hide one of the angles and ask pupils to work out its value. Add another angle to make the problem more difficult.

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**Angles on a straight line**

Angles on a line add up to 180. a b This should formally summarize the rule that the pupils deduced using the previous interactive slide. a + b = 180° because there are 180° in a half turn.

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Angles around a point Move the points to change the values of the angles. Show that these will always add up to 360º. Hide one of the angles, move the points and ask pupils to calculate the size of the missing angle.

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**Angles around a point Angles around a point add up to 360. b a c d**

This should formally summarize the rule that the pupils deduced using the previous interactive slide. a + b + c + d = 360 because there are 360 in a full turn.

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**Calculating angles around a point**

Use geometrical reasoning to find the size of the labelled angles. 69° 68° d 167° a 43° c 103° 43° b Point out that that there are two intersecting lines in the second diagram. Click to reveal the solutions. 137°

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Complementary angles When two angles add up to 90° they are called complementary angles. a b Ask pupils to give examples of pairs of complementary angles. For example, 32° and 58º. Give pupils an acute angle and ask them to calculate the complement to this angle. a + b = 90° Angle a and angle b are complementary angles.

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Supplementary angles When two angles add up to 180° they are called supplementary angles. b a Ask pupils to give examples of pairs of supplementary angles. For example, 113° and 67º. Give pupils an angle and ask them to calculate the supplement to this angle. a + b = 180° Angle a and angle b are supplementary angles.

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**Angles made with parallel lines**

When a straight line crosses two parallel lines eight angles are formed. a b d c e f h Ask pupils to give any pairs of angles that they think are equal and to explain their choices. g Which angles are equal to each other?

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**Angles made with parallel lines**

Use this activity to show that when a line crosses a pair of parallel lines eight angles are produced. The four acute angles are equal and the four obtuse angles are equal. The obtuse angle and the acute angle form a pair of supplementary angles. Hide all but one of the angles, move the end points to change the angles and ask pupils to find the value of each hidden angle.

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Corresponding angles There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines. g g d = h because Corresponding angles are equal

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

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

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

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**Alternate angles There are two pairs of alternate angles, or Z-angles.**

b d d c e f f h g d = f because Alternate angles are equal

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**Alternate angles There are two pairs of alternate angles, or Z-angles.**

b d c c e e f h g c = e because Alternate angles are equal

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**Calculating angles Calculate the size of angle a. 29º a**

Hint: Add another line. 46º Ask pupils to explain how we can calculate the size of angle a using what we have learnt about angles formed when lines cross parallel lines. If pupils are unsure reveal the hint. When a third parallel line is added we can deduce that a = 29º + 46º = 75º using the equality of alternate angles. a = 29º + 46º = 75º

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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) = 560° The sum of the interior angles in a pentagon is 560°.

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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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