Ks3 Mathematics Homework Pack F Level 8 Answers

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KS3 Mathematics. S1 Lines and Angles. S1 Lines and angles. Contents. S1.2 Parallel and perpendicular lines. S1.1 Labelling lines and angles. S1.3 Calculating angles. S1.4 Angles in polygons. Lines. In Mathematics, a straight line is defined as having infinite length and no width. - PowerPoint PPT Presentation

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PowerPoint PresentationThe 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 178-183.
© Boardworks Ltd 2004
S1 Lines and angles
S1.4 Angles in polygons
© Boardworks Ltd 2004
Lines
In Mathematics, a straight line is defined as having infinite length and no width.
Is this possible in real life?
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.
© Boardworks Ltd 2004
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
has end points A and B.
We can call this line, line segment AB.
A
B
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Labelling angles
When two lines meet at a point an angle is formed.
An angle is a measure of the rotation of one of the line segments to the other.
We label angles using capital letters.
A
B
C
or ABC
or B.
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.
© Boardworks Ltd 2004
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.
For example, the angles in an equilateral triangle are each 60°.
60°
60°
60°
Discuss the difference between a convention, a definition and a derived property.
© Boardworks Ltd 2004
Convention, definition or derived property?
Decide whether the information given is a convention, a definition or a derived property.
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S1.4 Angles in polygons
S1.3 Calculating angles
These lines cross, or intersect.
These lines do not intersect.
They are parallel.
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.
© Boardworks Ltd 2004
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.
© Boardworks Ltd 2004
We use arrow heads to show that lines are parallel.
Parallel lines will never meet. They stay an equal distance apart.
Where do you see parallel lines in everyday life?
This means that they are always equidistant.
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.
© Boardworks Ltd 2004
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Perpendicular lines
What is special about the angles at the point of intersection here?
a = b = c = d
Lines that intersect at right angles are called perpendicular lines.
a
b
c
d
Each angle is 90. We show this with a small square in each corner.
Pupils should be able to explain that perpendicular lines intersect at right angles.
© Boardworks Ltd 2004
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
The shortest distance from a point to a line is always the perpendicular distance.
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.
© Boardworks Ltd 2004
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.
© Boardworks Ltd 2004
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.
© Boardworks Ltd 2004
S1 Lines and angles
© Boardworks Ltd 2004
It is called a right angle.
We label a right angle with a small square.
90°
180°
270°
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 = c
a
b
c
d
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 line add up to 180.
a + b = 180°
because there are 180° in a half turn.
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
© Boardworks Ltd 2004
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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a + b + c + d = 360
a
b
c
d
because there are 360 in a full turn.
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
© Boardworks Ltd 2004
Calculating angles around a point
Use geometrical reasoning to find the size of the labelled angles.
103°
a
167°
137°
69°
Point out that that there are two intersecting lines in the second diagram.
Click to reveal the solutions.
© Boardworks Ltd 2004
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Complementary angles
When two angles add up to 90° they are called complementary angles.
a
b
Angle a and angle b are complementary angles.
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.
© Boardworks Ltd 2004
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Supplementary angles
When two angles add up to 180° they are called supplementary angles.
a
b
Angle a and angle b are supplementary angles.
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.
© Boardworks Ltd 2004
Angles made with parallel lines
When a straight line crosses two parallel lines eight angles are formed.
Which angles are equal to each other?
a
b
c
d
e
f
g
h
Ask pupils to give any pairs of angles that they think are equal and to explain their choices.
© Boardworks Ltd 2004
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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a
b
c
e
f
g
Corresponding angles are equal
Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines.
© Boardworks Ltd 2004
b
c
d
f
g
h
c = g because
b = f because
d = f because
c = e because
a
29º
46º
75º
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.
© Boardworks Ltd 2004
S1.3 Calculating angles
© Boardworks Ltd 2004
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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a
b
c
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.
These angles are equal because they are alternate angles.
a
a
b
b
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°.
Discuss this proof that angles in a triangle have a sum of 180º.
© Boardworks Ltd 2004
Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles.
233°
82°
31°
116°
326°
43°
49°
28°
a
b
c
d
33°
64°
88°
25°
Ask pupils to calculate the size of the missing angles before revealing them.
© Boardworks Ltd 2004
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.
© Boardworks Ltd 2004
For example,
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°
46°
46°
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.
© Boardworks Ltd 2004
Polygons
A polygon is a 2-D shape made when line segments enclose a region.
A
B
C
D
E
The line segments are called sides.
The end points are called vertices. One of these is called a vertex.
2-D stands for two-dimensional.
These two dimensions are length and width. A polygon has no height.
© Boardworks Ltd 2004
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Naming polygons
Polygons are named according to the number of sides they have.
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
The angles inside a polygon are called interior angles.
The sum of the interior angles of a triangle is 180°.
c
a
b
f
d
e
When we extend the sides of a polygon outside the shape
exterior angles are formed.
a
b
c
Any exterior angle in a triangle is equal to the sum of the two opposite interior angles.
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.
b
c
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.
82°
31°
64°
34°
a
b
33°
116°
152°
d
25°
127°
131°
c
272°
43°
Calculate the size of the lettered angles in this diagram.
56°
a
73°
b
86°
69°
104°
= 76º ÷ 2
= 86º
= 69º
c
a
b
What is the sum of the interior angles in a quadrilateral?
We can work this out by dividing the quadrilateral into two triangles.
d
f
e
(a + b + c) + (d + e + f ) = 360°
The sum of the interior angles in a quadrilateral is 360°.
© Boardworks Ltd 2004
Sum of interior angles in a polygon
We already know that the sum of the interior angles in any triangle is 180°.
a + b + c = 180 °
Do you know the sum of the interior angles for any other polygons?
a
b
c
a
b
c
d
We have just shown that the sum of the interior angles in any quadrilateral is 360°.
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
© Boardworks Ltd 2004
Sum of the interior angles in a pentagon
What is the sum of the interior angles in a pentagon?
We can work this out by using lines from one vertex to divide the pentagon into three triangles .
a + b + c = 180°
(a + b + c) + (d + e + f ) + (g + h + i) = 560°
The sum of the interior angles in a pentagon is 560°.
c
a
b
and
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.
How many triangles can a hexagon be divided into?
A hexagon can be divided into four triangles.
© Boardworks Ltd 2004
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°.
© Boardworks Ltd 2004
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
Sum of the interior angles
Size of each interior angle
Ask pupils to complete the table for regular polygons with up to 10 sides.
© Boardworks Ltd 2004
In an equilateral triangle,
The sum of the interior angles is 3 × 60° = 180°.
The sum of the exterior angles is 3 × 120° = 360°.
60°
60°
120°
120°
60°
120°
In a square,
The sum of the interior angles is 4 × 90° = 360°.
The sum of the exterior angles is 4 × 90° = 360°.
90°
90°
90°
90°
90°
90°
90°
90°
In a regular pentagon,
The sum of the interior angles is 5 × 108° = 540°.
The sum of the exterior angles is 5 × 72° = 360°.
108°
108°
108°
108°
108°
72°
72°
72°
72°
72°
In a regular hexagon,
The sum of the interior angles is 6 × 120° = 720°.
The sum of the exterior angles is 6 × 60° = 360°.
120°
120°
120°
120°
120°
120°
60°
60°
60°
60°
60°
60°
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°.
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°.
Discuss this algebraic proof that the sum of the exterior angles in a polygon is always 360°.
© Boardworks Ltd 2004
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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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
=…