Download Yr8-GeometricalReasoning (Slides)

Year 8: Geometric Reasoning
Dr J Frost ([email protected])
Objectives: Be able to classify shapes, reason about sides and
angles, and find interior/exterior angles of polygons.
Last modified: 13th April 2016
STARTER: Identifying 2D polygons
!
A polygon is a 2D shape with?straight sides.
Sides:
3
Triangle
?
Equilateral
4
Quadrilateral
?
?
Square
?
Parallelogram
5
6
7
Pentagon
?
Hexagon
?
Heptagon
?
8
9
10
?
?
Isosceles
Scalene
?
Rectangle
?
Trapezium
Octagon
?
Nonagon
?
Decagon
?
?
Rhombus
?
Arrowhead
12
20
Dodecagon
?
Kite
?
Icosagon
?
A big debate we had in the maths office…
Is a square a trapezium?
It depends on the definition:
• “A trapezium is a quadrilateral with exactly one pair of parallel sides.” or
• “A trapezium is a quadrilateral with at least one pair of parallel sides.”
A square would be a trapezium under the second definition. It seems odd that we
might consider shapes with two pairs of parallel as trapeziums (as we’d usually call
them parallelograms), but otherwise we’d have the following situation…
Trapezium?
YES
NO
Click to Start
Bromanimation
This lack of continuity (i.e. not being a trapezium for
one specific case) is considered a bad thing in maths, so
the weight of evidence is that a square IS a trapezium,
and thus the second definition is the correct one.
Categorising Activity
Shape
Description
2D Shape
Trapezium
We’ve seen that all squares are trapeziums. We
might use the arrow to mean “is a type of”.
Using a full page of your book, form a ‘tree’ of
all the shape classifications on the right. You
may work in pairs.
‘2D Shape’ should be at the top of your
tree/page, i.e.:
Polygon
A 2D shape with straight edges.
Ellipse
(Also known as an oval)
Circle
Square
2D Shape
Polygon
Bro Tip: To see if
for example all
squares are
rectangles, see if a
square satisfies the
definition of a
rectangle.
Quadrilateral
Polygon with four edges.
Square
Regular polygon with four edges.
Rectangle
Two pairs of parallel sides, all
angles equal.
Oblong
All rectangles which are not
squares.
Trapezium
Quadrilateral with at least one
pair of parallel sides.
Parallelogram
Quadrilateral with two pairs of
parallel sides.
Rhombus
Quadrilateral with all edges the
same length.
Kite
Quadrilateral where sides in
adjacent pairs are equal in
length.
Arrowhead
Kite with a reflex angle.
Solution
2D Shape
Polygon
Ellipse
Quadrilateral
Circle
Trapezium
Kite
Parallelogram
Arrowhead
Rhombus
Square
Rectangle
Oblong
Properties of quadrilaterals
Shape
Name
Lines of
Num pairs of
symmetry parallel sides
Diagonals
always
equal in
length?
Diagonals
perpendicular?
Square
4
?
2
?
Yes?
Yes?
Rectangle
2
?
2
?
Yes?
No ?
Kite
1
?
0
?
No ?
Yes?
Rhombus
2
?
2
?
No ?
Yes?
Parallelogram
0
?
2
?
No ?
No ?
Arrowhead
1
?
0
?
No ?
Yes?
RECAP: Interior angles of quadrilateral
The interior angles of a quadrilateral add up to 360.
?
1
Parallelogram
2
y
100°
x
x
50°
x = 130°
?
3
? y = 80°
?
x = 100°
Trapezium
x
x = 120°
?
4
Kite
x
95°
55°
60°
x = 105°
?
Sum of interior angles
n=3
Total of interior
angles = 180°
n=4
Total of interior
angles = 360°
Can you guess what the angles add up to in a
pentagon? How would you prove it?
Sum of interior angles
Click to
Bromanimate
We can cut a pentagon
into three triangles.
The sum of the interior
angles of the triangles is:
3 x 180° = 540°
! For an n-sided shape, the sum of the interior angles is:
180(n-2)
?
Test Your Understanding
A regular decagon (10 sides).
160°
x
x
80°
130°
120°
x = 140°
?
x = 144°
?
120°
40°
x = 240°
?
x
100°
40°
Exercise 1
1a
b
c
b = 260
?
x = 75
?
x = 25
?
d
e
a = 100
?
f
x = 222
?
x = 309
?
Exercise 1
g
h
x = 54
?
2
i
x = 120
?
x = 252
?
The total of the interior angles of a polygon is 1260°. How
many sides does it have?
?
N1
The interior angle of a regular polygon is 179°. How many sides
does it have?
?
Exercise 1
N2
If a n-sided polygon has exactly 3 obtuse angles (i.e. 90 <  < 180), then
determine the possible values of (Hint: determine the possible range for
the sum of the interior angles, and use these inequalities to solve).
?
Interior Angles
An exterior angle of a polygon is an angle
between the line extended from one side,
and an adjacent side.
Which of these are exterior angles of the polygon?
?
NO
?
YES
NO?
Interior Angles
To defeat Kim Jon Il, Matt Damon
must encircle his pentagonal palace.
What angle does Matt Damon turn
in total?
360°
?
! The sum of the exterior angles of
any polygon is 360°.
Click to Start
Damonimation
Interior Angles
If the pentagon is regular, then all the
exterior angles are clearly the same.
Therefore:
Exterior angle of pentagon
= 360 / 5?= 72°
Interior angle of pentagon
= 180 – 72
? = 108°
Angles in Regular Polygons
Num Sides
Name of
Exterior
Regular Polygon Angle
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
?
90° ?
72° ?
60° ?
51.4° ?
45° ?
40° ?
36° ?
120°
Interior
Angle
?
90° ?
108° ?
120° ?
128.6°?
135° ?
140° ?
144° ?
60°
Bonus Question: What is the largest number of sides a shape can
have such that its interior angle is an integer?
360 sides. The interior
? angle will be 179°.
Test Your Understanding
GCSE question
The diagram shows a regular hexagon and a regular octagon.
Calculate the size of the angle marked x.
You must show all your working.
x = 105°
?
GCSE Question
Hint: Fill in what angles
you do know. You can
work out what the
interior angle of Tile A
will be.
Question: The pattern is made from two types of tiles, tile A and tile B.
Both tile A and tile B are regular polygons.
Work out the number of sides tile A has.
Sides = 12
?
Test Your Understanding
Q1
Q2
50
85
a
75
80
80
80
c
a = 110°
?
Q3
What is the exterior
angle of a 180-sided
regular polygon?
360 ÷ 180
? = 2
c = 70°?
Q4
The interior angle of a regular
polygon is 165. How many sides
does it have?
Interior angle = 180 – 165 = 15
n = 360 ÷ 15 = 24
?
Alternative method:
Total interior angle = 165n
Then solve 180(n – 2) = 165n
Exercise 2
Q1 Determine how many sides a regular
Q3
polygon with the following exterior
angle would have:
30
12 sides
45
8 sides
12
30 sides
9
40 sides
?
?
?
?
The diagram shows a regular hexagon and a
regular octagon. Calculate the size of the angle
marked. You must show all your working.
Interior angle of hexagon: 180 – (360/6) = 120
Interior angle of octagon: 180 – (360/8) = 135
x = 360 – 120 – 135 = 105
Q2 Determine how many sides a regular
polygon with the following interior
angle would have:
156
15 sides
162
20 sides
144
10 sides
175
72 sides
?
?
?
?
?
Q4
The pattern is made
from two types of
tiles, tile A and tile B.
Both tile A and tile B
are regular polygons.
Work out the number
of sides tile A has.
Interior angle of A =
(360 – 60)/2 = 150
Exterior angle = 30
Sides = 360/30 = 12
?
Exercise 2
Q5
A regular polygon is
surrounded by squares and
regular hexagons, alternating
between the two. How many
sides does this shape have?
Q6
?
Equilateral triangle, square, hexagon.
N
Interior angle = 360 – 90 – 120 = 150
n = 360 / 30 = 12 sides
?
Find all regular polygons which tessellate (when
restricted only to one type of polygon).
By thinking about interior angles, prove that the
regular polygons you identified above are the only
regular polygons which tessellate.
Method 1: The possible exterior angles of a
regular polygons are the factors of 360 less than
180: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30,
36, 40, 45, 60, 72, 90, 120
This gives interior angles of 179, 178, ..., 140,
135, 120, 108, 90, 60.
To tessellate, the interior angle has to divide 360.
Only 120, 90 and 60 does. This corresponds to a
hexagon, square and equilateral triangle.
?
Method 2: 360 divided by the interior angle must
give a whole number, in order for the regular
polygon to tessellate. Interior angle is 180 –
(360/n), so 360 / (180 – (360/n)) = k for some
constant k. Simplifying this gives kn – 2k – 2n = 0
This factorises to (k – 2)(n – 2) = 4
This only numbers which multiply to give 4 are 1
x 4 or 2 x 2 or 4 x 1. This n = 6, 4 or 3 in each case.
TEST YOUR UNDERSTANDING
Vote with your diaries!
A
B
C
D
What is the total exterior angle of a
polygon in terms of the number of
sides n?
360
360
n
360n
180(n-2)
What is the total interior angle of a 20
sided polygon?
360
3600
3240
6480
The interior angle of a polygon is 178.
How many sides does it have?
20
40
90
180
What is the interior angle of a 90 sided
regular polygon?
172
176
178
179
Determine the angle
.
61
29
105
120
215
223
225
235