Angles and Geometry Dr J Frost jfrosttiffinkingstonschuk Last modified 5 th January 2014 Copy the diagram then determine x Starter 11 x 22 11 22 136 33 ID: 625437
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Slide1
GCSE: Angles and Geometry
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 5
th
January 2014Slide2
Copy the diagram, then determine x.
Starter
11
°
x
°
22
°
11
°
22
°
136
°
33
°
33
°
114
°
44
°
44
°
192
°
55
°
55
°
70
°
x
= 66°
?Slide3
Laws of Angles
a
a
a
a
a
a
Opposite
angles
Alternate
angles
Corresponding
angles
Bro Tip:
When asked to give a justification in an exam for how you determined an angle, write one of the above –
NOT
“Z angles” or “F angles”. You won’t get the mark.
?
?
?Slide4
Quick Exercises
74
°
93
°
b
a
115
°
a
b
°
77
°
c
70
°
50
°
c
68
°
a
The two squares are congruent.
Q1
Q2
Q3
Q4
a = 87°, b = 74°
a = 65°, b = 103°,
c = 38°
c = 20°
a = 136°
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?
Q5
150
85
y
x
x
= 150
°
, y = 95°
?Slide5
Angles in Polygons
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
?
?
?Slide6
Angles in Polygons
Click to Start
Damonimation
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°.
?Slide7
Angles in Polygons
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
°
?
?Slide8
Angles in Polygons
Num Sides
Name of Regular Polygon
Exterior
Angle
Interior Angle
3
Triangle
120
°
60
°
4
Quadrilateral90
°90°
5Pentagon72
°108°
6Hexagon60
°120°
7Heptagon51.4
°128.6°
8Octagon45
°135°9Nonagon40°140°
10Decagon36°
144°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°.
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?Slide9
Angles in Polygons
Example GCSE Question
Interior angle of hexagon = 180° – (360°/6) = 60°
Interior angle of octagon = 180° – (360°/8) = 135°
x = 360° – 120° – 135° = 105
°
?Slide10
Angles in Polygons
Copy the diagram and answer.
Given that B has interior angle 60:
Interior angle of A = (360 – 60)/2 = 150
Exterior angle of A
= 30
So A has 360 / 30
= 12 sides
?Slide11
Sum of Interior Angles
We know how to find the interior angle of a regular polygon.
But can we find the total angle if the polygon is not regular?
n sides
We could split the shape up into n – 2 triangles.
Now noticing that the interior angles of each triangle form the interior angles of the overall polygon, the total interior angle must be 180(n – 2)
!
Total interior angle = 180(n – 2)
?Slide12
Sum of Interior Angles
Num Sides
Name
Sum of Interior Angles
3
Triangle
180
°
4
Quadrilateral
360
°
5
Pentagon
540°6
Hexagon720°
?
?
?
?Slide13
Quick Exercises
80
80
a
a
= 110
°
110
100
80
b
60
110
b
= 260
°
50
75
85
80
c = 70
°
c
62
242
81
a
69
284
62
a
= 100
°
88
67
86
a
b
a = 119
°
, b = 25
°
The sum of the interior angles of a polygon is 3600
°
. How many sides does it have?
22
Q1
Q2
Q3
Q4
Q5
Q6
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