Download GCSE: Angles and Geometry

GCSE: Angles and Geometry
Dr J Frost ([email protected])
Last modified: 5th January 2014
Starter
Copy the diagram, then determine x.
55° 55°
33°
192°
x°
70°
114°
44° 44°
22°
33°
136° 11°
x = 66°?
22°
11°
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.
Quick Exercises
Q1
Q2
74°
a = 65°,?b = 103°,
c = 38°
c
b
93°
a
a
77°
a = 87°,?b = 74°
Q3
115°
Q4
50°
68°
b°
The two
squares are
congruent.
Q5
y
c
70°
x
85
a
c =?20°
a = 136°
?
x = 150°,
? y = 95°
150
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?
Angles in Polygons
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
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°
Angles in 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°.
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°
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
? / 30
So A has 360
= 12 sides
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?
We could split the shape up
into n – 2 triangles.
n sides
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)
Sum of Interior Angles
Num Sides
Name
Sum of Interior Angles
3
Triangle
180°
4
Quadrilateral
360°
5
Pentagon
540°
6
Hexagon
720°
?
?
?
?
Quick Exercises
Q1
Q2
Q3
110
a
b
80
a = 110°
?
c
b = 260°
?
Q4
Q6
62
242
81
284
69
a = 100°
?
a
a
b
80
c = 70°?
Q5
62
75
80
60
80
85
100
110
50
88
86
a = 119°,
? b = 25°
?
67
The sum of the interior
angles of a polygon is
3600°. How many sides
does it have?
22?