gcse: angles and geometry dr j frost ([email protected]) last modified: 5 th january...
TRANSCRIPT
Copy the diagram, then determine x.
Starter
11°x°
22°
11°
22°
136°33°
33°
114°44°44°
192°
55°55°
70°
x = 66° ?
Laws of Angles
aa
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.
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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
15085
y
x
x = 150°, y = 95° ?
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
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°.
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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°
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Angles in Polygons
Num Sides Name of Regular Polygon
Exterior Angle
Interior Angle
3 Triangle 120° 60°
4 Quadrilateral 90° 90°
5 Pentagon 72° 108°
6 Hexagon 60° 120°
7 Heptagon 51.4° 128.6°
8 Octagon 45° 135°
9 Nonagon 40° 140°
10 Decagon 36° 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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Angles in PolygonsExample GCSE Question
Interior angle of hexagon = 180° – (360°/6) = 60°Interior angle of octagon = 180° – (360°/8) = 135°x = 360° – 120° – 135° = 105°
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Angles in PolygonsCopy 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
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Sum of Interior AnglesWe 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) ?
Sum of Interior Angles
Num Sides Name Sum of Interior Angles
3 Triangle 180°
4 Quadrilateral 360°
5 Pentagon 540°
6 Hexagon 720°
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Quick Exercises
80
80
a
a = 110°
110100
80b
60
110
b = 260°
50
7585
80
c = 70°
c
62242
81
a69
284
62
a = 100°
88
6786
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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