1. GCSE: Angles and Geometry
Dr J Frost
Last modified: 5th January 2014

2. x = 66° Starter ? Copy the diagram, then determine x. 55° 55° 33° 192°
136°
11°
70° x°
114°
44°
44°
22°
22°
11° ?
x = 66°

3. 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.

4. Quick Exercises ? ? ? ? ? Q1 74° Q2 a = 65°, b = 103°, c = 38° c b 93°
77°
a = 87°, b = 74° ?
115° b° Q3 Q4
The two squares are congruent. Q5
50°
68° y c
150 x 85
70° a
c = 20° ?
a = 136° ? ?
x = 150°, y = 95°

5. Angles in Polygons NO YES NO ? ? ?
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

6. Click to Start Damonimation
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

7. Angles in Polygons ? ? Exterior angle of pentagon = 360 / 5 = 72°
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° ? ?

8. Angles in Polygons ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Num Sides
Name of Regular Polygon
Exterior Angle
Interior Angle 3
Triangle
120°
60° 4
Quadrilateral
90° 5
Pentagon
72°
108° 6
Hexagon 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°. ?

9. 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°

10. 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

11. ! Total interior angle = 180(n – 2)
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) ?

12. Sum of Interior Angles ? ? ? ? Num Sides Name Sum of Interior Angles 3
Triangle
180° 4
Quadrilateral
360° 5
Pentagon
540° 6
Hexagon
720° ? ? ? ?

13. Quick Exercises Q1 Q2 Q3 50
110 85 75
100
110 a b 80 80 60 80 80 c
a = 110° ?
b = 260° ?
c = 70° ? Q4 Q5 Q6
The sum of the interior angles of a polygon is 3600°. How many sides does it have? 22 62
242 62 81
284 88 ? a a 67 b 86 69
a = 119°, b = 25° ? ?
a = 100° ?