Year 11 GCSE Maths - Intermediate
Triangles and Interior and
Exterior Angles
In this lesson you will learn:
How to prove that the angles of a triangle
will always add up to 180º ;
How to use angle notation – the way we
refer to angles in complicated diagrams ;
How to work out the total of the interior
(inside) angles of any polygon.
How to prove the angles of a triangle = 180º
c
a
b
In the diagram above we have a triangle inbetween two parallel lines. At the top of the
triangle there are three angles: a, b and c.
Because these three angles make a straight line:
a + b + c = 180º
c
a
b
c
Because of the Z-rule, we see that this
angle here is also equal to c
Remember: this means the two angles
marked c are ALTERNATE angles!!
c
a
b
c
a
Because of the Z-rule again, we see
that this angle here is equal to a
Remember: this means the two angles
marked a are also ALTERNATE angles!!
c
a
b
c
a
Now we have a, b and c as the three
angles in the triangle……..
…. And we already know that a + b + c
= 180º so this proves the angles in a
triangle add up to 180º !!
Using Angle notation
Often we can get away
with referring to an
angle as just a, or b, or
c or even just x or y. But
sometimes this can be a
little unclear.
Copy the diagram on
the next slide…..
C
B
8
7
6
9
10
D
11
2
A
5
3
1
4
12
F
Just saying ‘the angle F’ could actually be
referring to one of ten possible angles at the
point F. If we actually mean angle 1, then we
give a three-letter code which starts at one
end of the angle, goes to F, and finishes at the
other end of the angle we want.
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
4
12
F
So for angle 1 we start at B, then go to F and
finish at A, and we write:
Angle 1 = BFA
(sometimes you write this as BFA)
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
4
12
F
BUT notice we could go the other way round
and start at A, then go to F and finish at B,
and we write:
Angle 1 = AFB instead.
Either answer is correct!!
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
12
4
F
Also for angle 4 we start at D, then go to F and
finish at E, and we write:
Angle 4 = DFE
(or EFD)
(sometimes you write this as DFE)
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
12
4
F
And for angle 9 we start at F, then go to C and
finish at D, and we write:
Angle 9 = FCD
(or DCF)
(sometimes you write this as FCD)
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
4
12
F
Now you have a go at writing the three-letter coding
for the following angles:
Angle 2
Angle 4
Angle 10
Angle 6
Angle 12
Angle 3+4
E
C
B
8
7
6
9
10
D
11
2
A
5
3
1
4
12
F
The answers are:
Angle 2 = BFC or CFB
Angle 10 = CDF or FDC
Angle 12 = FED or DEF
Angle 4 = DFE or EFD
Angle 6 = ABF or FBA
Angle 3+4 = CFE or EFC
E
Interior Angles of a Polygon
A polygon is any shape with straight
lines for sides, so a circle is NOT a
polygon.
A pentagon
Interior Angles of a Polygon
To find the total of the angles inside any polygon,
just pick a vertex (corner) and divide the polygon
into triangles, starting at that vertex:
VERTEX
Interior Angles of a Polygon
Now each triangle has a total of 180º, so with
three triangles, the pentagon has total interior
angles of 3 x 180º = 540º
Interior Angles of a Polygon
What about a heptagon? This has 7 sides.
Copy the one below into your book and label
the vertex shown:
VERTEX
Now divide it
into triangles…
Interior Angles of a Polygon
You can see now that the heptagon has been
divided into 5 triangles. That means the
interior angles of a heptagon must add up to
5 x 180º = 900º.
Interior Angles of a Polygon
Now copy this table and fill it in for the 2
polygons we have looked at so far:
Name of
Polygon
Triangle
Number Number of Working Total of
Interior
of sides triangles
out
angles
3
1
1 x 180
180º
7
8
10
5
5 x 180
900º
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Decagon
Interior Angles of a Polygon
Now complete your table – here’s a hint:
look for patterns in the numbers!!
Name of
Polygon
Triangle
Number Number of Working Total of
Interior
of sides triangles
out
angles
3
1
1 x 180
180º
7
8
10
5
5 x 180
900º
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Decagon
Interior Angles of a Polygon
Challenge Question: What would be
the total of the Interior angles of a 42sided polygon?
Answer:
The number of triangles that can be
drawn is always two less than the
number of sides in the polygon, so:
40 x 180 = 7200º !!