Year 11 GCSE Maths - IntermediateYear 11 GCSE Maths - IntermediateTriangles and Interior and Triangles and Interior and

Exterior AnglesExterior AnglesIn this lesson you will learn:In this lesson you will learn:

How to How to proveprove that the angles of a triangle that the angles of a triangle will will alwaysalways add up to 180 add up to 180º ;º ;How to use How to use angle notationangle notation – the way we – the way we refer to angles in complicated diagrams ;refer to angles in complicated diagrams ;How to work out the How to work out the total of the interiortotal of the interior (inside) angles of (inside) angles of anyany polygon. polygon.

ab

c

In the diagram above we have a triangle in-between 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

How to prove the angles of a triangle = 180º

ab

c

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!!

ab

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

ab

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 !!

c

Using Angle notationUsing Angle notation

Often we can get away Often we can get away with referring to an with referring to an angle as just angle as just a, or b, or a, or b, or cc or even just or even just x or yx or y. But . But sometimes this can be a sometimes this can be a little unclear.little unclear.Copy the diagram on Copy the diagram on the next slide…..the next slide…..

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.

12 3

4A

B C

D

EF

5

67 8

9

1011

12

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)

12 3

4A

B C

D

EF

5

67 8

9

1011

12

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!!

12 3

4A

B C

D

EF

5

67 8

9

1011

12

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)

12 3

4A

B C

D

EF

5

67 8

9

1011

12

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)

12 3

4A

B C

D

EF

5

67 8

9

1011

12

12 3

4A

B C

D

EF

5

67 8

9

1011

12

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

12 3

4A

B C

D

EF

5

67 8

9

1011

12

The answers are:

Angle 2 = BFC or CFB Angle 4 = DFE or EFD Angle 10 = CDF or FDC Angle 6 = ABF or FBA Angle 12 = FED or DEF Angle 3+4 = CFE or EFC

Interior Angles of a PolygonInterior Angles of a PolygonA polygon is A polygon is anyany shape with shape with straight straight

lineslines for sides, so a circle is for sides, so a circle is NOTNOT a a polygon.polygon.

A pentagon

Interior Angles of a PolygonInterior Angles of a PolygonTo find the total of the angles inside any polygon, To find the total of the angles inside any polygon,

just pick a vertex (corner) and divide the polygon just pick a vertex (corner) and divide the polygon into triangles, starting at that vertex:into triangles, starting at that vertex:

VERTEX

Interior Angles of a PolygonInterior Angles of a PolygonNow Now each triangleeach triangle has a total of 180 has a total of 180º, so with º, so with

three triangles, the pentagon has total interior three triangles, the pentagon has total interior angles of angles of 3 x 180º = 540º3 x 180º = 540º

Interior Angles of a PolygonInterior Angles of a PolygonWhat about a What about a heptagonheptagon? This has ? This has 77 sides. sides.

Copy the one below into your book and label Copy the one below into your book and label the vertex shown:the vertex shown:

VERTEX Now divide it into triangles…

Interior Angles of a PolygonInterior Angles of a PolygonYou can see now that the heptagon has been You can see now that the heptagon has been

divided into divided into 5 triangles5 triangles. That means the . That means the interior angles of a interior angles of a heptagonheptagon must add up to must add up to 5 x 1805 x 180º = 900º.º = 900º.

Interior Angles of a PolygonInterior Angles of a PolygonNow copy this table and fill it in for the 2 Now copy this table and fill it in for the 2

polygons we have looked at so far:polygons we have looked at so far:Name of Name of PolygonPolygon

Number Number of sidesof sides

Number of Number of trianglestriangles

Working Working outout

Total of Total of Interior Interior anglesangles

TriangleTriangle 33 11 1 x 1801 x 180 180180ººQuadrilateralQuadrilateral

PentagonPentagonHexagonHexagonHeptagonHeptagon 77 55 5 x 1805 x 180 900900ººOctagonOctagon 88DecagonDecagon 1010

Interior Angles of a PolygonInterior Angles of a PolygonNow complete your table – Now complete your table – here’s a hint: here’s a hint:

look for patterns in the numbers!!look for patterns in the numbers!!Name of Name of PolygonPolygon

Number Number of sidesof sides

Number of Number of trianglestriangles

Working Working outout

Total of Total of Interior Interior anglesangles

TriangleTriangle 33 11 1 x 1801 x 180 180180ººQuadrilateralQuadrilateral

PentagonPentagonHexagonHexagonHeptagonHeptagon 77 55 5 x 1805 x 180 900900ººOctagonOctagon 88DecagonDecagon 1010

Interior Angles of a PolygonInterior Angles of a PolygonChallenge QuestionChallenge Question: What would be : What would be

the total of the Interior angles of a 42-the total of the Interior angles of a 42-sided polygon?sided polygon?

Answer:Answer:The number of triangles that can be The number of triangles that can be

drawn is always two less than the drawn is always two less than the number of sides in the polygon, so:number of sides in the polygon, so:

40 x 180 = 720040 x 180 = 7200ºº !! !!