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Year 11 GCSE Maths - Intermediate Triangles and Interior and Exterior Angles

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 180 º ; How to use angle notation – the way we refer to angles in complicated diagrams ;

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Year 11 GCSE Maths - Intermediate Triangles and Interior and Exterior Angles

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  1. Year 11 GCSE Maths - IntermediateTriangles 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.

  2. How to prove the angles of a triangle = 180º c a b 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

  3. c a b c Because of the Z-rule, we see that this angle here is also equal toc Remember: this means the two angles marked c are ALTERNATE angles!!

  4. c a b c a Because of the Z-rule again, we see that this angle here is equal toa Remember: this means the two angles marked a are also ALTERNATE angles!!

  5. 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 !!

  6. 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…..

  7. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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.

  8. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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)

  9. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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!!

  10. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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)

  11. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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)

  12. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E 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

  13. C B 8 7 9 6 10 D 11 2 3 5 1 12 4 A E F 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

  14. Interior Angles of a Polygon A polygon is any shape with straight lines for sides, so a circle is NOT a polygon. A pentagon

  15. 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

  16. 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

  17. 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: Now divide it into triangles… VERTEX

  18. 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.

  19. Interior Angles of a Polygon Now copy this table and fill it in for the 2 polygons we have looked at so far:

  20. Interior Angles of a Polygon Now complete your table – here’s a hint: look for patterns in the numbers!!

  21. Interior Angles of a Polygon Challenge Question: What would be the total of the Interior angles of a 42-sided 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 !!

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