Key Stage 3

Measures, Shape & Space Dimension

Learning Unit : Angles related with Lines and Rectilinear Figures

Learning Objectives :

·explore and use the properties of lines and angles of triangles

·explore and use the formulas for the angle sum of the interior angles and exterior angles of polygons

·explore regular polygons that tessellate

Programme Title : Polygons

Programme Objectives

  1. Explore the angle sum of the interior angles of polygons that tessellate.
  2. Introduce the angle sum of the interior angles of triangles, the relation between an exterior angle and its interior opposite angles, and the calculation of the interior angles of isosceles triangles and equilateral triangles.
  3. Use examples to illustrate the calculation of the interior angles of polygons and deduce the formula for the angle sum of the interior angles of polygons.
  4. Use an intuitive approach to deduce the formula for the angle sum of the exterior angles of polygons .
  5. Introduce some methods for constructing tessellated patterns.

Programme Outline

The programme starts with the introduction of the interesting tessellated patterns drawn by Escher, the famous Holland artist. From the simple patterns tessellated with triangles, the angle sum of the interior angles of triangles, and the relation between an exterior angle and its interior opposite angles are investigated. Subsequently, the calculations of the interior angles of isosceles triangles and equilateral trianglesare discussed.

In identifying the various combinations of regular polygons that tessellate, it is guided to explore the relation between the sizes of the interior angles of regular polygons that tessellate. The calculations of the angle sum of the interior angles of a pentagon and a hexagon are carried out by dissecting the polygons into triangles. The same approach is used for deriving the general formula for the angle sum of the interior angles of an n-sided polygon.

The situation where a racing car running round a track is used as an example to illustrate intuitively that the angle sum of the exterior angles of an n-sided polygon is 360. An example is also used to emphasize the flexibility of applying either the formula for interior angles or that for exterior angles in solving problems.

Finally, the programme uses lively animations to illustratehow to apply translational transformation, rotational transformation and reflectional transformation to construct tessellated patterns.

Worksheet Answers

1. 150n，(n-2)180，150n=(n-2)180，

n=12，a regular 12-sided polygon.

  1. An interior angle is 150, that is an exterior angle is 30.

Since the angle sum of exterior angles is 360,

therefore, the number of exterior angles =36030=12，

that is the number of sides of the regular polygon is 12.

The calculation is easier for the exterior angle sum formula.

  1. 2y=2x50；y=x∠E，therefore∠E=yx；

∠E=25

Key Stage 3 ETV Programme《Polygons》

Worksheet

  1. As what you have seen in the programme, use the formula for the angle sum of the interior angles of polygons to find out the number of sides of the regular polygon drawn in the piece of paper below.

150

Solution:

Suppose it is a regular n-sided polygon.

Since an interior angle is 150，therefore the total sum of n interior angles is :

Using the formula for the angle sum of the interior angles of an n-sided polygon, the total sum of n interior angles can be written as :

Hence, the equation is :

Solving the equation :

We find that the polygon is a regular -sided polygon.

  1. Try to use the formula forthe

angle sum of the exterior angles of polygons

to solve the same problem.

150

Solution :

Comparing the two solutions, which formula, the interior angle sum formula or the exterior angle sum formula, gave easier calculations?

  1. In the figure,∠A50,

EBis the angle bisector of ∠ABC, A

EC is the angle bisector of ∠ACD.

Find the angle of intersection,∠E, of EB and EC. 50 E

Solution :

B C D

EB is the angle bisector of ∠ABC, that is EB divides ∠ABC into two equal portions.

Suppose each portion is x. Mark x in the appropriate positions of the figure.

Similarly, EC divides∠ACD into two equal portions.

Suppose each portion is y. Mark y in the appropriate positions of the figure.

In △ABC, since an exterior angle equals the sum of its interior opposite angles, therefore : 【Hint: find a relation between x,y.】

Similarly, for an exterior angle in△EBC : 【Hint: express∠E in x,y.】

Therefore, ∠E=

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