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
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triangles and equilateral triangles are 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 illustrate how 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.
2. 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.
3. 2y=2x50； y=x∠E，therefore∠E=yx；
∠E= 25
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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.
2. Try to use the formula for the
angle sum of the exterior angles of polygons
to solve the same problem.
150
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Solution :
Comparing the two solutions, which formula, the interior angle sum
formula or the exterior angle sum formula, gave easier calculations?
3. In the figure,∠A50,
EBis the angle bisector of ∠ABC,
EC is the angle bisector of ∠ACD.
Find the angle of intersection,∠E, of EB and EC.
A
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 :
Therefore, ∠E=
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【Hint: express∠E in x,y.】