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4. POLYGONS
4 - 1 Polygons in circles
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4 - 3 Convex and concave polygons
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4 - 4 The sum of the angles of convex polygons
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4 - 6 Finding unknown angles
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Activity 4 – 1
Polygons in circles
A regular triangle
∗ •∗
∗
Yes. The 3 small triangles are identical to each other.
They are isosceles triangles because 2 sides are equal (radii of the
same circle).
Each angle at the centre is 120°. This is because the sum of angles at a
point is 360°. There are 3 equal angles. 360°÷ 3 = 120°.
For one small triangle, each angle at the circumference is 30°. This is
because:
the sum of the angles of a triangle is 180°.
Therefore the two equal angles at the circumference must together be:
180° - 120° (the angle at the centre) = 60°.
Therefore each angle at the circumference must be:
60°÷ 2 = 30°.
The large triangle has each angle equal to 60°. This is because:
each angle is the sum of 2 equal angles of the smaller triangles.
30°+ 30°= 60°.
The large triangle is an equilateral triangle.
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A regular quadrilateral
•
Yes. All the angles at the centre of the circle the same size .
This is because:
one of the angles is 90°.
Therefore the vertically opposite angle is 90°.
The other angles are also 90° because they are supplementary to a 90°
angle (angles on a straight line).
Yes. The 4 small triangles that meet in the centre of the circle above.
are identical to each other.
They are isosceles triangles because they have 2 sides equal.
For one of the small triangles, the size of each angle at the
circumference of the circle is 45°. This is because:
the sum of the angles of a triangle is 180°.
Therefore the two equal angles at the circumference must together be:
180° - 90° (the angle at the centre) = 90°.
Therefore each angle at the circumference must be:
90°÷ 2 = 45°.
The circle around the quadrilateral called a circumcircle.
The centre is called a circumcentre
Each angle of the quadrilateral is 90°
This is because:
each angle is the sum of 2 equal angles of the smaller triangles.
45°+ 45°= 90°.
The quadrilateral is a square.
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A regular pentagon
72°
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All regular polygons can be divided into identical triangles which have
equal angles at the circumcentre.
To find the angle made by two adjacent radii of a regular pentagon:
Sum of the angles at a point = 360°
Angle made by adjacent radii = 360°÷ 5 = 72°.
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A regular hexagon
• P
6 triangles have been drawn.
Yes. All the triangles are the same.
They are equilateral triangles.
Each angle is 60°.
Reasons for this:
1) Sum of the angles at a point = 360°. 360°÷ 6 = 60°.
2) The triangles are equilateral. 180° ÷ 3 = 60°.
The 6 adjacent triangles make a hexagon.
Yes. It is a regular shape because all its sides are equal and all its
angles are equal.
Each angle of the larger shape (the hexagon) is 120° because it is
made by two adjacent angles that are 60°.
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Activity 4 – 3
Convex and concave polygons
The concave figures are coloured.
The concave polygons all have at least one interior angle greater than
180°.
A concave heptagon
A heptagon has 4 diagonals.
No. In a concave heptagon, the diagonals do not all lie inside the figure.
No. You cannot draw a regular polygon that is concave.
A concave polygon must have at least one angle that is greater than
180°. A regular polygon has all its angles the same size. Therefore, all
the interior angles of a concave regular polygon would have to be
greater than 180°. If this happened, the sides of the figure could not
meet.
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Activity 4 – 4
The sum of the angles of convex polygons
An irregular polygon is a polygon that has sides that are not all the same
length and angles that are not all the same size.
Triangle (3 sides)
Quadrilateral (4 sides)
Pentagon (5 sides)
Hexagon (6 sides)
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The interior angles of each triangle add to 180°.
Convex
polygon
Triangle
Number of
Number of
interior angles triangles made
3
1
Sum of the
interior angles
180°
Quadrilateral
4
2
2 x 180° = 360°
Pentagon
5
3
3 x 180° = 540°
Hexagon
6
4
4 x 180° = 720°
Heptagon
7
5
5 x 180° = 900°
A rule you can use to find out the sum of the interior angles of a convex
polygon:
Multiply 180° by 2 less than the number of sides of the polygon.
A convex heptagon
The angles should add up to 900°.
Yes. This rule works for all regular polygons because they are all convex
polygons.
For a regular hexagon:
Using the rule, the sum of the interior angles is 4 x 180º = 720°.
Each interior angle of a regular hexagon is 120°, so the sum of the 6
angles is 6 x 120° = 720°.
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Activity 4 – 6
Finding unknown angles
• The sum of angles on a straight line is 180°.
• Alternate angles made by a transversal of parallel lines are equal.
• Corresponding angles made by a transversal of parallel lines are
equal.
• The sum of co-interior angles made by a transversal of parallel lines
is 180°.
• The sum of the angles of a triangle is 180°.
• The exterior angle of a triangle is equal to the sum of the two remote
interior angles.
• In an isosceles triangle, the angles opposite the equal sides are
equal.
• In an equilateral triangle, all the angles are equal and have a size of
60°.
• The sum of the angles of a convex polygon is equal to 2 less than the
number of sides, multiplied by 180°.
A
α = 60°
Reason: ∆ ACD is isosceles (AC
and CD are radii of a circle), so
α= ∠CAD
60°
α
γ
D
β
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C
β = 120°
Reason: The sum of the remote
interior angles (∠CAD + ∠CDA)
γ = 30°
Reason: ∆ ABC is isosceles (AC
and CB are radii of a circle).
2γ + β = 180° (angle sum of a ∆).
2γ + 120° = 180°, so γ = 30°
B
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ε
ε
δ
γ
α
53°
33°
φ
χ
β
27°
86°
α = 180° - 33° - 53° = 94°
Reason: Sum of the angles of a triangle
β = 180° - 27° - 86° = 67°
Reason: Sum of the angles of a triangle
γ = 180° - 53° = 127°
Reason: Angles on a straight line
δ = 33°
Reason: Equal angles of an isosceles triangle
ε = 180° - 127° - 33° = 20°
Reason: Sum of the angles of a triangle
χ = 67°
Reason: Alternate angles made by a transversal of parallel lines
φ = 180° - 67° - 90° = 23°
Reason: Sum of the angles of a triangle
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