Triangles and
Quadrilateral
Polygon matchstick puzzle
This shape has been constructed with
16 matches.
1. Can you remove four matches
to leave exactly four
equilateral triangles of
the same size?
2. Can you remove three
matches to leave exactly
three parallelograms
of the same size?
The what-gons?
Study the table below.
These are accurate drawings, so you may wish to use your ruler and protractor to
check side lengths and angle sizes.
1. What reason has been used to group the polygons down columns 1, 2, 3 and 4?
2. There seems to be a missing group of polygons between columns 2 and 3, and also
between columns 3 and 4.
a) Which groups of polygons are missing?
b) Draw one polygon from each of these missing groups.
3. The polygons in row 1 are called regular polygons.What do these polygons have in
common to make them all regular polygons?
4. What is special about the polygons in row 2?
5. a) What is special about the polygons in row 3?
b) The last three polygons in row 3 are not regular polygons.Why not?
6. Compare your answers with other learners in your class.
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A polygon is a regular polygon if all its sides are equal in length and all its interior
angles are of equal size.
For example, in a regular hexagon all
the sides are equal and all the
interior angles are 120°.
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The polygons have been grouped down the columns according to their number of
sides.
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These groups have special names.We could call then 3-gons, 4-gons, 6-gons or
8-gons, according to their number of sides. However, their well-known names are
listed below:
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‘number of sides’-gon
common name
3-gon
triangle
4-gon
quadrilateral
5-gon
pentagon
6-gon
hexagon
7-gon
heptagon
8-gon
octagon
You will notice that the names for these polygons have similar Greek roots to the
polyhedron names.This time you also have quad, which means four, and hepta,
which means seven.
1. What reasons have been used to group these triangles?
2. In which group would you place each of the following triangles:
a) A triangle that has three equal sides
b) A triangle in which each side has a different length (i.e. none of the sides are
equal)
c) A triangle with two sides of the same length
3. Use the table and the questions above to explain what the following are:
a) An equilateral triangle
b) An isosceles triangle
c) A scalene triangle
4. Compare with other learners in your class.
Triangles are classified according to the number of sides of equal length.
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An equilateral triangle has all three sides equal.
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An isosceles triangle has only two equal sides.
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A scalene triangle has no equal sides.
Classifying triangles
Work with a friend and answer the questions that follow.
The triangles from activity 6 can also be classified as shown on the following page:
1. What is the classification rule here? How do you decide in which group to place
the triangles?
2. Give suitable names to each of the groups.
3. In which of the above groups would you place each of the following triangles:
a) A triangle with angles of 60°, 80°, and 40°
b) A triangle with angles of 20°, 120°, and 40°
c) A triangle with angles of 30°, 60° and 90°.
4. Compare your answers with other learners in your class.
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The triangles have been grouped according to their interior angles.
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An acute-angled triangle has only acute interior angles, as in group 1.
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An obtuse-angled triangle has one obtuse interior angle, as in group 3.
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A right-angled triangle has one right angle, as in group 2.
We can summarise
the classification
of triangles in the
diagram alongside:
1. Flat faces, which are 3-sided (triangles), 4-sided (quads) and 8-sided (octagons) are
in the designs. Draw one of each.
2. How are the shapes of the quads that you see different to the shapes of the
squares and the rectangles?
3. Why do you think the top shapes in both gems are octagons rather than triangles
or quads?
4. Which gemstone, do you think, will glitter the most? Discuss in class.
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You see again that geometric shapes are very often used in design activities.
Parallelograms, Kites, rhombuses
Work in groups of three or four and answer the questions that follow.
Draw four copies of the
same triangle onto stiff
paper and cut them out.
Label the sides as shown.
Use one pair of triangles to
form a parallelogram and
another pair to form a kite,
as shown alongside:
1. The parallelogram and kite are both quads.Why are they quads?
2. Describe how the two shapes are different.
3. Which sides of the parallelogram are equal? How do you know this?
4. Which sides of the kite are equal? How do you know this?
5. How do the positions of the equal sides in the parallelogram and the kite differ?
6. Do you think any of the shapes have parallel sides? How will you check this?
7. Do you think any of the shapes have interior angles that are equal? How will you
check this?
8. Compare answers to other groups in class.
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The opposite sides of the
parallelogram are equal in
length;they are formed by the
same sides of the two triangles.
The rhombus and the square
The tessellations of a square and a rhombus are shown below:
1. What is the same about the rhombus and the square?
2. What is different about the rhombus and the square?
3. How is the rhombus the same as a parallelogram? Answer by referring to its sides
and interior angles.
4. How is the square the same as a parallelogram?
5. What is special about the interior angles of a square?
6. What is the same about a square, a rhombus and a kite?
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In the square and the rhombus all 4 sides are equal.
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The short, double lines show the equal sides.
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The opposite sides of the square and the rhombus are also
parallel.
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In the rhombus only the opposite interior angles are equal.
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All the angles in a square are equal to 90°.
EXERCISE
1. Draw a parallelogram and show the sides that are parallel.
2. Draw a kite and show the sides that are equal.
3. Name each of the figures below and explain why you have named them so.
4. List all the properties that you can think of for:
a) a parallelogram
b) a rhombus
5. Two of the sides of a quad are 4 cm long, and the other two sides are each 3 cm
long. Is it a parallelogram or is it a kite? Explain your answer.
6. What is special about this quad?
Is it a parallelogram?
Is it a kite?
Explain your answer in each case.
7. One of the following quads is not a kite.Which one and why not?
8. Copy the tangram pieces
alongside onto paper, then
cut them out along the
edges.
Polygons: The many-sided figures
a) Form the biggest right-angled triangle you can by fitting suitable pieces together.
Is it also an isosceles triangle?
b) Form a quadrilateral with three pieces.
c) Form a pentagon by using three pieces.
9. Calculate the missing angles in the triangles: