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We use the symbol ||| to mean is similar to .

Similar Polygons

Two polygons are similar if:

  1. their corresponding angles are equal, and
  2. the ratios of corresponding sides are equal.

Show that the following two polygons are similar.

  1. We are required to show that the pair of polygons is similar. We can do this by showing that the ratio of corresponding sides is equal and by showing that corresponding angles are equal.

  2. We are given the angles. So, we can show that corresponding angles are equal.

  3. All angles are given to be 90 and

    A ^ = E ^ B ^ = F ^ C ^ = G ^ D ^ = H ^
  4. We first need to see which sides correspond. The rectangles have two equal long sides and two equal short sides. We need to compare the ratio of the long side lengths of the two different rectangles as well as the ratio of the short side lenghts.

    Long sides, large rectangle values over small rectangle values:

    Ratio = 2 L L = 2

    Short sides, large rectangle values over small rectangle values:

    Ratio = L 1 2 L = 1 1 2 = 2

    The ratios of the corresponding sides are equal, 2 in this case.

  5. Since corresponding angles are equal and the ratios of the corresponding sides are equal the polygons ABCD and EFGH are similar.

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All squares are similar.

If two pentagons ABCDE and GHJKL are similar, determine the lengths of the sides and angles labelled with letters:

  1. We are given that ABCDE and GHJKL are similar. This means that:

    AB GH = BC HJ = CD JK = DE KL = EA LG


    A ^ = G ^ B ^ = H ^ C ^ = J ^ D ^ = K ^ E ^ = L ^
  2. We are required to determine the

    1. a , b , c and d , and
    2. e , f and g .
  3. The corresponding angles are equal, so no calculation is needed. We are given one pair of sides D C and K J that correspond. D C K J = 4 , 5 3 = 1 , 5 so we know that all sides of K J H G L are 1,5 times smaller than A B C D E .

  4. a 2 = 1 , 5 a = 2 × 1 , 5 = 3 b 1 , 5 = 1 , 5 b = 1 , 5 × 1 , 5 = 2 , 25 6 c = 1 , 5 c = 6 ÷ 1 , 5 = 4 d = 3 1 , 5 d = 2
  5. e = 92 ( corresponds to H ) f = 120 ( corresponds to D ) g = 40 ( corresponds to E )
  6. a = 3 b = 2 , 25 c = 4 d = 2 e = 92 f = 120 g = 40
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Activity: similarity of equilateral triangles

Working in pairs, show that all equilateral triangles are similar.


  1. Find the values of the unknowns in each case. Give reasons.
  2. Find the angles and lengths marked with letters in the following figures:

Investigation: defining polygons

Investigate the different ways of defining polygons. Polygons that you should pay special attention to are:

  • Isoceles, equilateral and right-angled triangle
  • Kites, parallelograms, rectangles, rhombuses (or 'rhombi'), squares and trapeziums

Things to consider are how these figures have been defined in this book and what alternative definitions exist. For example, a triangle is a three-sided polygon or a figure having three sides and three angles. Triangles can be classified using either their sides or their angles. Could you also classify quadrilaterals in this way? What other names exist for these figures? For example, quadrilaterals can also be called tetragons.

Questions & Answers

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