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medium difficulty4 min

SSS similarity criterion

Two triangles are similar if all three corresponding sides are proportional.

In one line: Three sides, one ratio.

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Understand it

When every matching side ratio is the same, the triangle keeps the same shape and only the size changes. This is why three proportional sides are enough.

Example

Sides 7 cm, 9 cm, 12 cm and 14 cm, 18 cm, 24 cm are proportional in the ratio 1:2, so the triangles are similar.

Formula and rule

SSS similarity: all three corresponding sides are proportional.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1For triangles with sides 5, 12, 13 and 10, 24, 26, every ratio is 1:2.
  2. 2Therefore the triangles are similar by SSS.

Method you can reuse

  1. 1Arrange the corresponding sides.
  2. 2Compare all three ratios.
  3. 3If the ratios are equal, conclude similarity by SSS.

Sketch it in your answer

Write the triangles in the same vertex order and mark matching sides before comparing ratios.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What condition must be satisfied to use SSS similarity between two triangles?

Think first, then show answer
SSS similarity is used when all three corresponding sides are proportional in the same order. If the three ratios match, the triangles are similar.

Common wrong answer

Saying similar after checking only two side ratios. SSS needs all three corresponding sides.

Don't mix this up

Students sometimes check only two side ratios or mix up the side order.

Exam tip

Always compare sides in matching order before concluding SSS similarity.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

A question may give only side lengths and ask whether similarity can be proved without angle information.
  1. 1

    Enough for 1 mark

    Two triangles are similar if all three corresponding sides are proportional.

  2. 2

    Add for 2 marks

    Example: Sides 7 cm, 9 cm, 12 cm and 14 cm, 18 cm, 24 cm are proportional in the ratio 1:2, so the triangles are similar.

  3. 3

    Add for 3 marks

    When every matching side ratio is the same, the triangle keeps the same shape and only the size changes.

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