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Using similarity to prove relations

Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.

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Student-friendly explanation

Similarity is not only a naming result. It becomes a tool for proving unknown lengths, ratios, and geometric relations inside a figure.

How to write this in exams

  1. 1

    Start with the exact idea

    Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.

  2. 2

    Then show how to use it

    1. Prove the triangles similar. 2. Match the corresponding vertices. 3. Write the ratio of matching sides. 4. Substitute and solve the required relation.

  3. 3

    Add one concrete example

    If triangles ABC and DEF are similar, then AB/DE = BC/EF = AC/DF can be used to find a missing side or check a proportion.

  4. 4

    Avoid this incomplete answer

    Mixing non-corresponding sides, such as AB with EF, which breaks the ratio comparison.

Definition

Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.

Example

If triangles ABC and DEF are similar, then AB/DE = BC/EF = AC/DF can be used to find a missing side or check a proportion.

Rule to remember

If triangles are similar, then corresponding sides are proportional and corresponding angles are equal.

Memory hook

Prove similarity once, then ratios do the rest.

Examples and method

Worked example

If triangles ABC and DEF are similar with AB = 6 cm, DE = 9 cm and BC = 8 cm, then AB/DE = 2/3. So if EF is the matching side, BC/EF = 2/3 gives EF = 12 cm.

Method to apply

1. Prove the triangles similar. 2. Match the corresponding vertices. 3. Write the ratio of matching sides. 4. Substitute and solve the required relation.

Diagram support

Mark the two similar triangles clearly and keep the same vertex order while writing ratios.

How CBSE asks it

A proof question may ask you to derive a side length, show two segments are proportional, or establish a line relation after proving similarity.

Avoid common mistakes

Common confusion

Students often stop after writing the similarity statement and do not use the corresponding side ratios.

Common wrong answer

Mixing non-corresponding sides, such as AB with EF, which breaks the ratio comparison.

Exam tip

After proving similarity, write only matching side ratios and keep the vertex order correct.

Quick check

Why is proving triangle similarity useful in geometry questions?

Similarity is useful because it gives equal angles and proportional corresponding sides. That lets us prove missing lengths, side ratios, and other relations cleanly.

Answer writing and exam use

1-mark answer

Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results.

2-mark answer

Once two triangles are proved similar, you can use corresponding side ratios and equal angles to prove lengths, proportions, and parallel-line results. If triangles are similar, then corresponding sides are proportional and corresponding angles are equal. If triangles ABC and DEF are similar, then AB/DE = BC/EF = AC/DF can be used to find a missing side or check a proportion.

3-mark answer

Similarity is not only a naming result. It becomes a tool for proving unknown lengths, ratios, and geometric relations inside a figure. If triangles are similar, then corresponding sides are proportional and corresponding angles are equal. If triangles ABC and DEF are similar with AB = 6 cm, DE = 9 cm and BC = 8 cm, then AB/DE = 2/3. So if EF is the matching side, BC/EF = 2/3 gives EF = 12 cm. A proof question may ask you to derive a side length, show two segments are proportional, or establish a line relation after proving similarity. Mixing non-corresponding sides, such as AB with EF, which breaks the ratio comparison.
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