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SAS similarity criterion

Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.

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Student-friendly explanation

The included angle fixes the opening, and the two surrounding side ratios fix the scale. Together they are enough to match the triangles exactly in shape.

How to write this in exams

  1. 1

    Start with the exact idea

    Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.

  2. 2

    Then show how to use it

    1. Identify the included angle. 2. Compare the two sides around that angle. 3. Check whether the ratios match. 4. State SAS similarity.

  3. 3

    Add one concrete example

    If ∠A = ∠P and AB/PQ = AC/PR = 2/3, then triangles ABC and PQR are similar.

  4. 4

    Avoid this incomplete answer

    Using one equal angle and any two proportional sides that do not touch that angle. That is not SAS similarity.

Definition

Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.

Example

If ∠A = ∠P and AB/PQ = AC/PR = 2/3, then triangles ABC and PQR are similar.

Rule to remember

SAS similarity: one included angle equal and the two surrounding sides proportional.

Memory hook

Angle in the middle, sides on both sides.

Examples and method

Worked example

If ∠B = ∠E and AB/DE = BC/EF = 4/6 = 2/3, then triangles ABC and DEF are similar by SAS.

Method to apply

1. Identify the included angle. 2. Compare the two sides around that angle. 3. Check whether the ratios match. 4. State SAS similarity.

Diagram support

Draw the included angle first, then label the two adjacent sides and compare their ratios carefully.

How CBSE asks it

A figure may give one equal angle and two side lengths around it, then ask whether the triangles are similar.

Avoid common mistakes

Common confusion

Students often compare the wrong pair of sides or forget that the angle must lie between the compared sides.

Common wrong answer

Using one equal angle and any two proportional sides that do not touch that angle. That is not SAS similarity.

Exam tip

Check the angle that lies between the two sides you are comparing.

Quick check

When should you use SAS similarity for two triangles?

Use SAS similarity when one included angle is equal and the two sides touching that angle are in the same ratio. This is enough to prove that the triangles have the same shape.

Answer writing and exam use

1-mark answer

Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.

2-mark answer

Two triangles are similar if one included angle is equal and the two sides around that angle are proportional. SAS similarity: one included angle equal and the two surrounding sides proportional. If ∠A = ∠P and AB/PQ = AC/PR = 2/3, then triangles ABC and PQR are similar.

3-mark answer

The included angle fixes the opening, and the two surrounding side ratios fix the scale. Together they are enough to match the triangles exactly in shape. SAS similarity: one included angle equal and the two surrounding sides proportional. If ∠B = ∠E and AB/DE = BC/EF = 4/6 = 2/3, then triangles ABC and DEF are similar by SAS. A figure may give one equal angle and two side lengths around it, then ask whether the triangles are similar. Using one equal angle and any two proportional sides that do not touch that angle. That is not SAS similarity.
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