medium difficulty5 min

SAS similarity criterion

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

In one line: Angle in the middle, sides on both sides.

Understand it

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.

Formula and rule

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

See it worked out

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

Worked example

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

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

When should you use SAS similarity for two triangles?

Think first, then show answer

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.

Common wrong answer

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

Don't mix this up

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

Exam tip

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

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

1
Enough for 1 mark
Two triangles are similar if one included angle is equal and the two sides around that angle are proportional.
2
Add for 2 marks
Example: If ∠A = ∠P and AB/PQ = AC/PR = 2/3, then triangles ABC and PQR are similar.
3
Add for 3 marks
The included angle fixes the opening, and the two surrounding side ratios fix the scale.

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