Question

list out the similarity criterion for triangles and explain it.​

Answer 1

Answer:

There are three criteria for proving that triangles are similar: AA: If two triangles have two pairs of congruent angles, then the triangles are similar. SAS: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

Step-by-step explanation:

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Answer 2

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There are 3 main criteria for similarity of triangles

1) AAA or AA

2) SSS

3) SAS.

If in two triangles, (i)the corresponding angles are equal, then their corresponding sides are proportional (i.e. in the same ratio) and hence the triangles are similar.

In two triangles ABC and DEF are similar,if

(i) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F and

$\frac{ab}{de} = \frac{bc}{ef} = \frac{ca}{fd}$

In such a case, we write ΔABC ~ ΔDEF

Explain :-

1) AAA similarity :

If two triangles are equiangular( all three angles are equal to each other), then they are similar.

Example : In ΔABC and ΔDEF, ∠A = ∠D, ∠B = ∠E and ∠C= ∠F then ΔABC ~ ΔDEF by AAA criteria.

2) AA similarity :

If two angles of one triangle are respectively equal to tow angles of another triangle, then the two triangles are similar.

Example : In ΔPQR and ΔDEF, ∠P = ∠D, ∠R = ∠F then ΔPQR ~ ΔDEF by AA criteria.

3) SSS similarity :

If the corresponding sides of two triangles are proportional, then the two triangles are similar.

Example : In ΔXYZ and ΔLMN, XY = LM, YZ = MN and XZ = LN then

ΔXYZ ~ ΔLMN by SSS criteria.

Two triangles XYZ and LMN such that

$\frac{xy}{lm} = \frac{yz}{mn} = \frac{xz}{ln}$

Then the two triangles are similar by SSS similarity.

4) SAS similarity :

If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.

In triangle ABC and DEF, ∠A = ∠D

$\frac{ab}{de} = \frac{ac}{df}$

Then the two triangles ABC and DEF are similar by SAS.