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Criteria for Similarity of Triangles




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

(ii)
AB BC CA
---- = ------ = ------
DE EF FD

In such a case, we write ΔABC
~ ΔDEF

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
XY YZ XZ
---- = ------ = ------
LM MN 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 and AB BC CA
---- = ------ = ------
DE EF FD
Then the two triangles ABC and DEF are similar by SAS.


Criteria for Similarity

AAA Similarity
AA Similarity
SSS Similarity
SAS Similarity
Practice on Similarity

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