# 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) â~ez_circ~ A = â~ez_circ~ D, â~ez_circ~ B = â~ez_circ~ E, â~ez_circ~ C = â~ez_circ~ F and

$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$
In such a case, we write Î"ABC
~ Î"DEF

1) AAA similarity : If two triangles are equiangular( all three angles are equal to each other), then they are similar.
Example : In Î"ABC and Î"DEF, â~ez_circ~ A = â~ez_circ~ D, â~ez_circ~ B = â~ez_circ~ E and â~ez_circ~ C= â~ez_circ~ F then Î"ABC ~ Î"DEF 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 Î"PQR and Î"DEF, â~ez_circ~ P = â~ez_circ~ D, â~ez_circ~ R = â~ez_circ~ F then Î"PQR ~ Î"DEF by AA criteria.

3) SSS similarity : If the corresponding sides of two triangles are proportional, then the two triangles are similar.
Example : In Î"XYZ and Î"LMN, XY = LM, YZ = MN and XZ = LN then
Î"XYZ ~ Î"LMN 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, â~ez_circ~ A = â~ez_circ~ D
$\frac{AB}{DE}=\frac{AC}{DF}$
Then the two triangles ABC and DEF are similar by SAS.

Criteria for Similarity

â~ez_euro~¢ AAA Similarity
â~ez_euro~¢ AA Similarity
â~ez_euro~¢ SSS Similarity
â~ez_euro~¢ SAS Similarity
â~ez_euro~¢ Practice on Similarity

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