4.2 Similarity · Part 2

CA RP AB PQ BC QR CA RP = Same shape and same size. P Q R Fig. . D D ABC PQR  ∠ = ∠ ∠ = ∠ ∠ = ∠ P Q R AB PQ BC QR CA RP ¹ ¹ ¹ but AB PQ BC QR CA RP or > < Same shape but not same size. Thinking Corner . Are square and a rhombus similar or congruent. Discuss. . Are a rectangle and a parallelogram similar. Discuss. . . Criteria of Similarity The following criteria are sufficient to prove that two triangles are similar. AA Criterion of similarity If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar, because the third angle in both triangles must be equal. Therefore, AA similarity criterion is same as the AAA similarity criterion. So if ∠ = ∠ P and ∠ = ∠ Q then D D ABC PQR  SAS Criterion of similarity If one angle of a triangle is equal to one angle of another triangle and if the sides including them are proportional then the two triangles are similar. Thus if ∠ = ∠ P and AB PQ AC PR then D D ABC PQR 