Class 10 Maths Triangles Criteria for Similarity of Triangles

Criteria for Similarity of Triangles

The symbol ~ stands for ‘ is similar to’. As done in the case of congruency of two triangles, the similarity of two triangles should also be expressed symbolically, using correct correspondence of their vertices. For example, for the triangles ABC and DEF, we cannot write Δ ABC ~ Δ EDF or Δ ABC ~ Δ FED. However, we can write Δ BAC ~ Δ EDF. Two triangles are similar, if

• their corresponding angles are equal or
• their corresponding sides are in the same ratio (or proportion).

That is, Δ ABC and Δ DEF are similar , if

(i) ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F or

(ii) AB/ DE =  CA/FD = BC/EF

Theorem 3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

This criterion is referred to as the AAA (Angle–Angle–Angle) criterion of similarity of two triangles. Refer ExamFear video lesson for Proof for this theorem.

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This may be referred to as the AA similarity criterion for two triangles.

Theorem 4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are  similiar. This criterion is referred to as the SSS (Side–Side–Side) similarity criterion for two triangles. Refer ExamFear video lesson for Proof for this theorem.

Theorem 5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles. Refer ExamFear video lesson for Proof for this theorem.

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