Theorem 6.4 Class 10 (SSS Criteria) :
If in two triangles, sides of one triangle are proportional to (i.e., the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
Given: In ΔABC and ΔDEF it is given that
To Prove: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F and ΔABC ~ ΔDEF.
Construction: Draw P and Q on DE & DF such that DP = AB and DQ = AC respectively and join PQ.
So, BC=PQ
Thus, ΔABC ≅ ΔDEF (by SSS Congruence)
So, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
from eq (i) & (ii), we get
∠P = ∠E
and ∠Q = ∠F
therefore,
∠C = ∠Q = ∠F …….(iii)
and, ∠B = ∠P = ∠E ……..(iv)
So, in ΔABC & ΔDEF
∠B = ∠E
∠C = ∠F
ΔABC ~ ΔDEF (by AA similarity criteria)
Hence Proved