Theorem 6.3 Class 10 (AAA Criteria) : 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.
Given: In two triangles ΔABC and ΔDEF
Such that
∠A = ∠D, ∠B = ∠E and ∠C = ∠F
To Prove: ΔABC ~ ΔDEF
Construction: Draw line PQ on DE and DF such that DP=AB and DQ=AC.
Proof: In ΔABC and ΔDPQ
⇒ AB=DP (By construction)
⇒ AC=DQ (By construction)
⇒ ∠A = ∠D (Given)
So, ΔABC ≅ ΔDPQ (By SAS rule)
⇒ ∠B = ∠P (CPCT) …… (i)
⇒ ∠B = ∠E (Given) …….(ii)
from eq. (i) & (ii)
⇒ ∠P = ∠E
Hence, PQ || EF.
By Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.