Theorem 6.3 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 Class 10

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.

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