the TSA are said ........... if corresponding angles of 2∆ are equal
Answers
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 triangle are similar.
it can be prove in two methods
We have to prove the corresponding sides are proportional and hence Method 1: (Based on Euclid's)
Place triangle DEF such that E falls on C and EF forms a straight line with BC as in diagram. Produce BA and FD to meet at G.
By suitable pairs of corresponding angles, CA||FG; DC||GB; GACD (or GAED) is a parallelogram.
So AG = CD (or ED); DG = CA
CA is a parallel line in the triangle BFG and so divides BG and BF in same ratio.
BA : AG = BC : CF
So BA : BC = AG : CF
AB : BC = DE : EF [as AG = CD and C and E mark the same point] - - - - (1)
DC is a parallel line in the triangle FGB and so divides FB and FG in same ratio.
BC : CF = GD : DF
BC : CF = CA : FD
BC : CA = CF : FD - - - - (2)
Similarly prove
CA : AB = FD : DE
Method 2:
We know, a line parallel to a side of a triangle divides the other two sides in the same ratio; So
Two sets of corresponding sides are proportional.
Similarly proving another pair of ratios we get
Hence the equiangular triangles are similar.