untitled

Proposition : Let in 'ABC and 'DEF,

EF

BC

DF

AC

DE

AB

.

It is to prove that,

‘A ‘D,‘B ‘E,‘C ‘F.

Construction: Consider the matching sides of the

triangles ABC and DEF unequal. Take two points

P and QonABand AC respectively so that AP =

DE and AQ = DF. Join PandQ.

Proof:

Steps Justificaltin

(1) Since

DF

AC

DE

AB

, so,

AQ

AC

AP

AB

.

Therefore, PQllBC

? ‘ABC ‘APQ and ‘ACB ‘AQP

? Triangles ABC and APQ are equiangular.

Therefore,

PQ

BC

AP

AB

, so,

AQ

BC

DE

AB

.

PQ

BC

EF

BC

? [supposition] ;?

EF

BC

DE

AB

? EF PQ

Therefore, 'APQ and 'DEF are congruent.

?

‘PAQ ‘EDF,‘APQ ‘DEF.‘AQP ‘DFE,

?‘APQ ‘ABC and ‘AQP ‘ACB

‘A ‘D,‘B ‘E,‘C ‘F.

[Theorem 2]

[Corresponding angles made

by the transversal AB]

[Corresponding angles made

by the transversal AC]

[Theorem 5]

[ SSS Theorem]

Theorem 7

If one angle of a triangle is equal to an angle of the other and the sides adjacent

to the equal angles are proportional, the triangles are similar.

Proposition : Let in 'ABC and 'DEF,‘A=‘D

and
DF

AC

DE

AB

.

It is to be proved that the triangles 'ABC

and'DEF are similar.

Construction: Consider the matching sides

of'ABC and 'DEF unequal. Take two pointsP and