Is CB||ED? (1) ABC and AED are both isosceles triangles (2) C = E

This question asks us to check the parallel line & it's transversal.

Here, BC & ED are the lines to be checked for parallelism and CD & BE are the transversals.

BC|| ED when the alternate angles in the given figure are equal. i.e, \(\angle{CBA}\)=\(\angle{AED}\) and \(\angle{ACB}\)=\(\angle{ADE}\)

Question stem: BC || ED? (Y/N)

Statement1:-∆ABC and ∆AED are both isosceles triangles

we only know that vertically opposite angles are equal, i.e, \(\angle{CAB}\)=\(\angle{EAD}\) and we don't know their measure in degrees.-----------(a)

Remaining two angles of both the isosceles triangles can have 'n' no of possible combinations.

hence st1 is not sufficient.

Statement1:-∠C = ∠E --------(b)

In ∆ABC and ∆AED, from (a) and (b) ,we have \(\angle{CBA}\)=\(\angle{ADE}\)

So, we can't say whether \(\angle{CBA}\)=\(\angle{AED}\) and \(\angle{ACB}\)=\(\angle{ADE}\) or not.

Hence statement2 is not sufficient.

(1)+(2), we have

(i) ∠C = ∠E
(ii)\(\angle{CAB}\)=\(\angle{EAD}\)
(iii) \(\angle{CBA}\)=\(\angle{ADE}\)
(iv) Both the triangles are isosceles; it has 2 cases:-

1.when \(\angle{E}\)=\(\angle{D}\)

then \(\angle{C}=\angle{D}\) & \(\angle{B}=\angle{E}\)

So, Is BC || ED? Yes (As alternate angles are found equal)

2.when \(\angle{E}=\angle{A}\)

then \(\angle{C}\neq\angle{D}\) & \(\angle{B}\neq\angle{E}\)

So, Is BC || ED? No (As alternate angles are not equal)

Ans. E