I was doing the below question from the Manhattan 5lb book (Triangles chapter) and I realized that there are 2 ways to solve this:

Angle ABD = 180-2x (supplementary angles) and hence angle ADB =x.

Approach 1: We can see that angle x is opp to AB and angle 2x is opp to DC. 2x> x so DC>AB and hence choose (A)

Approach 2: If we use the isosceles triangle property, then AB=BD (as angle DAB=ADB=x). Since angle DBC= angle DCB, so BD=DC. Therefore, AB=BD=DC so go with ( C ).

Manhattan has given the second approach in their answer guide. I thought of both the approaches and ended up choosing (D) because of conflicting answers. Iām sure I am missing some concept. Could someone explain why my 1st approach is wrong?