SOLUTION: (31) If AB = BC = CD and ∠BCA = 54°, then find the measure of angle ABD. Diagram: https://ibb.co/zVZdT58

In triangle ABC, AB = BC (given). 

It means that triangle ABC is isisceles, with base angles BAC and ACB of 54 degrees.

Hence, the angle ABC at the vertex is  180 - 54 - 54 = 180 - 108 = 72 degrees.



In triangle BCD, BC = CD (given).

It means that triangle BCD is isosceles.

Hence, its angles CBD and CDB are congruent, as the base angles.

The sum of these angles, CBD and CDB, is equal to the exterior angle ACD, which is 54 degrees.



Hence, angle CBD is half of 54 degrees, i.e. 27 degrees.



Now, angle ABD is the sum of angles ABC (72 degrees) and CBD (27 degrees)

So,  angle ABD = 72 + 27 = 99 degrees.


At this point, the problem is solved completely.


ANSWER.  Angle ABD is 99 degrees.