Given the condition that point O lies in the interior of angle ABC, and the equation:
\[ \text{measure of angle OBC} + \text{measure of angle ABO} = \text{measure of angle ABC} \]
This implies that the angles OBC and ABO together form the angle ABC without any overlap, meaning O is indeed within angle ABC.
To analyze the provided options:
A. BA is between BO and BC: This could be true or not based on the specific configuration of points A, B, C, and O. Without more information about the relative positions of points B, A, and C, we cannot conclude this.
B. measure of angle OBC is smaller than measure of angle ABO: This is not necessarily true; the angles could be equal or OBC could even be larger.
C. BO is between BA and BC: Again, this cannot be definitively concluded from the given information without additional geometric context.
D. Angle ABO and angle OBC are congruent angles: Since the sum of angle OBC and angle ABO equals angle ABC, we cannot directly conclude they are congruent unless they are each half of angle ABC, which is not stated.
E. BC is between BA and BO: This is also not substantiated by the information given.
None of the statements can be conclusively proven as true based on the given information. However, we can infer that the relationship \( \text{measure of angle OBC} + \text{measure of angle ABO} = \text{measure of angle ABC} \) suggests that angle OBC and angle ABO are part of the angle ABC, implying a specific order of angles typically consistent with how angles are formed.
Since none of the options can be definitively supported, we conclude that the options presented do not yield a clear, logically-supported choice based on the given information. Thus, the answer is that it's not possible to conclude one of the options is true without more geometric context.