ABC is an acute angle triangle with points D and E on BC and AC, respectively such that BE and AD are altitudes. AD and BE intersect at H. If \angle BCA = 39 ^\circ and \angle EBA = 2 \angle DAB , what is the measure of \angle ABC(in degrees)?

To find the measure of angle ABC, we need to use the given information and apply some angle relationships.

Let's start with the information we are given:
- ABC is an acute angle triangle, which means that all three angles are less than 90 degrees.
- BE and AD are altitudes, which means that they are perpendicular to BC and AC, respectively.
- AD and BE intersect at H.

Now, let's consider the given angle measures:
- \angle BCA = 39 degrees
- \angle EBA = 2 \angle DAB

To find the measure of angle ABC, we need to use the fact that the three angles of any triangle add up to 180 degrees. So, we can start by finding the measure of angle BAC:

\angle BAC = 180 - \angle BCA - \angle ABC

We know that \angle BCA = 39 degrees, so we can substitute it into the equation:

\angle BAC = 180 - 39 - \angle ABC
\angle BAC = 141 - \angle ABC

Now, let's consider the relationship between angles EBA and DAB. We are given that \angle EBA = 2 \angle DAB. Substituting this into the equation, we get:

\angle BAC = 141 - (2 \angle DAB)

Since we know that AD is an altitude, it is perpendicular to BC. This means that \angle DAB is a right angle, which is 90 degrees. Substituting this into the equation, we get:

\angle BAC = 141 - (2 * 90)
\angle BAC = 141 - 180
\angle BAC = -39

Uh-oh! We have a problem here. Since ABC is an acute triangle, all three angles must be less than 90 degrees. However, we obtained a negative angle for \angle BAC, which is not possible.

There might be an error or inconsistency in the problem statement or the given information. Please double-check the given data and constraints.

To find the measure of angle ABC, we can start by considering the relationship between angle EBA and angle DAB.

Given that angle EBA is twice the measure of angle DAB, we can represent this relationship as:

∠EBA = 2 * ∠DAB

Since BE is an altitude, it forms a right angle with the base BC. Therefore, we have:

∠EBA + ∠ABC = 90°

Substituting the relationship between angles EBA and DAB, we have:

2 * ∠DAB + ∠ABC = 90°

We also know that angle BCA is 39°, and since ABC and BCA are both angles in triangle ABC, the sum of their measures must be 180°. Therefore:

∠ABC + ∠BCA = 180°

Rearranging this equation for ∠ABC, we have:

∠ABC = 180° - ∠BCA

Substituting the given measure of ∠BCA (39°), we get:

∠ABC = 180° - 39°

Simplifying, we find:

∠ABC = 141°

Therefore, the measure of angle ABC is 141 degrees.