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 ∠BCA=39∘ and ∠EBA=2∠DAB, what is the measure of ∠ABC (in degrees)?

To find the measure of angle ABC, we need to use the given information about the acute triangle ABC.

Let's break down the given information:
1. ABC is an acute angle triangle.
2. Points D and E are on sides BC and AC, respectively, such that BE and AD are altitudes.
3. AD and BE intersect at point H.
4. ∠BCA = 39° (given)
5. ∠EBA = 2∠DAB (given)

Let's start by analyzing the information given.

Since AD is an altitude, it is perpendicular to BC, and since BE is an altitude, it is perpendicular to AC. This means that triangles ABD and BEC are right triangles.

Let's consider the angle relations:
∠DAB (angle ADB) is a right angle because AD is perpendicular to BC.
∠BEA (angle BEC) is also a right angle because BE is perpendicular to AC.

We are given that ∠BCA = 39°. We also know that the sum of the angles in any triangle is 180°. Therefore, the sum of ∠BCA and ∠ACB must be 180 - 39 = 141°.

Now, let's look at the relation ∠EBA = 2∠DAB. Since ∠DAB is a right angle, we can rewrite it as ∠EBA = 2 * 90°.

This means that ∠EBA = 180°. Since ∠EBA is the same as ∠ABC, we have ∠ABC = 180°.

Therefore, the measure of angle ABC is 180 degrees.