To solve the question of determining the greatest number of acute angles in a right triangle, let's start by understanding the properties of right triangles and angles.
1. Properties of a Right Triangle:
- A right triangle contains one angle that is exactly [tex]\(90^\circ\)[/tex].
- A triangle has three angles that sum up to [tex]\(180^\circ\)[/tex].
2. Understanding Acute Angles:
- An acute angle is any angle that is less than [tex]\(90^\circ\)[/tex].
Given that one angle in a right triangle is always [tex]\(90^\circ\)[/tex], we focus on the other two angles.
3. Calculating the Other Two Angles:
- Let [tex]\(A\)[/tex] and [tex]\(B\)[/tex] be the measures of the other two angles in the right triangle.
- Since the sum of the angles in any triangle is [tex]\(180^\circ\)[/tex], we have the equation:
[tex]\[
A + B + 90^\circ = 180^\circ
\][/tex]
- Simplifying the above equation:
[tex]\[
A + B = 90^\circ
\][/tex]
4. Determining the Nature of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- Both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] must be less than [tex]\(90^\circ\)[/tex] to satisfy the equation [tex]\(A + B = 90^\circ\)[/tex].
- Hence, both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are acute angles (they are less than [tex]\(90^\circ\)[/tex]).
5. Greatest Number of Acute Angles:
- Since both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are acute and there are no other angles in the right triangle apart from the [tex]\(90^\circ\)[/tex] angle, the greatest number of acute angles in a right triangle is [tex]\(\boxed{2}\)[/tex].
Thus, the correct answer to the question is [tex]\( \text{A. 2} \)[/tex].
