Question

what is the greatest number of acute angles that a right triangle can contain? a. 0 b. 1 c. 3 d. 2

Explanation:

Step1: Recall triangle angle sum

A triangle's angles sum to \(180^\circ\). A right triangle has one \(90^\circ\) angle.

Step2: Analyze acute angles

Let the right angle be \(90^\circ\). The remaining two angles sum to \(180^\circ - 90^\circ = 90^\circ\). Each of these must be less than \(90^\circ\) (acute), so a right triangle can have 2 acute angles. Wait, no—wait, in a triangle, can there be three acute? No, because right triangle has one right angle. Wait, wait, no: wait, the sum of angles in a triangle is \(180^\circ\). A right triangle has one \(90^\circ\) angle. The other two angles must add up to \(90^\circ\), so each is less than \(90^\circ\) (acute). But wait, is there a case with three? No, because one is \(90^\circ\). Wait, no—wait, maybe I made a mistake. Wait, no: a right triangle has one right angle (\(90^\circ\)) and the other two angles are acute (since their sum is \(90^\circ\), so each is less than \(90^\circ\)). But wait, can a triangle have three acute angles? Yes, an acute triangle. But a right triangle has one right angle, so the other two are acute. Wait, but the question is the greatest number of acute angles in a right triangle. So the right triangle has one right angle, and the other two are acute. Wait, but wait—no, wait: the sum of angles is \(180^\circ\). If a triangle is right-angled, one angle is \(90^\circ\), so the other two sum to \(90^\circ\), so both are acute (less than \(90^\circ\)). So the maximum number of acute angles in a right triangle is 2? Wait, no, wait—wait, no, that's wrong. Wait, no: in a triangle, the sum of angles is \(180^\circ\). A right triangle has one \(90^\circ\) angle. The other two angles must add up to \(90^\circ\), so each is less than \(90^\circ\) (acute). So two acute angles. But wait, the options have C as 3, D as 2. Wait, maybe I messed up. Wait, no—wait, no, a right triangle cannot have three acute angles because one angle is \(90^\circ\). So the maximum number of acute angles in a right triangle is 2? But wait, the options: A.0, B.1, C.3, D.2. Wait, no—wait, maybe I made a mistake. Wait, no: let's re-express. The sum of angles in a triangle is \(180^\circ\). A right triangle has one angle of \(90^\circ\). The other two angles must sum to \(90^\circ\). Therefore, each of these two angles must be less than \(90^\circ\) (since their sum is \(90^\circ\), so neither can be \(90^\circ\) or more). Therefore, a right triangle has two acute angles. Wait, but the option D is 2. But wait, the question is "greatest number of acute angles that a right triangle can contain". So the answer is 2? Wait, no—wait, no, wait: in a triangle, can there be three acute angles? Yes, in an acute triangle. But a right triangle has one right angle, so the other two are acute. So the maximum number of acute angles in a right triangle is 2. So the correct option is D.2? Wait, but wait, the options: A.0, B.1, C.3, D.2. So the answer is D.2.

Answer:

D. 2