What is the greatest number of acute angles that a triangle can contain?A.0B.3C.2D.1SUBMITarrow_backPREVIOUS

Break Down the Problem

1. Identify the types of angles in a triangle: acute, right, and obtuse.
2. Understand the relationship between the angles in a triangle, which must sum to $180^\circ$.

Relevant Concepts

1. An acute angle is an angle less than $90^\circ$.
2. A triangle can contain different types of angles, but the sum of all angles in a triangle adheres to the rule: $\text{Sum of angles} = 180^\circ$

Analysis and Detail

1. If a triangle were to contain three acute angles, then each angle $A$, $B$, and $C$ must satisfy: $A < 90^\circ, \quad B < 90^\circ, \quad C < 90^\circ$ So, the maximum would be: $A + B + C < 270^\circ$ This contradicts the triangle angle sum property. Therefore, having three acute angles is impossible.

2. If a triangle contains one obtuse angle (greater than $90^\circ$), the remaining two angles must be acute to satisfy the triangle's angle sum: $\text{Obtuse Angle} > 90^\circ \implies \text{Two Acute Angles} < 90^\circ$ Hence, it is possible to have two acute angles while having one obtuse angle.

3. A triangle can also contain one right angle (equal to $90^\circ$), allowing the other two angles to also be acute: $\text{Right Angle} = 90^\circ \implies A + B < 90^\circ$ So, two angles can still be acute if one angle is right.

Verify and Summarize

Evaluating these possibilities:

• Three Acute Angles: Not possible (angle sum exceeds $180^\circ$).
• Two Acute Angles: Possible with one obtuse angle.
• One Acute Angle: Possible, but also more limiting than two acute angles.

Thus, the maximum number of acute angles in a triangle is two.

Final Answer

The greatest number of acute angles that a triangle can contain is 2 (C).