Given that $\alpha$ is an acute angle, determine the range of $2\alpha$. The options are: [A] First quadrant angle [B] Second quadrant angle [C] Positive angle less than $180^\circ$ [D] Positive angle no greater than a right angle

1. Problem Description

Given that $\alpha$ is an acute angle, determine the range of $2\alpha$. The options are:

[A] First quadrant angle

[B] Second quadrant angle

[C] Positive angle less than $180^\circ$

[D] Positive angle no greater than a right angle

2. Solution Steps

Since $\alpha$ is an acute angle, we have $0^\circ < \alpha < 90^\circ$.

Multiplying the inequality by 2, we get $0^\circ < 2\alpha < 180^\circ$.

Thus, $2\alpha$ is a positive angle less than $180^\circ$.

Also, since $0^\circ < 2\alpha < 180^\circ$, $2\alpha$ can be in the first or second quadrant.

Option A: If $0^\circ < 2\alpha < 90^\circ$, then $2\alpha$ is in the first quadrant. This is possible.

Option B: If $90^\circ < 2\alpha < 180^\circ$, then $2\alpha$ is in the second quadrant. This is also possible.

Option C: Since $0^\circ < 2\alpha < 180^\circ$, $2\alpha$ is a positive angle less than $180^\circ$.

Option D: If $0^\circ < 2\alpha \le 90^\circ$, then $2\alpha$ is a positive angle no greater than a right angle. However, $2\alpha$ can be greater than $90^\circ$, so this option is not always true.

Since $2\alpha$ can be in the first quadrant ($0 < 2\alpha < 90$) or in the second quadrant ($90 < 2\alpha < 180$).

However, the range $0^\circ < 2\alpha < 180^\circ$ implies that it is a positive angle less than 180 degrees.

3. Final Answer

[C] 小于180度的正角

Positive angle less than $180^\circ$.