To solve for angle $\alpha$ in this right triangle, we can use trigonometric ratios. Since we are given the lengths of the opposite side $a = 3$ and the adjacent side $b = 7$, we will use the tangent function:

$\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{3}{7}$

Now, to find $\alpha$, we take the arctangent (inverse tangent) of the ratio:

$\alpha = \tan^{-1} \left( \frac{3}{7} \right)$

Let me calculate that for you.The measure of angle $\alpha$ is approximately $23.2^\circ$ when rounded to the nearest tenth.

Would you like further details on this solution, or do you have any questions?

Here are some related questions for further practice:

1. What is the value of angle $\beta$ in this triangle?
2. How would you calculate the area of the triangle?
3. Can you use the sine or cosine functions to verify $\alpha$?
4. How would the values change if $a$ were doubled while $b$ stayed the same?
5. What is the relationship between the angles in a right triangle?

Tip: Always check your results by ensuring the angles in a triangle add up to 180°.