Isolate the angle by taking the inverse tangent.

We are given the equation:

$$\tan (\theta + 25^\circ) = 5.145$$

To find the angle $(\theta + 25^\circ)$, we take the inverse tangent (arctan) of both sides:

$$\theta + 25^\circ = \arctan(5.145)$$

Using a calculator, we find:

$$\arctan(5.145) \approx 79.00^\circ$$

So,

$$\theta + 25^\circ \approx 79.00^\circ$$

Subtract $25^\circ$ from both sides:

$$\theta \approx 79.00^\circ - 25^\circ$$ $$\theta \approx 54.00^\circ$$

$$\theta \approx 54.0^\circ$$

This value is within the given range $0^\circ \le \theta \le 90^\circ$.

The final answer is $54.0^\circ$.