## Answer :

1.

**Convert the angle from degrees to radians:**

The formula to convert degrees to radians is:

[tex]\[ \text{angle in radians} = \left(\frac{\text{angle in degrees} \times \pi}{180}\right) \][/tex]

Given the angle is [tex]\(85^\circ\)[/tex]:

[tex]\[ \text{angle in radians} = \left(\frac{85 \times \pi}{180}\right) \][/tex]

Simplifying this expression:

[tex]\[ \text{angle in radians} = \left(\frac{85}{180}\right) \pi \approx 1.4835298641951802 \text{ radians} \][/tex]

2.

**Determine the range category for the angle in radians:**

We need to check within which of the provided ranges [tex]\(1.4835298641951802\)[/tex] radians falls:

- [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex] radians: This range is approximately 0 to 1.5708 radians.

- [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex] radians: This range is approximately 1.5708 to 3.1416 radians.

- [tex]\(\pi \leq \theta < \frac{3\pi}{2}\)[/tex] radians: This range is approximately 3.1416 to 4.7124 radians.

- [tex]\(\frac{3\pi}{2} \leq \theta < 2\pi\)[/tex] radians: This range is approximately 4.7124 to 6.2832 radians.

Comparing [tex]\(1.4835298641951802\)[/tex] radians with these ranges:

- It is not within the range [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex] radians because [tex]\(1.4835298641951802\)[/tex] is slightly less than [tex]\(\frac{\pi}{2}\)[/tex] radians.

- It falls within the range [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex] radians as [tex]\(1.4835298641951802\)[/tex] is less than [tex]\(\pi\)[/tex] but more than [tex]\(\frac{\pi}{2}\)[/tex].

Therefore, the measure of the central angle [tex]\(85^\circ\)[/tex] in radians falls within the range:

[tex]\[ 0 \leq \theta < \frac{\pi}{2} \text{ radians} \][/tex]