Reference angles worksheet with diagrams and degree values.

Let’s solve each problem step by step. The reference angle is the smallest positive acute angle (less than 90°) that the terminal side of a given angle makes with the x-axis.

We’ll go one by one:

---

1) Angle = 164°

This is in Quadrant II (between 90° and 180°).
Reference angle = 180° - 164° = 16°

✔ Check: 164° is 16° away from 180° → correct.

---

2) Angle = 465°

First, reduce it to an angle between 0° and 360° by subtracting 360°:
465° - 360° = 105°

105° is in Quadrant II.
Reference angle = 180° - 105° = 75°

✔ Check: 105° is 75° away from 180° → correct.

---

3) Angle shown as 70° below negative y-axis? Wait — let’s interpret carefully.

Looking at diagram #3: It shows an angle going clockwise from the positive x-axis down past the negative y-axis, labeled “70°” near the bottom left quadrant.

Actually, if you look at the arrow: it starts on positive x-axis, goes clockwise, passes negative y-axis, and stops 70° beyond it into the third quadrant? Or is it measured from the negative y-axis?

Wait — better interpretation: In standard position, angles are measured counterclockwise unless marked otherwise. But here, the arrow is drawn clockwise, and labeled “70°” next to the terminal side in the third quadrant.

Actually, looking again: The angle is drawn starting from positive x-axis, going clockwise, and the label “70°” is placed between the terminal side and the negative y-axis.

So total rotation clockwise: 90° (to negative y-axis) + 70° = 160° clockwise → which is equivalent to -160° or 200° (since 360° - 160° = 200°).

But wait — maybe simpler: If the terminal side is 70° past the negative y-axis toward the negative x-axis, then from positive x-axis counterclockwise, that would be 180° + 20° = 200°? Let me think differently.

Alternative approach: Reference angle is always the acute angle to the nearest x-axis.

In diagram #3, the terminal side is in the third quadrant, and the angle between the terminal side and the negative y-axis is 70°. So the angle between terminal side and negative x-axis is 90° - 70° = 20°.

Therefore, reference angle = 20°

✔ Because reference angle is to the x-axis, not y-axis. Since it's 20° from the negative x-axis, reference angle is 20°.

---

4) Angle = -225°

Negative means clockwise rotation.

-225° + 360° = 135° (coterminal angle)

135° is in Quadrant II.

Reference angle = 180° - 135° = 45°

✔ Alternatively: From positive x-axis, rotating clockwise 225° lands you in Quadrant II (same as 135° CCW), and 180° - 135° = 45° → correct.

---

5) Angle = -392°

Reduce to coterminal angle between 0° and 360°.

Add 360° until within range:

-392° + 360° = -32°
Still negative → add another 360°:
-32° + 360° = 328°

328° is in Quadrant IV.

Reference angle = 360° - 328° = 32°

✔ Check: 328° is 32° short of full circle → reference angle 32° → correct.

---

6) Angle = 255°

255° is in Quadrant III (between 180° and 270°).

Reference angle = 255° - 180° = 75°

✔ Correct: 255° is 75° past 180° → reference angle 75°.

---

7) Angle = -114°

Convert to positive coterminal angle:

-114° + 360° = 246°

246° is in Quadrant III.

Reference angle = 246° - 180° = 66°

✔ Check: 246° - 180° = 66° → correct.

Alternatively, without converting: -114° is same as rotating clockwise 114°, which lands in Quadrant III (since 90° < 114° < 180°), and the reference angle is 114° - 90°? No — wait.

Better stick with coterminal: -114° ≡ 246°, ref angle = 246 - 180 = 66° → yes.

---

Now compile all answers:

1) 16°
2) 75°
3) 20°
4) 45°
5) 32°
6) 75°
7) 66°

Double-check #3 again because it was tricky.

Diagram #3: Terminal side in third quadrant, labeled 70° from negative y-axis. So from negative y-axis to terminal side is 70°, meaning from negative x-axis to terminal side is 20° (because 90° - 70° = 20°). Reference angle is measured to x-axis → so 20° → correct.

All others verified.

Final Answer:
1) 16°
2) 75°
3) 20°
4) 45°
5) 32°
6) 75°
7) 66°