Find the missing angle using the angle addition postulate.

Let's solve each problem using the Angle Addition Postulate, which states:

> If point $ B $ lies in the interior of $ \angle AOC $, then
> $$
> \angle AOB + \angle BOC = \angle AOC
> $$

In other words, when two adjacent angles form a larger angle, the sum of the smaller angles equals the measure of the larger angle.

We will use this principle to find the missing angle measurements.

---

1)

Given:
- $ \angle DEG = 66^\circ $
- $ \angle GEF = ? $
- $ \angle DEF = 140^\circ $

Since $ \angle DEF $ is the total angle and $ \angle DEG $ and $ \angle GEF $ are adjacent parts:
$$
\angle DEG + \angle GEF = \angle DEF
$$
$$
66^\circ + \angle GEF = 140^\circ
$$
$$
\angle GEF = 140^\circ - 66^\circ = 74^\circ
$$

✔ Answer: $ \angle GEF = 74^\circ $

---

2)

Given:
- $ \angle BCE = 77^\circ $
- $ \angle ECD = 58^\circ $
- $ \angle BCD = ? $

$ \angle BCD $ is the total angle made by $ \angle BCE $ and $ \angle ECD $:
$$
\angle BCD = \angle BCE + \angle ECD = 77^\circ + 58^\circ = 135^\circ
$$

✔ Answer: $ \angle BCD = 135^\circ $

---

3)

Given:
- $ \angle GHK = 71^\circ $
- $ \angle KHJ = 19^\circ $
- $ \angle GHJ = ? $

These two angles make up $ \angle GHJ $:
$$
\angle GHJ = \angle GHK + \angle KHJ = 71^\circ + 19^\circ = 90^\circ
$$

✔ Answer: $ \angle GHJ = 90^\circ $

---

4)

Given:
- $ \angle ABD = ? $
- $ \angle DBC = 65^\circ $
- $ \angle ABC = 130^\circ $

Since $ \angle ABC = \angle ABD + \angle DBC $:
$$
\angle ABD + 65^\circ = 130^\circ
$$
$$
\angle ABD = 130^\circ - 65^\circ = 65^\circ
$$

✔ Answer: $ \angle ABD = 65^\circ $

---

5)

Given:
- $ \angle HUL = ? $
- $ \angle LJK = 17^\circ $
- $ \angle HJK = 100^\circ $

Wait — looking at the diagram: $ \angle HJK $ is the total angle, and it’s composed of $ \angle HUL $? That doesn’t seem right. Let's check the labels.

Actually, from the diagram:
- $ \angle HJK $ is the big angle.
- It's split into $ \angle HUL $ and $ \angle LJK $?

But the points don't match. Let's assume the angle is split at point $ L $, so:
- $ \angle HJK = \angle HJL + \angle LJK $
- But we’re given $ \angle HJK = 100^\circ $, $ \angle LJK = 17^\circ $
- So $ \angle HJL = 100^\circ - 17^\circ = 83^\circ $

But the question asks for $ \angle HUL $. That seems like a typo or mislabeling.

Wait — perhaps $ \angle HUL $ is meant to be $ \angle HJL $? Or maybe the diagram shows $ \angle HUL $ as part of the angle.

Looking carefully: The vertex is $ J $, with rays $ JH $, $ JL $, and $ JK $. So:

- $ \angle HJK = \angle HJL + \angle LJK $
- $ \angle HJK = 100^\circ $
- $ \angle LJK = 17^\circ $
- So $ \angle HJL = 100^\circ - 17^\circ = 83^\circ $

But the blank is for $ \angle HUL $. Unless $ U $ is a typo for $ J $, or $ L $ is labeled incorrectly.

Alternatively, maybe $ \angle HUL $ is supposed to be $ \angle HJL $. But since the label says $ HUL $, and $ U $ is not on the ray, this might be a labeling error.

Wait — looking at the diagram again: Point $ H $, $ U $, $ L $, $ J $, $ K $. Maybe $ U $ is a point on $ HJ $? Then $ \angle HUL $ would be the same as $ \angle HJL $ if $ U $ is on $ HJ $.

Assuming that $ \angle HUL $ is the angle from $ H $ to $ U $ to $ L $, but $ U $ is likely just a point on $ HJ $, so $ \angle HUL $ is the same as $ \angle HJL $.

So:
$$
\angle HUL = \angle HJL = \angle HJK - \angle LJK = 100^\circ - 17^\circ = 83^\circ
$$

✔ Answer: $ \angle HUL = 83^\circ $

---

6)

Given:
- $ \angle OPR = 84^\circ $
- $ \angle RPQ = 26^\circ $
- $ \angle OPQ = ? $

These two angles are adjacent and form $ \angle OPQ $:
$$
\angle OPQ = \angle OPR + \angle RPQ = 84^\circ + 26^\circ = 110^\circ
$$

✔ Answer: $ \angle OPQ = 110^\circ $

---

7)

Given:
- $ \angle PQS = 113^\circ $
- $ \angle SQR = ? $
- $ \angle PQR = 150^\circ $

So:
$$
\angle PQR = \angle PQS + \angle SQR
$$
$$
150^\circ = 113^\circ + \angle SQR
$$
$$
\angle SQR = 150^\circ - 113^\circ = 37^\circ
$$

✔ Answer: $ \angle SQR = 37^\circ $

---

8)

Given:
- $ \angle QRT = 92^\circ $
- $ \angle TRS = ? $
- $ \angle QRS = 126^\circ $

So:
$$
\angle QRS = \angle QRT + \angle TRS
$$
$$
126^\circ = 92^\circ + \angle TRS
$$
$$
\angle TRS = 126^\circ - 92^\circ = 34^\circ
$$

✔ Answer: $ \angle TRS = 34^\circ $

---

9)

Given:
- $ \angle EFH = 110^\circ $
- $ \angle HFG = 36^\circ $
- $ \angle EFG = ? $

So:
$$
\angle EFG = \angle EFH + \angle HFG = 110^\circ + 36^\circ = 146^\circ
$$

✔ Answer: $ \angle EFG = 146^\circ $

---

✔ Final Answers:

| Problem | Missing Angle | Answer |
|--------|----------------|--------|
| 1) | $ \angle GEF $ | $ 74^\circ $ |
| 2) | $ \angle BCD $ | $ 135^\circ $ |
| 3) | $ \angle GHJ $ | $ 90^\circ $ |
| 4) | $ \angle ABD $ | $ 65^\circ $ |
| 5) | $ \angle HUL $ | $ 83^\circ $ |
| 6) | $ \angle OPQ $ | $ 110^\circ $ |
| 7) | $ \angle SQR $ | $ 37^\circ $ |
| 8) | $ \angle TRS $ | $ 34^\circ $ |
| 9) | $ \angle EFG $ | $ 146^\circ $ |

---

📝 Summary of Method:

For each problem, identify whether the unknown angle is:
- Part of a larger angle → subtract known part from total.
- Total of two adjacent angles → add them.

Use the Angle Addition Postulate:
> Adjacent angles add to form the total angle.

All solutions are based on this principle.

Let me know if you'd like this formatted for printing or need diagrams explained!