Find missing angles 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, their measures add up to the measure of the larger angle.

We'll go through each question one by one.

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1)

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

Since $ \angle DEF $ is made up of $ \angle DEG $ and $ \angle GEF $, we can write:
$$
\angle DEG + \angle GEF = \angle DEF
$$
$$
66^\circ + \angle GEF = 140^\circ
$$
$$
\angle GEF = 140^\circ - 66^\circ = 74^\circ
$$

✔ Answer: $ \angle GEF = 74^\circ $

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2)

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

$ \angle BCD $ is the whole angle formed by $ \angle BCE $ and $ \angle ECD $. So:
$$
\angle BCD = \angle BCE + \angle ECD = 77^\circ + 58^\circ = 135^\circ
$$

✔ Answer: $ \angle BCD = 135^\circ $

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3)

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

$ \angle GHJ $ is the sum of $ \angle GHK $ and $ \angle KHJ $:
$$
\angle GHJ = \angle GHK + \angle KHJ = 71^\circ + 19^\circ = 90^\circ
$$

✔ Answer: $ \angle GHJ = 90^\circ $

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4)

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

Note: $ \angle ABC $ is the total angle, composed of $ \angle ABD $ and $ \angle DBC $. But wait — the diagram shows ray $ BD $ inside $ \angle ABC $, so:
$$
\angle ABD + \angle DBC = \angle ABC
$$
$$
\angle ABD + 65^\circ = 130^\circ
$$
$$
\angle ABD = 130^\circ - 65^\circ = 65^\circ
$$

✔ Answer: $ \angle ABD = 65^\circ $

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5)

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

Wait — let’s look carefully. The angle $ \angle HJK $ is made up of $ \angle HUL $? That doesn’t make sense unless points are aligned properly.

But from the diagram (based on naming), it seems that point $ L $ is between $ H $ and $ J $, and $ K $ is another point such that $ \angle HJK $ is split into $ \angle HUL $ and $ \angle LJK $? Wait — the notation may be off.

Actually, looking at the labels:
- $ \angle HJK = 100^\circ $
- $ \angle LJK = 17^\circ $
- $ \angle HUL = ? $

Wait — this might be a typo or mislabeling. Let’s assume the correct structure is that $ \angle HJK $ is split into $ \angle HJL $ and $ \angle LJK $, but here it's written as $ \angle HUL $. That suggests point $ U $ is on the ray from $ H $, and $ L $ is somewhere else.

But given the values:

If $ \angle HJK = 100^\circ $, and $ \angle LJK = 17^\circ $, and assuming $ \angle HJL $ is the rest, then:
$$
\angle HJL = \angle HJK - \angle LJK = 100^\circ - 17^\circ = 83^\circ
$$

But the question asks for $ \angle HUL $. Unless $ U $ is the same as $ L $, this is confusing.

Alternatively, perhaps there's a typo — maybe it should be $ \angle HJL $ instead of $ \angle HUL $. But since $ \angle HUL $ is listed, and $ \angle HJK $ is the large angle, maybe $ U $ is a point along $ HJ $, and $ L $ is a point along $ JK $, forming $ \angle HUL $?

This is ambiguous.

Wait — perhaps the intended meaning is that $ \angle HJK $ is made up of $ \angle HUL $ and $ \angle LJK $, but that doesn't make sense because $ U $ and $ L $ are not on the same ray.

Let me reevaluate: likely, the angle $ \angle HJK $ is split by ray $ JL $, so:
- $ \angle HJL + \angle LJK = \angle HJK $
- $ \angle HJL + 17^\circ = 100^\circ $
- $ \angle HJL = 83^\circ $

But the question asks for $ \angle HUL $. Unless $ U $ is a typo for $ L $, or $ HUL $ refers to $ \angle HJL $, it's unclear.

But given that $ \angle HUL $ is listed as missing, and $ \angle HJK = 100^\circ $, $ \angle LJK = 17^\circ $, and if $ \angle HUL $ is the part from $ H $ to $ U $ to $ L $, and $ U $ is on $ HJ $, $ L $ on $ JK $, then $ \angle HUL $ would be the same as $ \angle HJL $, which is $ 83^\circ $.

So likely, $ \angle HUL $ is meant to be $ \angle HJL $, or there's a labeling issue.

Assuming $ \angle HUL = \angle HJL $, then:
$$
\angle HUL = 100^\circ - 17^\circ = 83^\circ
$$

✔ Answer: $ \angle HUL = 83^\circ $

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6)

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

$ \angle OPQ $ is the total angle from $ O $ to $ P $ to $ Q $, which includes $ \angle OPR $ and $ \angle RPQ $. So:
$$
\angle OPQ = \angle OPR + \angle RPQ = 84^\circ + 26^\circ = 110^\circ
$$

✔ Answer: $ \angle OPQ = 110^\circ $

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7)

Given:
- $ \angle PQS = 113^\circ $
- $ \angle SQR = ? $
- $ \angle PQR = 160^\circ $

$ \angle PQR $ is the total angle, made up of $ \angle PQS $ and $ \angle SQR $:
$$
\angle PQS + \angle SQR = \angle PQR
$$
$$
113^\circ + \angle SQR = 160^\circ
$$
$$
\angle SQR = 160^\circ - 113^\circ = 47^\circ
$$

✔ Answer: $ \angle SQR = 47^\circ $

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8)

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

$ \angle QRS $ is the total angle, made up of $ \angle QRT $ and $ \angle TRS $:
$$
\angle QRT + \angle TRS = \angle QRS
$$
$$
92^\circ + \angle TRS = 126^\circ
$$
$$
\angle TRS = 126^\circ - 92^\circ = 34^\circ
$$

✔ Answer: $ \angle TRS = 34^\circ $

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9)

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

$ \angle EFG $ is the total angle from $ E $ to $ F $ to $ G $, made up of $ \angle EFH $ and $ \angle HFG $. So:
$$
\angle EFG = \angle EFH + \angle HFG = 110^\circ + 36^\circ = 146^\circ
$$

✔ Answer: $ \angle EFG = 146^\circ $

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✔ 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 $ | $ 47^\circ $ |
| 8) | $ \angle TRS $ | $ 34^\circ $ |
| 9) | $ \angle EFG $ | $ 146^\circ $ |

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Summary: All problems use the Angle Addition Postulate — adding or subtracting adjacent angles to find missing parts.

Let me know if you'd like a visual explanation or want these filled into the worksheet!