Math worksheet for finding missing angles using given angle measurements.

Let's solve each of the six angle problems in the image step by step. The goal is to find all angle measurements using the given angles and properties of parallel lines, transversals, and angle relationships (such as vertical angles, corresponding angles, alternate interior/exterior angles, supplementary angles, etc.).

We'll go through each diagram one at a time.

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Diagram 1:

- Given: ∠LMQ = 105°
- Lines: RQ and SP are horizontal and parallel.
- Line LO is a transversal cutting through both.

#### Step-by-step:
- ∠LMQ = 105° → this is the angle between line LO and RQ at point M.
- Since RQ || SP, and LO is a transversal:
- ∠LMQ and ∠LNS are corresponding angles → so ∠LNS = 105°
- ∠LMQ and ∠NMS are vertical angles → so ∠NMS = 105°
- ∠NMS and ∠LNP are corresponding angles → ∠LNP = 105°

Now look at adjacent angles:
- ∠LMQ + ∠QMO = 180° (linear pair) → ∠QMO = 180° – 105° = 75°
- So, ∠QMO = 75° → then its corresponding angle ∠PNO = 75°
- ∠QMO and ∠MNS are vertical angles → ∠MNS = 75°
- Similarly, ∠PNO and ∠ONM are vertical → ∠ONM = 75°

So angles at M:
- ∠LMQ = 105°
- ∠QMO = 75°
- ∠NMS = 105°
- ∠SMO = 75°? Wait — let’s label carefully.

Actually, at point M:
- ∠LMQ = 105° (top-left)
- Adjacent angle on straight line RQ: ∠QMO = 75°
- At intersection M, the vertical opposite of ∠LMQ is ∠SNO → wait, no.

Better to list angles at each point:

At point M:
- ∠LMQ = 105° (between L-M-Q)
- ∠QMO = 75° (adjacent, forms linear pair)
- Vertical angle to ∠LMQ is ∠SMN = 105°
- Vertical angle to ∠QMO is ∠RML = 75°

Wait, let’s define points clearly.

Assume:
- RQ and SP are parallel lines.
- Transversal LO intersects RQ at M and SP at N.
- So we have two intersections: M and N.

At M:
- ∠LMQ = 105° (angle between LM and MQ)
- Then ∠QMO = 180° – 105° = 75° (since they form a straight line)
- ∠LMQ and ∠NMS are vertical angles? No — ∠NMS is at N.

Wait — actually, ∠LMQ and ∠SNL are not directly connected.

Let’s use standard notation.

Let’s name the angles around point M:

- ∠LMQ = 105° → this is the angle above the line RQ, between L-M and M-Q.
- Then, the angle below RQ, between L-M and M-O is ∠LMO = 180° – 105° = 75°
- But since LO is a straight line, angles on either side of it add up.

Actually, better to think:

At point M:
- The transversal LO crosses RQ.
- ∠LMQ = 105° → this is one angle formed.
- Its adjacent angle (on the same side of LO) along RQ is ∠QMO = 180° – 105° = 75°
- The vertical angle to ∠LMQ is ∠SMN = 105° (opposite angle across LO)
- The vertical angle to ∠QMO is ∠RML = 75°

Similarly, at point N (where LO crosses SP):
- ∠LNS = 105° (corresponding to ∠LMQ)
- ∠PNO = 75° (corresponding to ∠QMO)
- ∠LNS and ∠MNS are adjacent → ∠MNS = 75°? No.

Wait: ∠LNS is the angle between LN and NS → that’s the top angle at N.
Then the adjacent angle along SP is ∠PNO = 75°
And vertical angles:
- ∠LNS = 105° → vertical angle is ∠MNS = 105°? No — vertical angle would be on the other side of LO.

Actually, at point N:
- ∠LNS = 105° (top-right)
- Its vertical angle is ∠MNP = 105° (bottom-left)
- Adjacent angles: ∠SNP = 75° (top-left), ∠PNL = 75° (bottom-right)

But let’s organize.

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✔ Standard approach:

When a transversal cuts two parallel lines, we use:

- Vertical angles: equal
- Corresponding angles: equal
- Alternate interior angles: equal
- Supplementary angles: add to 180° (linear pairs)

Let’s do each diagram systematically.

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## ✔ Diagram 1: ∠LMQ = 105°

Lines RQ || SP, transversal LO.

At point M:
- ∠LMQ = 105° → angle between L-M and M-Q (upper right)
- Then:
- ∠QMO = 180° – 105° = 75° (adjacent, linear pair)
- ∠SMN = 105° (vertical angle to ∠LMQ)
- ∠RML = 75° (vertical angle to ∠QMO)

At point N:
- Since RQ || SP, and LO is transversal:
- ∠LNS = ∠LMQ = 105° (corresponding angles)
- ∠PNO = ∠QMO = 75° (corresponding)
- ∠MNS = ∠LMQ = 105°? No — ∠MNS is the angle between M-N and N-S → that’s the same as ∠LNS?

Wait: Let's define:

At N:
- ∠LNS = 105° → angle between L-N and N-S (top-left)
- Then ∠PNO = 75° (adjacent, bottom-right)
- Vertical angles:
- ∠LNS and ∠MNP are vertical → ∠MNP = 105°
- ∠PNO and ∠SNL are vertical → ∠SNL = 75°

But ∠SNL is same as ∠LNS? No — need to avoid confusion.

Let’s assign:

At M:
- ∠1 = ∠LMQ = 105°
- ∠2 = ∠QMO = 75° (adjacent)
- ∠3 = ∠SMN = 105° (vertical to ∠1)
- ∠4 = ∠RML = 75° (vertical to ∠2)

At N:
- ∠5 = ∠LNS = 105° (corresponding to ∠1)
- ∠6 = ∠PNO = 75° (corresponding to ∠2)
- ∠7 = ∠MNP = 105° (vertical to ∠5)
- ∠8 = ∠SNL = 75° (vertical to ∠6)

So all angles are:
- 105°, 75°, 105°, 75°, 105°, 75°, 105°, 75°

So total: Four angles of 105°, four of 75°

But only 8 angles total.

✔ Answer for Diagram 1:
All angles are either 105° or 75°, alternating.

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## ✔ Diagram 2: ∠LMQ = 92°

Same setup: RQ || SP, transversal LO.

Given: ∠LMQ = 92°

Then:
- ∠QMO = 180° – 92° = 88°
- ∠SMN = 92° (vertical angle)
- ∠RML = 88° (vertical angle)

At N:
- ∠LNS = 92° (corresponding to ∠LMQ)
- ∠PNO = 88° (corresponding to ∠QMO)
- ∠MNP = 92° (vertical to ∠LNS)
- ∠SNL = 88° (vertical to ∠PNO)

✔ Answer: Angles are 92° and 88°

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## ✔ Diagram 3: ∠LMQ = 129°

Given: ∠LMQ = 129°

Then:
- ∠QMO = 180° – 129° = 51°
- ∠SMN = 129° (vertical)
- ∠RML = 51° (vertical)

At N:
- ∠LNS = 129° (corresponding)
- ∠PNO = 51° (corresponding)
- ∠MNP = 129° (vertical)
- ∠SNL = 51° (vertical)

✔ Answer: Angles are 129° and 51°

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## ✔ Diagram 4: ∠LMQ = 41°

Given: ∠LMQ = 41°

Then:
- ∠QMO = 180° – 41° = 139°
- ∠SMN = 41° (vertical)
- ∠RML = 139° (vertical)

At N:
- ∠LNS = 41° (corresponding)
- ∠PNO = 139° (corresponding)
- ∠MNP = 41° (vertical)
- ∠SNL = 139° (vertical)

✔ Answer: Angles are 41° and 139°

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## ✔ Diagram 5: ∠LMQ = 55°

Given: ∠LMQ = 55°

Then:
- ∠QMO = 180° – 55° = 125°
- ∠SMN = 55° (vertical)
- ∠RML = 125° (vertical)

At N:
- ∠LNS = 55° (corresponding)
- ∠PNO = 125° (corresponding)
- ∠MNP = 55° (vertical)
- ∠SNL = 125° (vertical)

✔ Answer: Angles are 55° and 125°

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## ✔ Diagram 6: ∠LMQ = 111°

Given: ∠LMQ = 111°

Then:
- ∠QMO = 180° – 111° = 69°
- ∠SMN = 111° (vertical)
- ∠RML = 69° (vertical)

At N:
- ∠LNS = 111° (corresponding)
- ∠PNO = 69° (corresponding)
- ∠MNP = 111° (vertical)
- ∠SNL = 69° (vertical)

✔ Answer: Angles are 111° and 69°

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## ✔ Summary of All Answers:

| Diagram | Given Angle | Other Angle | All Angles |
|--------|-------------|-------------|------------|
| 1 | 105° | 75° | 105°, 75° (alternating) |
| 2 | 92° | 88° | 92°, 88° |
| 3 | 129° | 51° | 129°, 51° |
| 4 | 41° | 139° | 41°, 139° |
| 5 | 55° | 125° | 55°, 125° |
| 6 | 111° | 69° | 111°, 69° |

> Note: In each case, there are eight angles formed (4 at each intersection), but only two distinct measures: the given angle and its supplement (180° minus the given). These repeat due to vertical and corresponding angles.

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✔ Final Answer:

For each diagram, the angles are:

1. 105° and 75°
2. 92° and 88°
3. 129° and 51°
4. 41° and 139°
5. 55° and 125°
6. 111° and 69°

Each diagram has four angles of one measure and four of the other, depending on position.

Let me know if you'd like a labeled diagram or explanation with angle names!