Find the missing angles in intersecting lines worksheet.

Let's solve each of these problems step by step. These are all parallel lines cut by a transversal problems, and we'll use the following key angle relationships:

Key Angle Relationships:

1. Vertical angles are equal.
2. Corresponding angles are equal.
3. Alternate interior angles are equal.
4. Alternate exterior angles are equal.
5. Same-side interior angles are supplementary (add to 180°).
6. Linear pairs (adjacent angles on a straight line) add to 180°.

We’ll label the angles in each diagram as follows:

- The top horizontal line has angles: ∠1, ∠2, ∠3, ∠4 (clockwise from top-left)
- The bottom horizontal line has angles: ∠5, ∠6, ∠7, ∠8 (same order)

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Problem 1:

Given: ∠6 = 133.9°

#### Step 1: Use vertical angles
∠6 and ∠8 are vertical → ∠8 = 133.9°
∠5 and ∠7 are vertical → ∠5 = ∠7

#### Step 2: Linear pair
∠6 + ∠5 = 180° → ∠5 = 180 - 133.9 = 46.1°
So ∠7 = 46.1°

#### Step 3: Corresponding angles
Since lines are parallel:
- ∠1 corresponds to ∠5 → ∠1 = 46.1°
- ∠2 corresponds to ∠6 → ∠2 = 133.9°
- ∠3 corresponds to ∠7 → ∠3 = 46.1°
- ∠4 corresponds to ∠8 → ∠4 = 133.9°

#### Final Answers:
- ∠1 = 46.1°
- ∠2 = 133.9°
- ∠3 = 46.1°
- ∠4 = 133.9°
- ∠5 = 46.1°
- ∠6 = 133.9°
- ∠7 = 46.1°
- ∠8 = 133.9°

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Problem 2:

Given: ∠8 = 106.8°

#### Step 1: Vertical angles
∠8 and ∠6 are vertical → ∠6 = 106.8°
∠5 and ∠7 are vertical → ∠5 = ∠7

#### Step 2: Linear pair
∠8 + ∠7 = 180° → ∠7 = 180 - 106.8 = 73.2°
So ∠5 = 73.2°

#### Step 3: Corresponding angles
- ∠1 ↔ ∠5 → ∠1 = 73.2°
- ∠2 ↔ ∠6 → ∠2 = 106.8°
- ∠3 ↔ ∠7 → ∠3 = 73.2°
- ∠4 ↔ ∠8 → ∠4 = 106.8°

#### Final Answers:
- ∠1 = 73.2°
- ∠2 = 106.8°
- ∠3 = 73.2°
- ∠4 = 106.8°
- ∠5 = 73.2°
- ∠6 = 106.8°
- ∠7 = 73.2°
- ∠8 = 106.8°

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Problem 3:

Given: ∠4 = 76.2°

#### Step 1: Vertical angles
∠4 and ∠6 are vertical → ∠6 = 76.2°
∠1 and ∠3 are vertical → ∠1 = ∠3
∠2 and ∠4 are vertical → ∠2 = 76.2°? Wait — no!

Wait: ∠1 and ∠3 are opposite at the top intersection; they are vertical → ∠1 = ∠3
But ∠2 and ∠4 are adjacent → they form a linear pair?

Wait: At the top intersection:
- ∠1 and ∠2 are adjacent (linear pair)
- ∠1 and ∠4 are vertical?
No — let’s clarify labeling.

From standard notation:
At the top intersection:
- ∠1 is top-left
- ∠2 is top-right
- ∠3 is bottom-left
- ∠4 is bottom-right

Then:
- ∠1 and ∠4 are vertical → equal
- ∠2 and ∠3 are vertical → equal
- ∠1 and ∠2 are linear pair → sum to 180°

So given ∠4 = 76.2° → ∠1 = 76.2° (vertical)
Then ∠2 = 180 - 76.2 = 103.8°
And ∠3 = ∠2 = 103.8° (vertical)

Now at bottom intersection:
- ∠5 and ∠8 are vertical
- ∠6 and ∠7 are vertical

But ∠6 corresponds to ∠2 → ∠6 = 103.8°
∠5 corresponds to ∠1 → ∠5 = 76.2°
∠7 corresponds to ∠3 → ∠7 = 103.8°
∠8 corresponds to ∠4 → ∠8 = 76.2°

Also, check: ∠6 and ∠5 are linear pair → 103.8 + 76.2 = 180 → ✔

#### Final Answers:
- ∠1 = 76.2°
- ∠2 = 103.8°
- ∠3 = 103.8°
- ∠4 = 76.2°
- ∠5 = 76.2°
- ∠6 = 103.8°
- ∠7 = 103.8°
- ∠8 = 76.2°

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Problem 4:

Given: ∠3 = 133.8°

At top intersection:
- ∠3 is bottom-left
- ∠1 is top-left → vertical to ∠3 → ∠1 = 133.8°
- ∠2 is top-right → adjacent to ∠1 → ∠2 = 180 - 133.8 = 46.2°
- ∠4 is bottom-right → vertical to ∠2 → ∠4 = 46.2°

Now bottom intersection:
- ∠5 ↔ ∠1 (corresponding) → ∠5 = 133.8°
- ∠6 ↔ ∠2 → ∠6 = 46.2°
- ∠7 ↔ ∠3 → ∠7 = 133.8°
- ∠8 ↔ ∠4 → ∠8 = 46.2°

Check: ∠6 and ∠5 are linear pair → 46.2 + 133.8 = 180 → ✔

#### Final Answers:
- ∠1 = 133.8°
- ∠2 = 46.2°
- ∠3 = 133.8°
- ∠4 = 46.2°
- ∠5 = 133.8°
- ∠6 = 46.2°
- ∠7 = 133.8°
- ∠8 = 46.2°

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Problem 5:

Given: ∠2 = 65.2°

At top intersection:
- ∠2 = 65.2° → ∠4 is vertical → ∠4 = 65.2°
- ∠1 is adjacent to ∠2 → ∠1 = 180 - 65.2 = 114.8°
- ∠3 is vertical to ∠1 → ∠3 = 114.8°

Bottom intersection:
- ∠5 ↔ ∠1 (corresponding) → ∠5 = 114.8°
- ∠6 ↔ ∠2 → ∠6 = 65.2°
- ∠7 ↔ ∠3 → ∠7 = 114.8°
- ∠8 ↔ ∠4 → ∠8 = 65.2°

Check: ∠5 + ∠6 = 114.8 + 65.2 = 180 → ✔

#### Final Answers:
- ∠1 = 114.8°
- ∠2 = 65.2°
- ∠3 = 114.8°
- ∠4 = 65.2°
- ∠5 = 114.8°
- ∠6 = 65.2°
- ∠7 = 114.8°
- ∠8 = 65.2°

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Problem 6:

Given: ∠7 = 103.7°

At bottom intersection:
- ∠7 = 103.7° → ∠3 is corresponding → ∠3 = 103.7°
- ∠3 and ∠1 are vertical → ∠1 = 103.7°
- ∠2 is adjacent to ∠1 → ∠2 = 180 - 103.7 = 76.3°
- ∠4 is vertical to ∠2 → ∠4 = 76.3°

Now bottom:
- ∠7 = 103.7° → ∠5 is vertical → ∠5 = 103.7°
- ∠6 is adjacent to ∠5 → ∠6 = 180 - 103.7 = 76.3°
- ∠8 is vertical to ∠6 → ∠8 = 76.3°

Check correspondence:
- ∠1 ↔ ∠5 → both 103.7° → ✔
- ∠2 ↔ ∠6 → both 76.3° → ✔
- ∠3 ↔ ∠7 → both 103.7° → ✔
- ∠4 ↔ ∠8 → both 76.3° → ✔

#### Final Answers:
- ∠1 = 103.7°
- ∠2 = 76.3°
- ∠3 = 103.7°
- ∠4 = 76.3°
- ∠5 = 103.7°
- ∠6 = 76.3°
- ∠7 = 103.7°
- ∠8 = 76.3°

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Summary Table:

| Problem | ∠1 | ∠2 | ∠3 | ∠4 | ∠5 | ∠6 | ∠7 | ∠8 |
|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| 1 | 46.1° | 133.9° | 46.1° | 133.9° | 46.1° | 133.9° | 46.1° | 133.9° |
| 2 | 73.2° | 106.8° | 73.2° | 106.8° | 73.2° | 106.8° | 73.2° | 106.8° |
| 3 | 76.2° | 103.8° | 103.8° | 76.2° | 76.2° | 103.8° | 103.8° | 76.2° |
| 4 | 133.8° | 46.2° | 133.8° | 46.2° | 133.8° | 46.2° | 133.8° | 46.2° |
| 5 | 114.8° | 65.2° | 114.8° | 65.2° | 114.8° | 65.2° | 114.8° | 65.2° |
| 6 | 103.7° | 76.3° | 103.7° | 76.3° | 103.7° | 76.3° | 103.7° | 76.3° |

> ✔ All angles have been found using vertical angles, linear pairs, and corresponding angles with parallel lines.

Let me know if you'd like this formatted as a printable answer sheet!