Find the missing alternate angles in these geometry problems.

Let's solve each problem step by step using the properties of alternate angles and angles formed by parallel lines and a transversal.

🔍 Key Concepts:

- Alternate interior angles are equal when two parallel lines are cut by a transversal.
- Alternate exterior angles are also equal.
- Supplementary angles (on a straight line) add up to 180°.
- Vertical angles (opposite angles at an intersection) are equal.

We’ll assume that the horizontal lines in each diagram are parallel, as is typical in such problems unless stated otherwise.

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Let’s go through each question:

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1)

Given:
- ∠2 = 123.8°
- ∠3 = 56.2°

We need to find ∠1 and ∠4.

#### Step-by-step:
- ∠3 and ∠1 are alternate interior angles → So ∠1 = ∠3 = 56.2°
- ∠2 and ∠4 are alternate interior angles → So ∠4 = ∠2 = 123.8°

✔ Answer:
- ∠1 = 56.2°
- ∠2 = 123.8°
- ∠3 = 56.2°
- ∠4 = 123.8°

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2)

Given:
- ∠3 = 123°
- ∠4 = 56.1°

Find ∠1 and ∠2.

#### Step-by-step:
- ∠3 and ∠1 are alternate interior angles → ∠1 = ∠3 = 123°
- ∠4 and ∠2 are alternate interior angles → ∠2 = ∠4 = 56.1°

✔ Answer:
- ∠1 = 123°
- ∠2 = 56.1°
- ∠3 = 123°
- ∠4 = 56.1°

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3)

Given:
- ∠1 = 123.3°
- ∠2 = 56.7°

Find ∠3 and ∠4.

#### Step-by-step:
- ∠1 and ∠3 are alternate interior angles → ∠3 = ∠1 = 123.3°
- ∠2 and ∠4 are alternate interior angles → ∠4 = ∠2 = 56.7°

✔ Answer:
- ∠1 = 123.3°
- ∠2 = 56.7°
- ∠3 = 123.3°
- ∠4 = 56.7°

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4)

Given:
- ∠2 = 75.2°
- ∠3 = 104.8°

Find ∠1 and ∠4.

#### Step-by-step:
- ∠2 and ∠4 are alternate interior angles → ∠4 = ∠2 = 75.2°
- ∠3 and ∠1 are alternate interior angles → ∠1 = ∠3 = 104.8°

✔ Answer:
- ∠1 = 104.8°
- ∠2 = 75.2°
- ∠3 = 104.8°
- ∠4 = 75.2°

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5)

Given:
- ∠1 = 74.6°
- ∠4 = 106.4°

Find ∠2 and ∠3.

#### Step-by-step:
- ∠1 and ∠3 are alternate interior angles → ∠3 = ∠1 = 74.6°
- ∠4 and ∠2 are alternate interior angles → ∠2 = ∠4 = 106.4°

✔ Answer:
- ∠1 = 74.6°
- ∠2 = 106.4°
- ∠3 = 74.6°
- ∠4 = 106.4°

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6)

Given:
- ∠1 = 45.2°
- ∠2 = 134.8°

Find ∠3 and ∠4.

#### Step-by-step:
- ∠1 and ∠3 are alternate interior angles → ∠3 = ∠1 = 45.2°
- ∠2 and ∠4 are alternate interior angles → ∠4 = ∠2 = 134.8°

✔ Answer:
- ∠1 = 45.2°
- ∠2 = 134.8°
- ∠3 = 45.2°
- ∠4 = 134.8°

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✔ Final Answers Summary:

| Problem | ∠1 | ∠2 | ∠3 | ∠4 |
|--------|----------|----------|----------|----------|
| 1 | 56.2° | 123.8° | 56.2° | 123.8° |
| 2 | 123° | 56.1° | 123° | 56.1° |
| 3 | 123.3° | 56.7° | 123.3° | 56.7° |
| 4 | 104.8° | 75.2° | 104.8° | 75.2° |
| 5 | 74.6° | 106.4° | 74.6° | 106.4° |
| 6 | 45.2° | 134.8° | 45.2° | 134.8° |

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📌 Explanation Recap:

- Alternate angles are equal when two parallel lines are cut by a transversal.
- In all cases, we used the alternate interior angle rule: angles on opposite sides of the transversal but inside the parallel lines are equal.
- The labeling shows that ∠1 and ∠3 are alternate, and ∠2 and ∠4 are alternate.

✔ All answers are consistent with this rule.

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