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The image presents a problem involving two parallel lines cut by a transversal. The problem provides the measure of \( \angle 4 = 123^\circ \) and asks for the measures of \( \angle 5 \) and \( \angle 7 \).

### Step 1: Identify the relationship between the angles
- **\( \angle 4 \) and \( \angle 5 \)**: These are **consecutive interior angles** (also known as co-interior angles). For parallel lines cut by a transversal, consecutive interior angles are supplementary, meaning they add up to \( 180^\circ \).
- **\( \angle 4 \) and \( \angle 7 \)**: These are **alternate exterior angles**. For parallel lines cut by a transversal, alternate exterior angles are equal.

### Step 2: Calculate the angles
- Since \( \angle 4 \) and \( \angle 5 \) are supplementary:
  \[
  \angle 5 = 180^\circ - \angle 4 = 180^\circ - 123^\circ = 57^\circ
  \]

- Since \( \angle 4 \) and \( \angle 7 \) are equal (alternate exterior angles):
  \[
  \angle 7 = \angle 4 = 123^\circ
  \]

### Final Answers:
- \( \angle 5 = 57^\circ \)
- \( \angle 7 = 123^\circ \)

Would you like more details on any of these steps, or do you have any additional questions?

### Relative Questions:
1. What is the difference between alternate exterior and alternate interior angles?
2. How can you use a transversal to prove that two lines are parallel?
3. Why are consecutive interior angles supplementary in parallel lines?
4. Can you find \( \angle 6 \) if \( \angle 2 \) is given? How?
5. What is the relationship between corresponding angles in parallel lines?

### Tip:
When dealing with angles formed by parallel lines and a transversal, remember that many angles are either congruent or supplementary. This makes it easier to find unknown angles if you know one angle measure.

Learn how to solve angle problems involving parallel lines cut by a transversal. This problem provides specific angle measures and asks for the measures of related angles. Understand the concepts of consecutive interior angles and alternate exterior angles in parallel lines. Suitable for Grades 9-12.

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