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Finding angle measures given two parallel lines cut by a transversal

The image you've provided shows two parallel lines cut by a transversal. You are asked to find the measures of angles \(\angle 5\) and \(\angle 7\) given that \(m \angle 2 = 104^\circ\).

Let's analyze the situation step by step.

### Step 1: Identify the relationship between the angles.
- Since the lines are parallel and the transversal cuts through them, the angles form several pairs of corresponding, alternate interior, and consecutive interior angles.

### Step 2: Use the properties of the angles.
- **Corresponding angles**: These angles are equal. For example, \(\angle 2\) and \(\angle 6\) are corresponding angles.
- **Alternate interior angles**: These angles are equal. For example, \(\angle 2\) and \(\angle 7\) are alternate interior angles.
- **Consecutive interior angles**: These angles are supplementary, meaning their sum is \(180^\circ\). For example, \(\angle 2\) and \(\angle 5\) are consecutive interior angles.

### Step 3: Solve for the requested angles.
1. **Finding \(m \angle 7\)**:
   - Since \(\angle 2\) and \(\angle 7\) are alternate interior angles, \(m \angle 7 = m \angle 2\).
   - Therefore, \(m \angle 7 = 104^\circ\).

2. **Finding \(m \angle 5\)**:
   - \(\angle 2\) and \(\angle 5\) are consecutive interior angles, so they must add up to \(180^\circ\).
   - Therefore, \(m \angle 5 = 180^\circ - m \angle 2\).
   - \(m \angle 5 = 180^\circ - 104^\circ = 76^\circ\).

### Final Answer:
- \(m \angle 5 = 76^\circ\)
- \(m \angle 7 = 104^\circ\)

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What are corresponding angles, and how are they identified?
2. How do alternate interior angles behave when a transversal crosses parallel lines?
3. What is the sum of the angles on a straight line?
4. Can you explain the difference between consecutive interior and alternate exterior angles?
5. How do you calculate the measures of all eight angles formed when two parallel lines are cut by a transversal?

**Tip:** Remember, when two parallel lines are cut by a transversal, many angle pairs have equal measures, which can simplify finding unknown angles!

Learn how to find angle measures \(\angle 5\) and \(\angle 7\) given two parallel lines cut by a transversal. This problem covers key geometric concepts, including corresponding angles, alternate interior angles, and consecutive interior angles. Suitable for Grades 7-9.

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