https://file.aiquickdraw.com/mathbot/file/1728514397773fpoaiblm.jpg

Let's break down the problem:

### Given Equations:
1. The line \( \overline{AB} \) is represented by the equation:
   \[
   y = \frac{3}{5}x + 3
   \]
2. The line \( \overline{CD} \) is represented by the equation:
   \[
   y = \frac{3}{5}x - 1
   \]
   
### Problem Parts:
1. **(a)** You are asked to graph both lines, as well as points \( A \), \( B \), \( C \), \( D \), and \( M \), where \( M \) is the midpoint of \( \overline{AB} \).
2. **(b)** Given that \( \angle BMC = 74^\circ \), you are asked to find \( \angle MCD \).
3. **(c)** You are asked to thoroughly explain why your answer in part (b) must be true.

### Part (a): Graphing the lines
- The slope for both lines is \( \frac{3}{5} \), so the lines are parallel. 
  - For \( \overline{AB} \), the y-intercept is 3.
  - For \( \overline{CD} \), the y-intercept is -1.
- To graph:
  - Start by plotting points for both lines based on their equations.
  - Plot \( A \) and \( B \) as points along the line \( y = \frac{3}{5}x + 3 \).
  - Plot \( C \) and \( D \) as points along the line \( y = \frac{3}{5}x - 1 \).
  - Find the midpoint \( M \) of line \( \overline{AB} \).

### Part (b): Angle Calculation
- Given \( \angle BMC = 74^\circ \), we are asked to find \( \angle MCD \). Since the lines \( \overline{AB} \) and \( \overline{CD} \) are parallel and both intersect at point \( M \), we can use angle properties of parallel lines and the fact that the interior angles on the same side of the transversal sum to 180°.

  Since:
  \[
  \angle BMC + \angle MCD = 180^\circ
  \]
  Therefore:
  \[
  \angle MCD = 180^\circ - 74^\circ = 106^\circ
  \]

### Part (c): Justification
This result is due to the fact that when two parallel lines are intersected by a transversal (line through point \( M \)), the sum of the interior angles on the same side of the transversal is always 180°. Thus, if \( \angle BMC = 74^\circ \), then \( \angle MCD \) must be \( 180^\circ - 74^\circ = 106^\circ \).

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Would you like further details on any specific part or any questions? Here are some additional related questions:

1. How do you find the midpoint of a line segment?
2. Why are the slopes of parallel lines equal?
3. What are alternate interior angles, and how do they apply to parallel lines?
4. Can two different lines have the same slope and not be parallel?
5. What other angle properties can be derived when dealing with parallel lines and transversals?

**Tip:** When dealing with angles formed by parallel lines, always consider angle pairs like corresponding angles and alternate interior angles for quick calculations.

AB is represented by the equation y = (3/5)x + 3, CD is represented by the equation y = (3/5)x - 1, and M is the midpoint of AB. (a) Graph both lines and the points A, B, C, D, and M. (b) If ∠BMC = 74°, what is ∠MCD? (c) Explain why the answer to part (b) must be true.

This problem involves graphing parallel lines and calculating angles formed by a transversal. The lines are represented by linear equations with identical slopes. The problem includes concepts such as the midpoint formula, angle properties, and parallel line theorems. Suitable for students in Grades 8-10.

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