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The image shows two parallel lines, \(m\) and \(n\), with a transversal cutting through them. There are angle expressions involving \(x\) and \(y\) in degrees marked on the diagram. Based on the given diagram, the angles involve a combination of:

- \( (6x + 13)^\circ \) on line \(m\),
- \( y^\circ \) on line \(n\),
- and \( (7x - 5)^\circ \) also on line \(n\).

To solve for \(x\) and \(y\), we can use the properties of angles formed by parallel lines and a transversal:

1. **Corresponding Angles:** These angles are equal when a transversal cuts through parallel lines.
   
   The angle \( (6x + 13)^\circ \) on line \(m\) and the angle \( (7x - 5)^\circ \) on line \(n\) are corresponding angles. So, we can set them equal:
   \[
   6x + 13 = 7x - 5
   \]

2. **Linear Pair (Supplementary Angles):** The angles on a straight line (like \(y^\circ\) and \( (7x - 5)^\circ \)) add up to 180°.

Let's solve for \(x\) first from the corresponding angles equation. I'll work that out now.From the equation \(6x + 13 = 7x - 5\), we find that \(x = 18\).

Now that we know the value of \(x\), we can substitute it into the expression for \(y^\circ\) using the supplementary angles property:
\[
y^\circ + (7x - 5)^\circ = 180^\circ
\]
Let's calculate \(y\) next.We have calculated that the angle \(7x - 5\) when \(x = 18\) is \(121^\circ\). Since \(y^\circ + 121^\circ = 180^\circ\), solving for \(y\) gives \(y = 59^\circ\).

Thus, the values are:
- \(x = 18\)
- \(y = 59^\circ\)

Would you like further details on any steps? Here are a few related questions to explore:

1. How do corresponding angles relate to parallel lines and transversals?
2. Why are the angles on a straight line supplementary?
3. What are alternate interior angles, and how do they behave in parallel line situations?
4. Can you explain the significance of transversal lines in geometry?
5. How would you solve the problem if the lines were not parallel?

**Tip:** Remember, parallel lines maintain specific angle relationships, which are key to solving for unknown variables in geometric figures.

Two parallel lines m and n are cut by a transversal. One angle on line m is given as (6x + 13)°, and two angles on line n are given as y° and (7x - 5)°. Solve for x and y.

This geometry problem involves solving for the values of x and y using corresponding and supplementary angle properties in parallel lines cut by a transversal. The angle expressions are (6x + 13)°, (7x - 5)°, and y°. Suitable for students in Grades 8-10.

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