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Let's analyze the given problem step by step.

We are asked to find the value of \(x\), and we are given two parallel lines, \(m\) and \(n\), cut by a transversal. The labeled angles are:
- \((6x - 10)^\circ\)
- \((7x - 27)^\circ\)

### Step 1: Identify the relationship between the angles
Since the lines \(m \parallel n\) are parallel, and the angles are on opposite sides of the transversal, these are **corresponding angles**. When lines are parallel, corresponding angles are **equal**.

Thus, we have the equation:
\[
6x - 10 = 7x - 27
\]

### Step 2: Solve for \(x\)
1. Rearrange the equation:
   \[
   6x - 7x = -27 + 10
   \]
   \[
   -x = -17
   \]

2. Divide both sides by \(-1\):
   \[
   x = 17
   \]

### Step 3: Verify the solution
Plugging \(x = 17\) back into the expressions for the angles:
- \(6x - 10 = 6(17) - 10 = 102 - 10 = 92^\circ\)
- \(7x - 27 = 7(17) - 27 = 119 - 27 = 92^\circ\)

Since both angles are equal, the solution \(x = 17\) is correct.

### Final Answer
\[
x = 17
\]

Would you like any further details or clarification?

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Here are 5 related questions for further practice:
1. What is the sum of consecutive interior angles formed by a transversal and parallel lines?
2. How do alternate exterior angles behave when lines are parallel?
3. If the angle expressions were \(4x + 5\) and \(8x - 35\), what would \(x\) be?
4. How can we prove that two lines are parallel using corresponding angles?
5. What happens to angle relationships if the lines are not parallel?

**Tip:** Always look out for the special relationships between angles formed by parallel lines and transversals—they can simplify many problems.

Given m parallel n, find the value of x. The angles are (6x - 10)° and (7x - 27)°.

Learn how to solve for x when given two corresponding angles (6x - 10)° and (7x - 27)° formed by a transversal intersecting parallel lines. Step-by-step solution provided, suitable for Grades 8-10.

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