To solve for $x$ given that lines $m$ and $n$ are parallel $(m \parallel n)$, and that a transversal $t$ intersects them forming angles $(3x - 30)^\circ$ and $(8x - 10)^\circ$, we need to determine the relationship between these angles.

Common angle relationships when two parallel lines are cut by a transversal:

1. Corresponding angles are congruent.
2. Alternate interior angles are congruent.
3. Same-side interior angles are supplementary (sum to 180°).

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Step 1: Identify the relationship between the angles

Without an image, I’ll assume the angles $(3x - 30)^\circ$ and $(8x - 10)^\circ$ are either:

• Corresponding angles or alternate interior angles: Set them equal
  $3x - 30 = 8x - 10$
• Same-side interior angles: Set them as supplementary
  $(3x - 30) + (8x - 10) = 180$

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Case 1: Angles are Congruent

$3x - 30 = 8x - 10$ Subtract $3x$ from both sides: $-30 = 5x - 10$ Add $10$ to both sides: $-20 = 5x$ $x = -4$

However, a negative value for $x$ is unusual in geometry. Let’s check Case 2.

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Case 2: Angles are Supplementary

$(3x - 30) + (8x - 10) = 180$ Combine like terms: $11x - 40 = 180$ Add $40$ to both sides: $11x = 220$ $x = 20$

This value is more reasonable in a geometric context.

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Final Answer:

$\boxed{x = 20}$

Would you like a diagram to visualize the angles or a step-by-step explanation of the angle relationships?

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Related Questions:

1. How do corresponding angles behave when lines are parallel?
2. What is the difference between alternate interior and alternate exterior angles?
3. How do you determine if angles are supplementary or congruent in parallel line problems?
4. Can you use the same approach for non-parallel lines? Why or why not?
5. How do transversal angles apply in real-life scenarios (e.g., engineering, architecture)?

Tip:

Always sketch the figure when working with parallel lines and transversals—it helps identify angle relationships more easily!