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The problem involves two intersecting lines, and we are given:

\[
\text{m} \angle 1 = x - 6 \quad \text{and} \quad \text{m} \angle 2 = 5x + 6
\]

To solve this, follow these steps:

1. **Vertical Angle Theorem**: Angles 1 and 3 are vertical angles, so they are equal. This means:

   \[
   \text{m} \angle 1 = \text{m} \angle 3 = x - 6
   \]

2. **Linear Pair**: Angles 1 and 2 form a linear pair, meaning their measures add up to 180°. Therefore:

   \[
   \text{m} \angle 1 + \text{m} \angle 2 = 180^\circ
   \]

   Substituting the expressions for the angles:

   \[
   (x - 6) + (5x + 6) = 180
   \]

3. **Simplify the equation**:

   \[
   x - 6 + 5x + 6 = 180
   \]

   Combine like terms:

   \[
   6x = 180
   \]

4. **Solve for \(x\)**:

   \[
   x = \frac{180}{6} = 30
   \]

5. **Find the measure of \( \angle 3 \)**: Since \( \text{m} \angle 3 = x - 6 \):

   \[
   \text{m} \angle 3 = 30 - 6 = 24^\circ
   \]

So, the measure of \( \angle 3 \) is \( 24^\circ \).

If you need any clarification, feel free to ask! Here are some related questions:

1. What is the value of \( \angle 2 \)?
2. Can you explain the Vertical Angle Theorem in detail?
3. How does the Linear Pair Postulate work in this context?
4. What if the given angles were supplementary instead of vertical?
5. Can we use this method to solve other angle relationships?

**Tip**: Always look for angle relationships (vertical, linear pair, etc.) when dealing with intersecting lines!

What is the measure of ∠3 given m∠1 = x - 6 and m∠2 = 5x + 6?

Solve the problem of finding the measure of angle 3 when given m∠1 = x - 6 and m∠2 = 5x + 6 using algebra and geometry. This solution explores key concepts like vertical angles and linear pairs, making it suitable for students in Grades 8-10.

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