∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, 4, x, minus, 4, right parenthesis, degrees∠1=(4x−4) 
∘
  and mangle, 2, equals, left parenthesis, 5, x, minus, 29, right parenthesis, degrees∠2=(5x−29) 
∘
 , then find the measure of angle, 1∠1.

Given that ∠1 and ∠2 are vertical angles, their measures are equal. Therefore, we can set up the equation:

\[
m∠1 = m∠2
\]

This means:

\[
4x - 4 = 5x - 29
\]

### Step-by-Step Solution:

1. **Set up the equation:**
   \[
   4x - 4 = 5x - 29
   \]

2. **Subtract \(4x\) from both sides:**
   \[
   -4 = x - 29
   \]

3. **Add 29 to both sides:**
   \[
   25 = x
   \]

4. **Now, substitute \(x = 25\) back into the expression for \(m∠1\):**
   \[
   m∠1 = 4x - 4 = 4(25) - 4 = 100 - 4 = 96^\circ
   \]

Thus, the measure of ∠1 is:

\[
m∠1 = 96^\circ
\]

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Would you like further details or have any questions?

### Related Questions:
1. How can you prove that vertical angles are always equal?
2. What are supplementary angles, and how do they relate to vertical angles?
3. What if ∠1 was expressed as \(3x - 2\) instead—how would that change the solution?
4. How does this approach change when solving for more complex angle relationships?
5. How do parallel lines influence the relationship between angles?

**Tip:** Always check if angles are supplementary or complementary when solving problems with intersecting lines!

∠1 and ∠2 are vertical angles. If m∠1 = (4x−4)° and m∠2 = (5x−29)°, then find the measure of ∠1.

Learn how to solve a vertical angles problem where ∠1 = (4x−4)° and ∠2 = (5x−29)°. This problem involves setting up a linear equation and using algebraic techniques to find the value of x and the measure of ∠1. Suitable for students in Grades 8-10.

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