∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, 6, x, plus, 29, right parenthesis, degrees∠1=(6x+29) 
∘
  and mangle, 2, equals, left parenthesis, 7, x, plus, 25, right parenthesis, degrees∠2=(7x+25) 
∘
 , then find the value of x.

Since \(\angle 1\) and \(\angle 2\) are vertical angles, they are equal. Therefore, we can set their measures equal to each other and solve for \(x\).

Given:
\[
m\angle 1 = (6x + 29)^\circ
\]
\[
m\angle 2 = (7x + 25)^\circ
\]

Since \(\angle 1 = \angle 2\), we can write the equation:

\[
6x + 29 = 7x + 25
\]

Now, let's solve for \(x\):

1. Subtract \(6x\) from both sides:
   \[
   29 = x + 25
   \]

2. Subtract 25 from both sides:
   \[
   4 = x
   \]

So, the value of \(x\) is:

\[
\boxed{4}
\]

Would you like more details or have any other questions?

### Related Questions:
1. What are vertical angles, and why are they always equal?
2. How can we check if our solution for \(x\) is correct?
3. What happens if vertical angles are not equal in a given problem?
4. How do we solve for angles when given expressions for their measures?
5. How can we find the measures of the angles after solving for \(x\)?

**Tip:** Vertical angles are always equal, making them useful in solving equations involving unknowns in geometry problems!

∠1 and ∠2 are vertical angles. If m∠1 = (6x + 29)° and m∠2 = (7x + 25)°, then find the value of x.

This problem involves finding the value of x in a vertical angles equation where m∠1 = (6x + 29)° and m∠2 = (7x + 25)°. Learn how to apply the Vertical Angles Theorem and algebraic methods to solve for x. Suitable for students in Grades 8-10.

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