∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, 3, x, plus, 22, right parenthesis, degrees∠1=(3x+22) 
∘
  and mangle, 2, equals, left parenthesis, 7, x, minus, 22, right parenthesis, degrees∠2=(7x−22) 
∘
 , then find the value of x.

To solve this problem, we need to find the value of \(x\) given that \(\angle 1\) and \(\angle 2\) are vertical angles. Vertical angles are always equal in measure. Therefore, we can set the expressions for \(\angle 1\) and \(\angle 2\) equal to each other:

\[
m\angle 1 = m\angle 2
\]

Substitute the given expressions:

\[
3x + 22 = 7x - 22
\]

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

1. Subtract \(3x\) from both sides:

\[
22 = 4x - 22
\]

2. Add 22 to both sides:

\[
44 = 4x
\]

3. Divide both sides by 4:

\[
x = 11
\]

Therefore, the value of \(x\) is \(11\).

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:

1. What are vertical angles, and why are they always equal?
2. How would you solve for \(x\) if the angles were not vertical angles?
3. What are some real-life applications of vertical angles in geometry?
4. How can we prove that vertical angles are always congruent?
5. If \(\angle 1\) and \(\angle 2\) were supplementary, how would the problem change?

**Tip:** When solving for a variable, always check your solution by substituting the value back into the original equations to ensure it satisfies both conditions.

∠1 and ∠2 are vertical angles. If m∠1 = (3x + 22)° and m∠2 = (7x − 22)°, then find the value of x.

Learn how to solve a problem involving vertical angles where m∠1 = (3x + 22)° and m∠2 = (7x − 22)°. This step-by-step solution uses algebraic techniques to find the value of x, applying the Vertical Angles Theorem. Suitable for students in Grades 7-9.

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