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The question asks to find the measure of \( \angle 1 \), given that:

- \( \angle 1 \) and \( \angle 2 \) are vertical angles,
- \( m\angle 1 = (7x + 22)^\circ \),
- \( m\angle 2 = (8x + 20)^\circ \).

### Step-by-step Solution:

Since \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are congruent, meaning their measures are equal. So, we can set up the following equation:

\[
7x + 22 = 8x + 20
\]

1. Subtract \( 7x \) from both sides:

\[
22 = x + 20
\]

2. Subtract 20 from both sides:

\[
2 = x
\]

Now that we have the value of \( x \), we can substitute it back into the expression for \( m\angle 1 \):

\[
m\angle 1 = 7x + 22 = 7(2) + 22 = 14 + 22 = 36^\circ
\]

Thus, the measure of \( \angle 1 \) is \( \boxed{36^\circ} \).

Let me know if you'd like further clarification or details!

### Related Questions:
1. How do we know that vertical angles are congruent?
2. What would happen if we swapped the expressions for \( \angle 1 \) and \( \angle 2 \)?
3. How would the equation change if \( \angle 1 \) and \( \angle 2 \) were supplementary instead of congruent?
4. Can vertical angles ever be obtuse or acute? Why?
5. What other angle pairs, like vertical angles, are always congruent?

### Tip:
Always remember that vertical angles are congruent, meaning they have the same measure!

∠1 and ∠2 are vertical angles. If m∠1 = (7x + 22)° and m∠2 = (8x + 20)°, then find the measure of ∠1.

Learn how to solve for the measure of a vertical angle using algebraic expressions. This problem involves key concepts like vertical angles and algebra, and is suitable for students in Grades 8-10. The problem gives m∠1 = (7x + 22)° and m∠2 = (8x + 20)°.

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