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To solve this problem, we need to use the fact that vertical angles are congruent, meaning they have equal measures. Therefore, the measure of ∠1 is equal to the measure of ∠2.

Given:
- \( m\angle 1 = (6x + 11)^\circ \)
- \( m\angle 2 = (10x - 9)^\circ \)

Since \( m\angle 1 = m\angle 2 \), we can set the two expressions equal to each other:

\[
6x + 11 = 10x - 9
\]

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

\[
6x + 11 = 10x - 9
\]
\[
11 + 9 = 10x - 6x
\]
\[
20 = 4x
\]
\[
x = 5
\]

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

\[
m\angle 1 = 6x + 11 = 6(5) + 11 = 30 + 11 = 41^\circ
\]

Thus, the measure of \( \angle 1 \) is \( 41^\circ \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What are other properties of vertical angles?
2. How would the problem change if the angles were complementary instead of vertical?
3. Can you derive the measure of \( \angle 2 \) after finding \( x \)?
4. What happens when two angles are supplementary instead of vertical?
5. How would you verify the solution algebraically?

**Tip:** When solving for unknowns in geometry, always use properties like congruence or relationships between angles to set up an equation.

∠1 and ∠2 are vertical angles. If m∠1 = (6x + 11)° and m∠2 = (10x − 9)°, find m∠1.

Learn how to solve for the measure of ∠1 in a problem involving vertical angles. This step-by-step guide explains how to use algebra to solve equations where vertical angles are equal. Suitable for students in Grades 8-10.

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