https://file.aiquickdraw.com/mathbot/file/1726697956882lzjtq174.jpg

The problem in the image involves two intersecting lines, where we are given the measurements of two angles, \( m\angle 3 = 6t + 30 \) and \( m\angle 2 = 8t - 60 \), and we are asked to find the measure of \( m\angle 4 \).

From the diagram, we know that angles 2 and 3 form a linear pair, meaning their measures add up to \( 180^\circ \). Therefore, we can write the equation:

\[
m\angle 3 + m\angle 2 = 180^\circ
\]

Substituting the given expressions for \( m\angle 3 \) and \( m\angle 2 \):

\[
(6t + 30) + (8t - 60) = 180
\]

Simplifying this equation:

\[
6t + 30 + 8t - 60 = 180
\]
\[
14t - 30 = 180
\]
\[
14t = 210
\]
\[
t = 15
\]

Now that we have \( t = 15 \), we can find the measures of the angles.

First, for \( m\angle 3 \):

\[
m\angle 3 = 6(15) + 30 = 90 + 30 = 120^\circ
\]

Next, for \( m\angle 2 \):

\[
m\angle 2 = 8(15) - 60 = 120 - 60 = 60^\circ
\]

Since \( m\angle 4 \) is vertically opposite to \( m\angle 2 \), and vertical angles are equal, we have:

\[
m\angle 4 = 60^\circ
\]

Thus, the measure of \( m\angle 4 \) is \( 60^\circ \).

Would you like further details on this solution or more questions? Here are some related problems to explore:

1. What is the measure of \( m\angle 1 \) in the diagram?
2. How would the calculation change if \( m\angle 2 \) was given as \( 10t - 50 \) instead?
3. Can you explain why vertical angles are always equal?
4. What are alternate interior angles, and how do they relate to intersecting lines?
5. How do linear pairs and their properties help in solving angle-related problems?

**Tip:** When two lines intersect, vertical angles are always congruent, and adjacent angles form a linear pair, summing to 180°.

In the accompanying figure, two lines intersect, m∠3 = 6t + 30, and m∠2 = 8t - 60. Find the number of degrees in m∠4.

Learn how to find the measure of m∠4 from the given linear pair and vertical angles, using algebraic equations like m∠3 = 6t + 30 and m∠2 = 8t - 60. This problem involves key geometry concepts and is suitable for students in Grades 9-10.

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