https://file.aiquickdraw.com/mathbot/file/172592702719448qbzzky.jpg

The problem asks to solve for the value of \(q\). The image shows two angles that are adjacent to each other, forming a linear pair. 

When two angles are on a straight line, they are supplementary, meaning their measures add up to \(180^\circ\).

The angles given are:
1. \(78^\circ\)
2. \((5q + 2)^\circ\)

To solve for \(q\), we set up the equation:
\[
78 + (5q + 2) = 180
\]

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

\[
78 + 5q + 2 = 180
\]

Combine like terms:

\[
80 + 5q = 180
\]

Subtract 80 from both sides:

\[
5q = 100
\]

Divide by 5:

\[
q = 20
\]

### Solution:
The value of \(q\) is **20**.

Would you like more details, or do you have any questions? Here are some related questions to expand on this:

1. How do we know when angles are supplementary?
2. What other properties of angles could be helpful in geometric problems?
3. How would this problem change if the two angles were not on a straight line?
4. Can angles have negative measures?
5. What are complementary angles, and how are they different from supplementary angles?

**Tip:** Always remember that supplementary angles add up to \(180^\circ\), while complementary angles add up to \(90^\circ\).

Solve for the value of q using the properties of supplementary angles in a linear pair. This problem involves understanding the relationship between adjacent angles on a straight line and algebraic manipulation to find the unknown variable q. Suitable for Grades 8-10.

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