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The diagram shows two angles, \( (6n + 5)^\circ \) and \( (4n - 5)^\circ \), with one of the angles forming a right angle with the line (indicated by the 90° mark). These angles appear to form a linear pair, meaning the sum of the angles along a straight line is \(180^\circ\).

We can solve for \( n \) by setting up the following equation based on the angle relationship:

\[
(6n + 5)^\circ + 90^\circ + (4n - 5)^\circ = 180^\circ
\]

Now let's solve this step by step:

\[
(6n + 5) + 90 + (4n - 5) = 180
\]

Simplifying the equation:

\[
6n + 5 + 90 + 4n - 5 = 180
\]

Combine like terms:

\[
6n + 4n + 90 = 180
\]

\[
10n + 90 = 180
\]

Subtract 90 from both sides:

\[
10n = 90
\]

Divide by 10:

\[
n = 9
\]

Thus, the value of \( n \) is \( \boxed{9} \).

Would you like further details on this solution or have any questions?

Here are 5 related questions you can explore:
1. What is the definition of a linear pair of angles?
2. How do you identify supplementary angles in geometry?
3. Can angles in any geometric figure exceed \(180^\circ\)?
4. How do you solve for \( n \) if the angles are complementary instead of supplementary?
5. What other angle relationships can you deduce from a straight line?

**Tip:** When solving for unknowns in geometry problems, always check for relationships such as supplementary or complementary angles, as they can simplify the problem significantly.

Learn how to solve for n in a geometry problem involving a linear pair of angles where one angle is a right angle. This problem includes algebraic expressions and supplementary angle relationships, making it suitable for students in Grades 8-10.

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