In the provided diagram, you are asked to solve for $x$ given that lines $m \perp n$, meaning they are perpendicular.

Key Points:

• Line $m$ is perpendicular to line $n$, implying a 90° angle between them.
• There are two angles given: $(x + 21)^\circ$ and $(3x + 9)^\circ$, which form a right angle because $m \perp n$.

Since the two angles are complementary (they add up to 90°), we can write the equation:

$(x + 21) + (3x + 9) = 90$

Now, let me solve for $x$:

$x + 21 + 3x + 9 = 90$ $4x + 30 = 90$ $4x = 90 - 30$ $4x = 60$ $x = \frac{60}{4} = 15$

Thus, $x = 15$.

Would you like further details or explanations on any part of the solution?

Related Questions:

1. How can complementary angles help in geometric proofs?
2. What are perpendicular lines, and how are they used in solving geometry problems?
3. Can we use the same approach if the lines were not perpendicular? Why or why not?
4. How would the problem change if the angles were supplementary instead of complementary?
5. Can this approach be generalized for any angle relationship in geometric figures?

Tip: Always check the relationship between angles when given specific line conditions like perpendicularity, parallelism, or bisectors. This can help you set up accurate equations!