The problem involves determining the measures of two angles, $m \angle 1$ and $m \angle 2$, given their algebraic expressions. Let's solve it step by step.

Given:

1. $m \angle 1 = (x - 6)^\circ$
2. $m \angle 2 = 5x^\circ$

From the diagram, we can see that angles $\angle 1$ and $\angle 2$ are complementary, meaning their measures add up to $90^\circ$: $m \angle 1 + m \angle 2 = 90^\circ$

Step 1: Set up the equation

Substitute the given expressions for $m \angle 1$ and $m \angle 2$: $(x - 6) + 5x = 90$

Step 2: Solve for $x$

Combine like terms: $6x - 6 = 90$

Add 6 to both sides: $6x = 96$

Divide by 6: $x = 16$

Step 3: Find $m \angle 1$ and $m \angle 2$

Now substitute $x = 16$ into the expressions for $m \angle 1$ and $m \angle 2$:

For $m \angle 1$: $m \angle 1 = x - 6 = 16 - 6 = 10^\circ$

For $m \angle 2$: $m \angle 2 = 5x = 5(16) = 80^\circ$

Final Answer:

$m \angle 1 = 10^\circ$ $m \angle 2 = 80^\circ$

Would you like me to go over the steps in more detail, or do you have further questions?

Related Questions:

1. What are complementary angles, and how do they relate to this problem?
2. How do you solve linear equations with variables on one side?
3. What happens if $\angle 1$ and $\angle 2$ were supplementary instead of complementary?
4. What is the geometric significance of complementary angles in right triangles?
5. Can you verify the result by substituting back into the original complementary equation?

Tip:

Always double-check your work by substituting the value of the variable back into the original expressions to ensure the results make sense.