Let's analyze the problem.

Given:

• $m \angle 1 = (5x)^\circ$
• $m \angle 2 = (x + 6)^\circ$

From the diagram, $\angle 1$ and $\angle 2$ are adjacent angles on a straight line, which means they form a linear pair. The sum of the angles in a linear pair is 180°.

Part (a):

To find $x$, we set up the equation based on the fact that: $m \angle 1 + m \angle 2 = 180^\circ$ Substitute the given expressions for $m \angle 1$ and $m \angle 2$: $5x + (x + 6) = 180$

Part (b):

Once $x$ is found, we substitute it back into the expressions for $m \angle 1$ and $m \angle 2$ to find their individual measures.

Let me now solve for $x$ and then the angle measures.

Step-by-step solution:

1. $5x + x + 6 = 180$
2. Combine like terms: $6x + 6 = 180$
3. Subtract 6 from both sides: $6x = 174$
4. Divide by 6: $x = 29$

Now, substitute $x = 29$ into the expressions for the angles:

• $m \angle 1 = 5x = 5(29) = 145^\circ$
• $m \angle 2 = x + 6 = 29 + 6 = 35^\circ$

Thus, the degree measures of the angles are:

• $m \angle 1 = 145^\circ$
• $m \angle 2 = 35^\circ$

Summary:

• Equation to find $x$: $6x + 6 = 180$
• $x = 29$
• $m \angle 1 = 145^\circ$
• $m \angle 2 = 35^\circ$

Would you like further details or explanations on any step?

Here are 5 questions to expand your understanding:

1. What other types of angle pairs have specific relationships like linear pairs?
2. How do you find the supplementary angle of a given angle?
3. Can two angles form a linear pair if they are both acute? Why or why not?
4. What are complementary angles, and how do they differ from linear pairs?
5. How would the solution change if the angles were vertical angles instead of a linear pair?

Tip: Always remember that angles on a straight line sum to 180°!