In this problem, we have a system of angles that involves both variables $x$ and $y$. Let's go through it step-by-step:

• The measure of $\angle 1 = 4x^\circ$,
• The measure of $\angle 2 = y^\circ$,
• The measure of $\angle 3 = 72^\circ$.

Observations:

• From the diagram, we can see that $\angle 1$ and $\angle 3$ are vertically opposite angles. This means they are equal, as vertical angles are congruent.
• Therefore, $4x^\circ = 72^\circ$.

Now, we can solve for $x$:

$4x = 72$

$x = \frac{72}{4} = 18$

So, $x = 18$.

Next Step:

• $\angle 2$ and $\angle 3$ are adjacent angles on a straight line, which means they are supplementary. Supplementary angles add up to $180^\circ$.

Thus, $\angle 2 + \angle 3 = 180^\circ$.

Substituting the value for $\angle 3$:

$y^\circ + 72^\circ = 180^\circ$

Solving for $y$:

$y = 180 - 72 = 108$

So, $y = 108$.

Final Answer:

• $x = 18$
• $y = 108$

Let me know if you would like more details on any part of the solution!

Here are some related questions:

1. What is the relationship between vertically opposite angles?
2. Why do adjacent angles on a straight line add up to 180°?
3. How would the problem change if $\angle 3$ were a different value?
4. What happens if the angles in the diagram were not supplementary?
5. How can you use this knowledge in real-life geometry problems?

Tip: When working with angles on straight lines, always remember that adjacent angles sum to 180°.