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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!

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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°.

In the figure below, the measure of angle 1 = 4x°, the measure of angle 2 = y°, and the measure of angle 3 = 72°. Find the value of each variable.

In this problem, you'll solve for x and y by applying the concepts of vertically opposite angles and supplementary angles. Vertically opposite angles are equal, and adjacent angles on a straight line sum to 180°. Suitable for students in Grades 6-8.

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