The problem involves a triangle where two of the angles are given: one is $45^\circ$ and the other is $y - 12^\circ$. We are asked to find the value of $y$.

The Triangle Angle Sum Theorem tells us that the sum of the angles in any triangle is always $180^\circ$. Let's denote the angles as follows:

1. First angle: $45^\circ$
2. Second angle: $y - 12^\circ$
3. Third angle: $y + 3^\circ$

Using the theorem: $45^\circ + (y - 12^\circ) + (y + 3^\circ) = 180^\circ$

Now, let's solve this equation step by step:

$45 + (y - 12) + (y + 3) = 180$

Simplifying: $45 + y - 12 + y + 3 = 180$ $2y + 36 = 180$

Subtract 36 from both sides: $2y = 144$

Divide by 2: $y = 72$

Thus, the value of $y$ is $72^\circ$.

Would you like further details on this solution? Let me know if you have any questions.

Here are 5 related questions:

1. What is the Triangle Angle Sum Theorem?
2. How do you solve for unknown angles in any triangle?
3. What is an exterior angle in a triangle, and how does it relate to interior angles?
4. Can a triangle have two angles greater than 90 degrees? Why or why not?
5. How do supplementary and complementary angles relate to triangle problems?

Tip: Always double-check your math when solving for unknown angles to ensure you properly apply the Triangle Angle Sum Theorem.