To solve for the unknown angle $x^\circ$ in the triangle, we can use the fact that the sum of the interior angles of a triangle is always $180^\circ$.

The given angles are:

• $75^\circ$
• $42^\circ$
• Unknown angle: $x^\circ$

Using the angle sum property of a triangle:

$x^\circ + 75^\circ + 42^\circ = 180^\circ$

Simplifying:

$x^\circ + 117^\circ = 180^\circ$

$x^\circ = 180^\circ - 117^\circ$

$x^\circ = 63^\circ$

Thus, the unknown angle $x$ is $63^\circ$.

Do you want more details or have any questions?

Here are 5 questions to expand on this:

1. How can the exterior angle theorem help verify this result?
2. What are the properties of a triangle with angles $42^\circ$, $75^\circ$, and $63^\circ$?
3. How would this problem change if one of the angles was obtuse?
4. Can we classify this triangle based on its angles?
5. How does the triangle inequality theorem apply to this problem?

Tip: Always check that the sum of all angles in any triangle is exactly 180 degrees to ensure accuracy when solving for unknown angles.