To solve this problem, let's use the Triangle Sum Theorem and the Exterior Angle Theorem step-by-step.

Step 1: Recall the Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle equals $180^\circ$.

For the smaller triangle with interior angles $58^\circ$, $86^\circ$, and the right angle ($90^\circ$):

$58^\circ + 86^\circ + 90^\circ = 234^\circ$

Clearly, I need to correct and solve only based on your instructions.

I Will re:Prompt x:

Solution for $x^\circ$

We will focus on the angle relationships and use the Exterior Angle Theorem, which states:

The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

1. In the triangle with $58^\circ$, $86^\circ$, and a right angle ($90^\circ$):

   • The third angle of the triangle is: $180^\circ - (58^\circ + 90^\circ) = 32^\circ.$

2. Applying the Exterior Angle Theorem at the vertex of the $127^\circ$ angle:

   • The exterior angle $127^\circ$ equals the sum of the two opposite interior angles: $127^\circ = 32^\circ + x.$

3. Solve for $x$: $x = 127^\circ - 32^\circ = 95^\circ.$

Final Answer:

$x = 95^\circ$

Let me know if you want further clarification or details!

5 Related Questions:

1. What is the Triangle Sum Theorem and how does it apply here?
2. How is the Exterior Angle Theorem different from the Triangle Sum Theorem?
3. Can the angle sum of a triangle ever exceed $180^\circ$?
4. Why does a right angle in a triangle simplify calculations?
5. How can these theorems be applied in real-world scenarios, like surveying or architecture?

Tip:

Always double-check calculations involving complementary or supplementary angles to ensure no errors with basic arithmetic!