To solve for the missing angle measures in the provided diagram, we will use the principle that vertical angles are equal and the fact that angles on a straight line sum to $180^\circ$.

Step-by-step solution:

1. Set the vertical angles equal:

   • $(65x - 12)^\circ = (43x + 10)^\circ$

2. Solve for $x$: $65x - 12 = 43x + 10$ Subtract $43x$ from both sides: $22x - 12 = 10$ Add $12$ to both sides: $22x = 22$ Divide by $22$: $x = 1$

3. Substitute $x = 1$ into the expressions for the angles:

   • First angle: $(65x - 12)^\circ = 65(1) - 12 = 53^\circ$
   • Vertical angle (equal to the first): $(43x + 10)^\circ = 43(1) + 10 = 53^\circ$

4. Find the other pair of angles (linear pair):

   • Angles on a straight line sum to $180^\circ$.
   • Complementary angle: $180^\circ - 53^\circ = 127^\circ$.
   • Therefore, the other two angles are $127^\circ$ each (opposite angles).

Final Angle Measures:

• $(65x - 12)^\circ = 53^\circ$
• $(43x + 10)^\circ = 53^\circ$
• Other two angles: $127^\circ$ and $127^\circ$.

Would you like further clarification or a more detailed explanation?

Here are 5 related questions for deeper understanding:

1. How do vertical angles relate to each other in any intersection of two lines?
2. What is the significance of supplementary angles in geometry problems like this?
3. How can you verify the accuracy of your calculated angles?
4. Are there any shortcuts to solving linear pair problems without fully substituting values?
5. Can you create a similar problem where the given expressions are more complex?

Tip: Always double-check your substitution to avoid small errors when solving equations.