Let's analyze the given diagram step by step to find the value of $x$.

We are given a triangle $QRT$ with the following information:

1. $QR \perp RT$, meaning $\angle QRT = 90^\circ$.
2. $\angle RTS = 64^\circ$ (at point $T$).
3. $\angle SQR = 110^\circ$ (at point $S$).
4. $\angle QRT = (2x - 10)^\circ$.

Steps to solve:

1. Since $\angle QRT$ is a right angle (perpendicular lines), the total sum of angles in triangle $QRT$ should be $180^\circ$.

   $\text{Sum of angles in triangle} \ QRT = 180^\circ$

   Thus, $\angle QRT + \angle RTS + \angle SQR = 180^\circ$

   Substituting the known values: $(2x - 10)^\circ + 64^\circ + 90^\circ = 180^\circ$

2. Simplifying the equation: $2x - 10 + 154 = 180$ $2x + 144 = 180$ $2x = 180 - 144$ $2x = 36$

3. Solving for $x$: $x = \frac{36}{2} = 18$

Thus, the value of $x$ is $\boxed{18}$.

Would you like more details or have any questions?

Related Questions:

1. What is the sum of interior angles in any triangle?
2. How would you find the value of an unknown angle in a triangle with two known angles?
3. What does it mean when two lines are perpendicular in terms of angles?
4. How can you verify your result in an angle calculation problem?
5. How do you calculate an angle when expressed as a variable?

Tip:

Always remember that the sum of angles in any triangle is 180°. This helps solve for unknown angles easily when given other angles.