The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is 180 degrees. Given the angle measures \(3x + 10\), \(2x - 3\), and \(5x\), we can use this property to find the values of \(x\) and the measures of the angles.
Set up the equation based on the Triangle Angle Sum Theorem:
\[ (3x + 10) + (2x - 3) + (5x) = 180 \]
Now, combine like terms:
\[ 3x + 10 + 2x - 3 + 5x = 180 \]
This simplifies to:
\[ (3x + 2x + 5x) + (10 - 3) = 180 \] \[ 10x + 7 = 180 \]
Next, subtract 7 from both sides:
\[ 10x = 180 - 7 \] \[ 10x = 173 \]
Now, divide by 10:
\[ x = 17.3 \]
Now we can find the measures of each angle by substituting \(x\) back into the angle expressions:
- For the angle \(3x + 10\):
\[ 3(17.3) + 10 = 51.9 + 10 = 61.9 \]
- For the angle \(2x - 3\):
\[ 2(17.3) - 3 = 34.6 - 3 = 31.6 \]
- For the angle \(5x\):
\[ 5(17.3) = 86.5 \]
Now that we have the angles:
- \(3x + 10 = 61.9^\circ\)
- \(2x - 3 = 31.6^\circ\)
- \(5x = 86.5^\circ\)
The largest angle is:
\[ \boxed{86.5^\circ} \]
So, the measure of the largest angle of the triangle is \(86.5^\circ\) (Option C).