For a quadrilateral to be a parallelogram, the sum of the opposite angles must be equal. That means ∠A + ∠C must equal 180 degrees and ∠B + ∠D must also equal 180 degrees.
Given the measures:
\[ \angle A = 3n + 3 \] \[ \angle B = 5m \] \[ \angle C = 3m + 30 \] \[ \angle D = 105 \]
We can start with the equation for angles A and C:
\[ \angle A + \angle C = 180 \] \[ (3n + 3) + (3m + 30) = 180 \] \[ 3n + 3m + 33 = 180 \] \[ 3n + 3m = 147 \] \[ n + m = 49 \quad \text{(1)} \]
Next, let's use the equation for angles B and D:
\[ \angle B + \angle D = 180 \] \[ 5m + 105 = 180 \] \[ 5m = 75 \] \[ m = 15 \quad \text{(2)} \]
Now substitute \( m = 15 \) into equation (1):
\[ n + 15 = 49 \] \[ n = 34 \]
Thus, Tiana is guaranteed that ABCD is a parallelogram for:
\[ m = 15 \quad \text{and} \quad n = 34. \]