To determine the values of \(m\) and \(n\) that guarantee quadrilateral \(ABCD\) is a parallelogram, we need to use the property that opposite angles in a parallelogram are equal; therefore, the sum of the interior angles must equal 360 degrees.
Given the measures:
- \( \angle A = 3n + 3 \)
- \( \angle B = 5m \)
- \( \angle C = 3m + 30 \)
- \( \angle D = 105 \)
To satisfy the properties of a parallelogram, we can set up the following equations based on the angles:
- \( \angle A + \angle C = 180^\circ \)
- \( \angle B + \angle D = 180^\circ \)
Starting with the first equation: \[ (3n + 3) + (3m + 30) = 180 \] \[ 3n + 3m + 33 = 180 \] \[ 3n + 3m = 180 - 33 \] \[ 3n + 3m = 147 \] \[ n + m = 49 \quad \text{(Equation 1)} \]
Now, for the second equation: \[ (5m) + 105 = 180 \] \[ 5m + 105 = 180 \] \[ 5m = 180 - 105 \] \[ 5m = 75 \] \[ m = 15 \quad \text{(Equation 2)} \]
Substituting Equation 2 into Equation 1: \[ n + 15 = 49 \] \[ n = 49 - 15 \] \[ n = 34 \quad \text{(Equation 3)} \]
Thus, the values of \(m\) and \(n\) for which Tiana can guarantee that quadrilateral \(ABCD\) is a parallelogram are: \[ m = 15, \quad n = 34 \]
So, if \( m = 15 \) and \( n = 34 \), then Tiana is guaranteed that \(ABCD\) is a parallelogram.