To determine the values of \(m\) and \(n\) that guarantee that quadrilateral \(ABCD\) is a parallelogram, we need to use the properties of a parallelogram: opposite angles are equal and the sum of consecutive angles is equal to \(180^\circ\).
Given the measures of the angles:
- \(A = 3n + 3\)
- \(B = 5m\)
- \(C = 3m + 30\)
- \(D = 105\)
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Checking that opposite angles are equal: Since in a parallelogram angles \(A\) and \(C\) are opposite to each other and \(B\) and \(D\) are opposite to each other, we set up the equations:
- \(A + C = 180^\circ\)
- \(B + D = 180^\circ\)
First, consider angles \(A\) and \(C\): \[ 3n + 3 + (3m + 30) = 180 \] Simplifying the equation gives: \[ 3n + 3m + 33 = 180 \] \[ 3n + 3m = 147 \] \[ n + m = 49 \quad \text{(Equation 1)} \]
Now, for angles \(B\) and \(D\): \[ 5m + 105 = 180 \] Simplifying gives: \[ 5m = 75 \] \[ m = 15 \quad \text{(Equation 2)} \]
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Substituting \(m\) back into Equation 1: Now substitute \(m = 15\) into Equation 1: \[ n + 15 = 49 \] Solving for \(n\): \[ n = 49 - 15 = 34 \]
Thus, the values of \(m\) and \(n\) that guarantee quadrilateral \(ABCD\) is a parallelogram are: \[ m = 15 \quad \text{and} \quad n = 34 \]
Therefore, the final answer is: \[ \text{If } m = 15 \text{ and } n = 34, \text{ then Tiana is guaranteed that } ABCD \text{ is a parallelogram.} \]
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