Question

tiana draws quadrilateral abcd as shown, with the measures of ∠a, ∠b, ∠c, and ∠d represented. for what values of m and n is tiana guaranteed that abcd is a parallelogram? (1 point) if m = and n =, then tiana is guaranteed that abcd is a parallelogram.

Explanation:

Step1: Use the property of adjacent angles in a parallelogram

In a parallelogram, adjacent angles are supplementary. So, \(\angle B+\angle C = 180^{\circ}\) and \(\angle D+\angle C=180^{\circ}\).
For \(\angle B = 5m\) and \(\angle C = 105^{\circ}\), we have \(5m+105 = 180\).
Subtract \(105\) from both sides: \(5m=180 - 105\), so \(5m = 75\).
Divide both sides by \(5\): \(m=\frac{75}{5}=15\).

Step2: Use the property of opposite angles in a parallelogram

In a parallelogram, opposite angles are equal. So, \(\angle A=\angle C\) and \(\angle B=\angle D\).
Since \(\angle D=(3m + 30)\) and \(m = 15\), then \(\angle D=3\times15+30=45 + 30=75^{\circ}\), and \(\angle B = 5\times15 = 75^{\circ}\) (verified the adjacent - angle property).
Also, \(\angle A=(3n + 3)\) and \(\angle C = 105^{\circ}\). Set \(3n+3=105\).
Subtract \(3\) from both sides: \(3n=105 - 3\), so \(3n = 102\).
Divide both sides by \(3\): \(n=\frac{102}{3}=34\).

Answer:

\(m = 15\) and \(n = 34\)