Question

use the image to answer the question.
tiana draws quadrilateral ( a b c d ) as shown, with the measures of ( angle a, angle b, angle c ), and ( angle d ) represented. for what values of ( m ) and ( n ) is tiana guaranteed that ( a b c d ) is a parallelogram?
(1 point)
if ( m=square ) and ( n=square ), then tiana is guaranteed that ( a b c d ) is a parallelogram.

Explanation:

Step1: Use the property of adjacent angles in a parallelogram

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

Step2: Use the property of adjacent angles for another pair

Now, since \(m = 15\), angle \(D=(3m + 30)=(3\times15+30)=45 + 30=75\).
In a parallelogram, adjacent angles are supplementary. For angles \(A\) and \(B\), \(A + B=180\).
Since \(B = 5m=75\), then \(A=(3n + 3)=105\) (because \(A + B=180\), so \(A=180 - 75 = 105\)).
Set up the equation \(3n+3 = 105\).
Subtract \(3\) from both sides: \(3n=105 - 3\).
\(3n=102\).
Divide both sides by \(3\): \(n=\frac{102}{3}=34\).

Answer:

If \(m = 15\) and \(n = 34\), then Tiana is guaranteed that \(ABCD\) is a parallelogram.