To determine the values of \( m \) and \( n \) for which quadrilateral \( ABCD \) is guaranteed to be a parallelogram, we can use the properties of opposite angles in a parallelogram.
A quadrilateral is a parallelogram if:
- Each pair of opposite angles are equal, or
- The sum of the measures of two adjacent angles is \( 180^\circ \).
If we label the angles as follows:
- \( \angle A = m \)
- \( \angle B = n \)
- \( \angle C = m \)
- \( \angle D = n \)
The conditions that would guarantee that \( ABCD \) is a parallelogram are:
- \( m + n = 180^\circ \)
- \( m + m = 180^\circ \) (or equivalently, \( 2m = 180^\circ \))
Given these properties, we can set the following equations:
- For adjacent angles, we have \( m + n = 180^\circ \).
- Opposite angles must also hold the same relationship, so if \( \angle A = \angle C \) and \( \angle B = \angle D \), we also should consider opposite angles being equal.
Based on the property mentioned, Tiana is guaranteed that quadrilateral ABCD is a parallelogram when:
- If \( m + n = 180^\circ \) (for adjacent angles), then:
- \( n = 180^\circ - m \).
This means any specific values for \( m \) or \( n \) must satisfy this relationship, ensuring that when one angle is known, the other can be deduced accordingly.
Therefore, Tiana is guaranteed that ABCD is a parallelogram if \( m \) and \( n \) satisfy the equation:
\[ m + n = 180^\circ. \]
In a specific example or case, if we pick \( m = 60^\circ \), then:
\[ n = 180^\circ - 60^\circ = 120^\circ. \]
So generalized, if \( m = x \) then \( n = 180 - x \) for any value of \( x \) such that \( x + n = 180 \).
Thus, for Tiana's quadrilateral ABCD to be a parallelogram, Tiana can use:
\[ m = m, \quad n = 180 - m. \]
Summarizing:
If \( m = m \) and \( n = 180 - m \), then Tiana is guaranteed that ABCD is a parallelogram.