To determine the values of \( m \) and \( n \) that guarantee quadrilateral \( ABCD \) is a parallelogram, we can use the properties of the angles in a parallelogram.
For a quadrilateral to be a parallelogram, one of the following conditions must be met:
- Both pairs of opposite angles are equal: \( \angle A = \angle C \) and \( \angle B = \angle D \).
- The sum of the measures of adjacent angles is \( 180^\circ \): \( \angle A + \angle B = 180^\circ \) and \( \angle C + \angle D = 180^\circ \).
- The measures of one pair of opposite angles are equal or the sum of one pair of adjacent angles is \( 180^\circ \).
Assuming \( \angle A = m \), \( \angle B = n \), \( \angle C = m \), and \( \angle D = n \), or if \( n = 180 - m \) guarantees that adjacent angles sum up to \( 180^\circ \).
So, we can set up the equations based on any of the conditions.
- Setting \( m = n \) leads to:
- \( \angle A + \angle B = 180^\circ \) implies \( m + n = 180^\circ \).
Thus, if \( m + n = 180^\circ \), Tiana is guaranteed that quadrilateral \( ABCD \) is a parallelogram.
So the values of \( m \) and \( n \) that guarantee \( ABCD \) to be a parallelogram would be:
- If \( m = x \) and \( n = 180-x \)
In summary, if
\( m = m \) and \( n = 180 - m \), then Tiana is guaranteed that \( ABCD \) is a parallelogram.
Can you please confirm or provide values for \( m \) and \( n \) so I can answer accordingly?
3 answers