In order for quadrilateral ABCD to be a parallelogram, certain angle relationships must hold true. Specifically, a quadrilateral is a parallelogram if:
- Opposite angles are equal, i.e., \( \angle A = \angle C \) and \( \angle B = \angle D \), or
- The sum of the measures of consecutive angles is supplementary, i.e., \( \angle A + \angle B = 180^\circ \) and \( \angle C + \angle D = 180^\circ \).
If Tiana has defined \( \angle A = m \), \( \angle B = n \), \( \angle C \), and \( \angle D \) in terms of \( m \) and \( n \), we can establish conditions based on the above definitions.
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If opposite angles are equal:
- For opposite angles:
- \( \angle A = m \) and \( \angle C = m \) suggests \( \angle C\) must also equal \( m \).
- And \( \angle B = n\) and \( \angle D = n\) suggests \( \angle D\) must also equal \( n \).
- For opposite angles:
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If the sum of consecutive angles is supplementary:
- \( m + n = 180^\circ \) guarantees that the two angles \( A + B\) are supplementary, implying that the figure must be a parallelogram.
Without more specific information about the angles in quadrilateral ABCD, we cannot determine exact numeric values for \( m \) and \( n \). Instead, we can provide conditions:
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For the case of equal opposite angles: \( m + n + m + n = 360^\circ \) thereby establishing conditions for A, B, C, D proportional to \( m\) and \( n \).
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For the case of supplementary angles: \( m + n = 180^\circ \).
If you can provide specific angle measures for \( m \) and \( n \) or more context to the angles in question, I can confirm the values. However, in a general case, if \( m + n = 180\), then ABCD will definitely be a parallelogram.
Thus, in summary:
- If \( m = 60^\circ \) then \( n = 120^\circ \) (or similarly for any other pairs summing to 180 degrees).
Hence,
- To guarantee that ABCD is a parallelogram, Tiana should establish that if \( m + n = 180^\circ \).
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