Tiana draws quadrilateral ABCD as shown, with the measures of ZA, ZB, ∠C, and ∠D represented. For what values of m and n is Tiana guaranteed that ABCD is a parallelogram?

Question

Tiana draws quadrilateral ABCD as shown, with the measures of ZA, ZB, ∠C, and ∠D represented. For what values of m and n is Tiana guaranteed that ABCD is a parallelogram?
(1 point

GPT-4o mini · Answer

To determine the values of $ m $ and $ n $ for which quadrilateral $ ABCD $ is a parallelogram, we can use properties of angles in a parallelogram: 1. **Opposite angles are equal**: $ \angle A = \angle C $ and $ \angle B = \angle D $. 2. **Consecutive angles are supplementary**: $ \angle A + \angle B = 180^\circ $ and $ \angle C + \angle D = 180^\circ $. Assuming $ ZA = m $, $ ZB = n $, and given the measures of $ \angle C $ and $ \angle D $, we want to set up a system of equations based on these properties. Now, we need to express the measures in a specific relationship. Whether $ \angle A $ and $ \angle B $ are equal to $ m $ and $ n $ (or something related), more information would typically be necessary to finalize $ m $ and $ n $. ### Case for Opposite angles: If $ m = \angle C $ and $ n = \angle D $ (which are opposite angles), then we can directly set $ m = n $. ### Case for Consecutive Angles: If we assume $ m + n = 180^\circ $ (which are consecutive angles), then we can directly relate them as supplementary. ### Conclusion: 1. If you compute $ m $ and $ n $ such that either: $$ m + n = 180^\circ \quad \text{(for consecutive angles)} $$ or: $$ m = n \quad \text{(for opposite angles)} $$ Thus, values of $ m $ and $ n $ can be derived based on their relationships in accordance with the properties of parallelograms. If you have specific measures or forms of angles $ \angle A $, $ \angle B $, $ \angle C $, and $ \angle D $ outlined already, please provide them for a precise solution.