1. **State the problem:** Given a parallelogram ABCD with AB \parallel DC and AD \parallel BC, prove that opposite angles are congruent: $\angle A \cong \angle C$ and $\angle B \cong \angle D$. 2. **Recall the properties of a parallelogram:** Consecutive angles in a parallelogram are supplementary, meaning their measures add up to 180 degrees. 3. **Express the supplementary angles:** Since ABCD is a parallelogram, $m\angle A + m\angle B = 180^\circ$ and $m\angle B + m\angle C = 180^\circ$. 4. **Use the supplementary angle equations:** From $m\angle A + m\angle B = 180^\circ$ and $m\angle B + m\angle C = 180^\circ$, set them equal: $$m\angle A + m\angle B = m\angle B + m\angle C$$ 5. **Subtract $m\angle B$ from both sides:** $$\cancel{m\angle A} + m\angle B - m\angle B = m\angle B + m\angle C - m\angle B$$ $$m\angle A = m\angle C$$ 6. **Conclude that $\angle A \cong \angle C$:** Since their measures are equal, the angles are congruent. 7. **Similarly, prove $\angle B \cong \angle D$:** Using the supplementary angles $m\angle B + m\angle C = 180^\circ$ and $m\angle C + m\angle D = 180^\circ$, set equal: $$m\angle B + m\angle C = m\angle C + m\angle D$$ 8. **Subtract $m\angle C$ from both sides:** $$m\angle B + \cancel{m\angle C} - \cancel{m\angle C} = \cancel{m\angle C} + m\angle D - \cancel{m\angle C}$$ $$m\angle B = m\angle D$$ 9. **Conclude that $\angle B \cong \angle D$:** Their measures are equal, so the angles are congruent. **Final conclusion:** In parallelogram ABCD, opposite angles are congruent: $\angle A \cong \angle C$ and $\angle B \cong \angle D$.