From the theorem above, we conclude that if a parallelogram has opposite angles that are not supplementary, then it cannot be inscribed in a circle. For instance, the next parallelogram cannot be inscribed in a circle.
However, we can have a parallelogram in which the opposite angles are supplementary. In this case, the measures of the angles of the parallelogram satisfy the following system.
m∠ A = m∠ C & (I) m∠ A + m∠ C = 180^(∘) & (II)
Let's substitute Equation (I) into Equation (II).
m∠ A + m∠ A = 180^(∘)
⇓
m∠ A = 90^(∘)
The final result implies that the parallelogram is a rectangle, which can be inscribed in a circle. Thus, when the parallelogram is a rectangle it can be inscribed in a circle.
In conclusion, a parallelogram can sometimes be inscribed in a circle.
