Question

proving the parallelogram angle theorem
given: abcd is a parallelogram.
prove: \\( \angle a \cong \angle c \\) and \\( \angle b \cong \angle d \\)
by the definition of a \\( \square, \overline{ad} \parallel \overline{bc} \\) and \\( \overline{ab} \parallel \overline{dc} \\).
using, \\( \overline{ad} \\) as a transversal, \\( \angle a \\) and \\( \angle \\) are same - side interior
angles, so they are transversal, \\( \angle b \\) and \\( \angle c \\) are same - side angles, so they are
supplementary. using \\( \overline{ab} \\) as a transve and \\( \angle b \\) are same - side
interior angles, so they are supplementary.
therefore, \\( \angle a \\) is congruent to \\( \angle c \\) because they are supplements of
the same angle. similarly, \\( \angle b \\) is congruent to \\( \angle \\)

Explanation:

Step1: Identify same - side interior angles

When \(\overline{AD}\) is a transversal for \(\overline{AB}\parallel\overline{DC}\), \(\angle A\) and \(\angle D\) are same - side interior angles. By the same - side interior angles theorem, same - side interior angles are supplementary.

Step2: Use the property of supplementary angles

Since \(\angle A+\angle D = 180^{\circ}\) (supplementary) and \(\angle B+\angle C=180^{\circ}\) (when \(\overline{BC}\) is a transversal for \(\overline{AB}\parallel\overline{DC}\)), and \(\angle A+\angle B = 180^{\circ}\) (when \(\overline{AD}\) is a transversal for \(\overline{AB}\parallel\overline{DC}\)).
We know that if \(\angle A+\angle D=180^{\circ}\) and \(\angle C+\angle D = 180^{\circ}\) (because \(\overline{AD}\parallel\overline{BC}\) and \(\overline{DC}\) is a transversal, \(\angle C\) and \(\angle D\) are same - side interior angles), then \(\angle A\cong\angle C\) (supplements of the same angle \(\angle D\) are congruent). Similarly, \(\angle B\) and \(\angle D\) are supplements of \(\angle A\) and \(\angle C\) respectively. Since \(\angle A\cong\angle C\), \(\angle B\cong\angle D\) (supplements of congruent angles are congruent).

Answer:

The first blank is \(D\), the second blank is supplementary. And \(\angle B\cong\angle D\)