Question

abcd is a quadrilateral in which opposite angles have the same measure, as indicated in the following figure. what kind of quadrilateral is abcd? justify your answer.
figure of quadrilateral abcd with vertices a, b, c, d (∠b, ∠c, ∠a, ∠d marked with congruency symbols)
complete the sentences below
because the sum of the interior angles is 360°. ∠a and ∠d are supplementary. also, ∠d and ∠c are supplementary. if we extend the side □ to form an alternate interior angle to ∠c, we see that these angles are congruent and thus, bc is parallel to ad. similarly, ab is parallel to dc. thus, abcd is a parallelogram
(type whole numbers.)

Explanation:

Step1: Recall Interior Angles Sum of Quadrilateral

The sum of the interior angles of any quadrilateral is \( 360^\circ \). For quadrilateral \( ABCD \), \( \angle A+\angle B+\angle C+\angle D = 360^\circ \). Given \( \angle A=\angle C \) and \( \angle B = \angle D \) (opposite angles equal), we can rewrite the sum as \( 2\angle A + 2\angle B=360^\circ \), or \( \angle A+\angle B = 180^\circ \), which means \( \angle A \) and \( \angle B \) are supplementary (similarly for other adjacent angles).

Step2: Use Supplementary Angles for Parallel Sides

If two interior angles on the same side of a transversal are supplementary, the lines are parallel. For \( \angle A \) and \( \angle D \) (supplementary), the transversal is \( AD \), so \( AB\parallel DC \). For \( \angle D \) and \( \angle C \) (supplementary), the transversal is \( DC \), so \( AD\parallel BC \).

Step3: Define the Quadrilateral

A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Also, a property of parallelograms is that opposite angles are equal, which matches the given condition.

Answer:

The quadrilateral \( ABCD \) is a parallelogram. Justification: The sum of interior angles of a quadrilateral is \( 360^\circ \). Given opposite angles are equal (\( \angle A=\angle C \), \( \angle B = \angle D \)), adjacent angles (e.g., \( \angle A \) and \( \angle D \), \( \angle D \) and \( \angle C \)) are supplementary. Supplementary adjacent angles imply opposite sides are parallel (by the converse of the same - side interior angles theorem). A quadrilateral with both pairs of opposite sides parallel is a parallelogram. For the "extend the side" part, to form an alternate interior angle to \( \angle C \) (to show \( BC\parallel AD \)), we extend side \( DC \) (or \( AB \), but \( DC \) is more direct as \( \angle C \) and the alternate angle would be formed with transversal \( DC \) when extending \( AD \) or \( BC \); here, extending \( DC \) (or \( AB \), but the logical side is \( DC \) to create alternate interior angles with \( \angle C \) for \( BC\parallel AD \)) shows congruent alternate interior angles, proving \( BC\parallel AD \), and similarly \( AB\parallel DC \), confirming it's a parallelogram.