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. 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 bc alternate interior angle to ∠c, we see that these angles are dropdown and thus, bc is parallel to ad. similarly, ab is parallel to dc. thus, abcd is a parallelogram. (type whole numbers.) dropdown options: complementary, alternate exterior angles, supplementary, congruent

Explanation:

Step1: Recall properties of quadrilaterals

In a quadrilateral, the sum of interior angles is \(360^\circ\). For a quadrilateral with opposite angles equal, we can use angle - related properties to determine the type of quadrilateral.

Step2: Analyze angle relationships

If \(\angle A=\angle C\) and \(\angle B = \angle D\) (opposite angles equal), and we know that consecutive angles (e.g., \(\angle A\) and \(\angle D\), \(\angle D\) and \(\angle C\)) are supplementary (their sum is \(180^\circ\)). When consecutive interior angles are supplementary, the sides are parallel (by the converse of the same - side interior angles theorem).
If \(\angle A+\angle D = 180^\circ\), then \(AB\parallel DC\) (since same - side interior angles are supplementary). If \(\angle D+\angle C=180^\circ\), then \(AD\parallel BC\) (same - side interior angles are supplementary).

Step3: Determine the type of quadrilateral

A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Also, when we consider the alternate interior angles (for example, when we extend a side and look at the alternate interior angle to \(\angle C\)), if those angles are congruent (because opposite angles of the quadrilateral are equal and we use the parallel - line angle relationships), it also supports the fact that the sides are parallel. But the key property here is that if opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. The supplementary angles (consecutive angles) imply that the sides are parallel, and a quadrilateral with two pairs of parallel sides is a parallelogram.

For the dropdown in the sentence "If we extend the side \(BC\), the alternate interior angle to \(\angle C\), we see that these angles are \(\boldsymbol{\text{congruent}}\) and thus, \(BC\) is parallel to \(AD\)". Because when two lines are cut by a transversal, if alternate interior angles are congruent, the lines are parallel. Since \(\angle A=\angle C\) (opposite angles of the quadrilateral) and using the transversal, the alternate interior angle to \(\angle C\) will be congruent to \(\angle A\) (or \(\angle C\)) which makes the lines parallel.

Answer:

The quadrilateral \(ABCD\) is a parallelogram. Justification: The sum of interior angles of a quadrilateral is \(360^\circ\). Given opposite angles are equal (\(\angle A=\angle C\), \(\angle B = \angle D\)), consecutive angles (e.g., \(\angle A\) and \(\angle D\), \(\angle D\) and \(\angle C\)) are supplementary. Supplementary consecutive interior angles imply \(AB\parallel DC\) and \(AD\parallel BC\) (by converse of same - side interior angles theorem). A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Also, when extending a side, alternate interior angles are congruent (due to equal opposite angles), further confirming the sides are parallel. For the dropdown, the correct option is "congruent".