Question

the figure below is a quadrilateral. find the indicated measures.
ad=
dc=
m∠a=°
m∠b=°

Explanation:

Step1: Identify the quadrilateral type

The figure is a parallelogram (opposite sides appear parallel). In a parallelogram, opposite sides are equal, and consecutive angles are supplementary.

Step2: Find \( AD \)

In a parallelogram, \( AD = BC \). Given \( BC = 8 \), so \( AD = 8 \).

Step3: Find \( DC \)

In a parallelogram, \( DC = AB \). Given \( AB = 15 \), so \( DC = 15 \).

Step4: Find \( m\angle A \)

Consecutive angles in a parallelogram are supplementary. \( \angle D = 68^\circ \), so \( m\angle A = 180^\circ - 68^\circ = 112^\circ \).

Step5: Find \( m\angle B \)

In a parallelogram, opposite angles are equal, and \( \angle B = \angle D \)? Wait, no, consecutive angles: \( \angle B \) and \( \angle A \) are supplementary? Wait, no, \( \angle B = \angle D \)? Wait, no, in a parallelogram, opposite angles are equal. \( \angle D = 68^\circ \), so \( \angle B = \angle D = 68^\circ \)? Wait, no, wait: \( \angle A \) and \( \angle D \) are consecutive, so supplementary. \( \angle A = 112^\circ \), then \( \angle B \) is equal to \( \angle D \)? Wait, no, let's correct. In a parallelogram, \( AB \parallel DC \), so \( \angle A + \angle D = 180^\circ \) (consecutive interior angles). So \( \angle A = 180 - 68 = 112^\circ \). Then \( \angle B \) is equal to \( \angle D \)? Wait, no, \( \angle B \) and \( \angle A \) are consecutive, so \( \angle B + \angle A = 180^\circ \)? No, wait, \( AB \parallel DC \), \( AD \) is transversal, so \( \angle A + \angle D = 180 \). \( AD \parallel BC \), \( AB \) is transversal, so \( \angle A + \angle B = 180 \). Wait, no, that can't be. Wait, no, in a parallelogram, opposite angles are equal. So \( \angle A = \angle C \), \( \angle B = \angle D \). So \( \angle B = \angle D = 68^\circ \). Wait, let's recheck: \( \angle D = 68^\circ \), so \( \angle B = 68^\circ \), \( \angle A = 180 - 68 = 112^\circ \), \( \angle C = 112^\circ \). Yes, that makes sense. So \( m\angle B = 68^\circ \).

Answer:

\( AD = 8 \)
\( DC = 15 \)
\( m\angle A = 112^\circ \)
\( m\angle B = 68^\circ \)