Question

question
answer
ad bc
ab dc
m a
m b
m d
dc
your answers are correct because the quadrilateral shown is a
this is true because it has

Explanation:

Step1: Identify the quadrilateral type

The quadrilateral has \( AD \parallel BC \) and \( AB \cong DC \), so it's an isosceles trapezoid (or a parallelogram? Wait, \( AB \cong DC \) and \( AD \parallel BC \), actually, if \( AD \parallel BC \) and \( AB \cong DC \), it's an isosceles trapezoid. In a trapezoid (or parallelogram, but here with \( AB \cong DC \) and \( AD \parallel BC \), let's check angles. In a parallelogram, opposite angles are equal and consecutive angles are supplementary. Wait, maybe it's a parallelogram? Wait, \( AD \parallel BC \) and if \( AB \parallel DC \), then it's a parallelogram. Wait, the problem says \( AB \cong DC \), not \( AB \parallel DC \). Wait, the given is \( AD \parallel BC \) and \( AB \cong DC \), so it's an isosceles trapezoid. In an isosceles trapezoid, base angles are equal, and consecutive angles between bases are supplementary.

Step2: Find \( m\angle A \)

In an isosceles trapezoid, \( \angle A \) and \( \angle D \) are adjacent to base \( AD \), and \( \angle B \) and \( \angle C \) are adjacent to base \( BC \). Wait, \( AD \parallel BC \), so \( \angle A + \angle B = 180^\circ \), \( \angle D + \angle C = 180^\circ \), and \( \angle A = \angle D \), \( \angle B = \angle C \)? Wait, no, in isosceles trapezoid, base angles are equal: \( \angle A = \angle D \), \( \angle B = \angle C \)? Wait, no, let's correct. In an isosceles trapezoid with bases \( AD \) and \( BC \), \( \angle A \) and \( \angle B \) are adjacent to leg \( AB \), \( \angle D \) and \( \angle C \) adjacent to leg \( DC \). Since \( AB \cong DC \), \( \angle A = \angle D \), \( \angle B = \angle C \). Wait, \( \angle C = 73^\circ \), so \( \angle B = 73^\circ \)? No, wait, \( AD \parallel BC \), so \( \angle A + \angle B = 180^\circ \), \( \angle D + \angle C = 180^\circ \). If it's an isosceles trapezoid, \( \angle A = \angle D \), \( \angle B = \angle C \). Wait, \( \angle C = 73^\circ \), so \( \angle B = 73^\circ \), then \( \angle A = 180^\circ - 73^\circ = 107^\circ \), \( \angle D = 107^\circ \).

Step3: Find \( m\angle B \)

Since \( \angle B \) and \( \angle C \) are base angles (if \( BC \) is the base), and in isosceles trapezoid, base angles are equal? Wait, no, maybe I mixed up. Wait, \( AD \parallel BC \), so \( \angle C + \angle D = 180^\circ \), \( \angle A + \angle B = 180^\circ \). If \( AB \cong DC \), then \( \angle A = \angle D \), \( \angle B = \angle C \). So \( \angle C = 73^\circ \), so \( \angle B = 73^\circ \), \( \angle D = 180^\circ - 73^\circ = 107^\circ \), \( \angle A = 107^\circ \).

Step4: Find \( DC \)

Since \( AB = 32 \) and \( AB \cong DC \) (given), so \( DC = 32 \).

Answer:

\( m\angle A = 107^\circ \), \( m\angle B = 73^\circ \), \( m\angle D = 107^\circ \), \( DC = 32 \)

(Note: The quadrilateral is an isosceles trapezoid because it has one pair of parallel sides (\( AD \parallel BC \)) and the non-parallel sides (\( AB \) and \( DC \)) are congruent. In an isosceles trapezoid, base angles are equal, and consecutive angles between the bases are supplementary.)