Question

in the figure below, (overline{ab} parallel overline{cd}). what is the measure of (angle dec)? select one answer a (48^circ) b (65^circ) c (67^circ) d (113^circ)

Explanation:

Step1: Identify Alternate Interior Angles

Since \( \overline{AB} \parallel \overline{CD} \), the angle \( \angle GEB = 48^\circ \) and its alternate interior angle \( \angle ECD \) should be equal. Wait, actually, let's look at the transversal. Wait, \( \angle GEB = 48^\circ \), and since \( AB \parallel CD \), the angle at \( E \) related to \( \angle D = 65^\circ \). Wait, in triangle \( DEC \), we can use the fact that the sum of angles in a triangle is \( 180^\circ \). Wait, first, find the angle \( \angle CED \)'s adjacent angles? Wait, no. Wait, the angle \( \angle GEB = 48^\circ \), so the vertical angle or alternate angle? Wait, \( AB \parallel CD \), so \( \angle AEF \) (but maybe better: the angle at \( E \) between \( AB \) and \( CD \)). Wait, actually, the angle \( \angle BEG = 48^\circ \), so the corresponding angle or alternate interior angle with \( \angle ECD \)? Wait, maybe another approach. The straight line \( FD \) is a transversal, and \( AB \parallel CD \), so \( \angle AEB \) (but no, let's look at the triangle \( DEC \). We know \( \angle D = 65^\circ \), and we need to find \( \angle DEC \). Wait, first, find the angle \( \angle ECD \). Since \( AB \parallel CD \), the angle \( \angle GEB = 48^\circ \) is equal to \( \angle ECD \) (alternate interior angles). So \( \angle ECD = 48^\circ \). Then in triangle \( DEC \), angles sum to \( 180^\circ \). So \( \angle DEC + \angle ECD + \angle D = 180^\circ \). So \( \angle DEC + 48^\circ + 65^\circ = 180^\circ \).

Step2: Calculate \( \angle DEC \)

\( \angle DEC = 180^\circ - 48^\circ - 65^\circ = 67^\circ \). Wait, but wait, maybe I made a mistake. Wait, no, let's check again. Wait, the angle \( \angle GEB = 48^\circ \), so the angle \( \angle AEC \) is vertical to something? Wait, no, let's re-express. The angle \( \angle BEG = 48^\circ \), so the alternate interior angle with \( \angle ECD \) is \( 48^\circ \), so \( \angle ECD = 48^\circ \). Then in triangle \( DEC \), \( \angle D = 65^\circ \), \( \angle ECD = 48^\circ \), so \( \angle DEC = 180 - 65 - 48 = 67^\circ \). Wait, but the options have 67 as option C. Wait, but let's check again. Wait, maybe the angle \( \angle BEG = 48^\circ \), so the angle \( \angle AED \) is supplementary? No, wait, maybe I messed up the alternate interior angles. Wait, \( AB \parallel CD \), and the transversal is \( EG \) or \( EB \). Wait, the line \( EB \) and \( CD \) are parallel? No, \( AB \parallel CD \), so the transversal is \( EC \). Wait, maybe the angle \( \angle BEG = 48^\circ \), so the angle \( \angle ECD = 48^\circ \) (alternate interior angles). Then in triangle \( DEC \), angles are \( \angle D = 65^\circ \), \( \angle ECD = 48^\circ \), so \( \angle DEC = 180 - 65 - 48 = 67^\circ \). So that's option C.

Answer:

C. \( 67^\circ \)