Question

use the angle addition postulate to find ( mangle cde ), if ( mangle cdy = 48^circ ) and ( mangle yde = 86^circ )
diagram: point d with rays: e (up), y (left), c (down - left).
options:
a ( 38^circ )
b ( 58^circ )
c ( 142^circ )
d ( 134^circ )

Explanation:

Step1: Recall Angle Addition Postulate

The Angle Addition Postulate states that if a point \( Y \) lies in the interior of \( \angle CDE \), then \( m\angle CDE = m\angle CDY + m\angle YDE \). Wait, no, looking at the diagram, actually, \( \angle CDE \) is composed of \( \angle CDY \) and \( \angle YDE \)? Wait, no, maybe I misread. Wait, the diagram: \( D \) is the vertex, with \( Y \) on a horizontal line, \( E \) on a vertical line, and \( C \) below. Wait, maybe \( \angle YDE \) is 90? No, the problem says \( m\angle YDE = 86^\circ \) and \( m\angle CDY = 48^\circ \). Wait, actually, maybe \( \angle CDE = \angle CDY + \angle YDE \)? Wait, no, that would be if \( Y \) is between \( C \) and \( E \), but the diagram shows \( Y \) horizontal, \( E \) vertical, \( C \) below. Wait, maybe it's a typo, or maybe I got the angles wrong. Wait, no, the Angle Addition Postulate: if two angles are adjacent (share a common side and vertex, and their interiors don't overlap), then the measure of the larger angle is the sum of the measures of the two smaller angles. Wait, maybe \( \angle YDE \) is 90? No, the problem says 86. Wait, maybe the correct approach is: Wait, the options include 134? Wait, 48 + 86 = 134? Wait, no, 48 + 86 is 134? Wait, 48 + 86: 40 + 80 = 120, 8 + 6 = 14, so 120 + 14 = 134. Wait, but let's check the diagram again. The diagram has \( D \) with \( Y \) to the left, \( E \) up, \( C \) down-left. So \( \angle CDY \) is between \( C \) and \( Y \), \( \angle YDE \) is between \( Y \) and \( E \). So \( \angle CDE \) is between \( C \) and \( E \), so by Angle Addition Postulate, \( m\angle CDE = m\angle CDY + m\angle YDE \)? Wait, no, that would be if \( Y \) is between \( C \) and \( E \), but in the diagram, \( Y \) is horizontal, \( E \) is vertical, so \( \angle YDE \) is 90? No, the problem says 86. Wait, maybe the problem has a typo, but according to the problem, \( m\angle CDY = 48^\circ \), \( m\angle YDE = 86^\circ \), so \( m\angle CDE = 48 + 86 = 134^\circ \)? Wait, but option D is 134? Wait, the options are A 38, B 58, C 142, D 134. Wait, 48 + 86 is 134, so D.

Wait, maybe I made a mistake. Wait, maybe the angle is \( \angle YDE \) is 90, but the problem says 86. Wait, the problem states \( m\angle YDE = 86^\circ \), so we have to go with that. So using Angle Addition Postulate: \( m\angle CDE = m\angle CDY + m\angle YDE \)? Wait, no, maybe it's \( m\angle CDE = m\angle YDE + m\angle CDY \)? Wait, 48 + 86 = 134, which is option D.

Step1: Apply Angle Addition Postulate

The Angle Addition Postulate states that if a point \( Y \) is in the interior of \( \angle CDE \), then \( m\angle CDE = m\angle CDY + m\angle YDE \).

Step2: Substitute the given values

We know \( m\angle CDY = 48^\circ \) and \( m\angle YDE = 86^\circ \). Substituting these values into the formula:

\( m\angle CDE = 48^\circ + 86^\circ \)

Step3: Calculate the sum

\( 48 + 86 = 134 \), so \( m\angle CDE = 134^\circ \).

Answer:

D. \( 134^\circ \)