Question

find the measure of the missing angles.
answer attempt 1 out of 5
b = \\(\square^\circ\\) \\(c = \square^\circ\\)

Explanation:

Step1: Find angle \( b \)

Angles on a straight line sum to \( 180^\circ \). So \( b + 64^\circ = 180^\circ \).
\( b = 180^\circ - 64^\circ = 116^\circ \)

Step2: Find angle \( c \)

Vertical angles are equal, or angles on a straight line. Also, \( c \) and \( 64^\circ \) are vertical angles? Wait, no, \( c \) and \( 64^\circ \) are adjacent? Wait, no, \( c \) and \( 64^\circ \): actually, \( c \) and the \( 64^\circ \) angle are vertical angles? Wait, no, looking at the diagram, \( c \) and \( 64^\circ \): wait, \( b \) and \( c \) are adjacent? Wait, no, let's re-examine. The angle \( c \) and the \( 64^\circ \) angle: actually, \( c \) is equal to \( 64^\circ \) because they are vertical angles? Wait, no, wait. Wait, the two lines intersect, so vertical angles are equal. Wait, the angle opposite to \( 64^\circ \) is \( c \)? Wait, no, the angle \( b \) is adjacent to \( 64^\circ \), and \( c \) is opposite to \( 64^\circ \)? Wait, no, let's see: the horizontal line and the slanted line intersect. So the angle \( 64^\circ \) and angle \( c \): are they vertical angles? Wait, no, \( c \) and \( 64^\circ \): actually, \( c \) is equal to \( 64^\circ \) because they are vertical angles? Wait, no, wait, the angle \( b \) and \( 64^\circ \) are supplementary (sum to \( 180^\circ \)), and angle \( c \) is equal to \( 64^\circ \) because they are vertical angles? Wait, no, let's correct. When two lines intersect, vertical angles are equal. So the angle opposite to \( 64^\circ \) is \( c \)? Wait, no, the angle \( c \) is adjacent to \( b \). Wait, maybe I made a mistake. Let's start over.

Wait, the horizontal line and the slanted line intersect. So the angle \( 64^\circ \) and angle \( b \) are adjacent and form a linear pair (sum to \( 180^\circ \)), so \( b = 180 - 64 = 116^\circ \). Then, angle \( c \) and the \( 64^\circ \) angle: are they vertical angles? Wait, no, angle \( c \) and \( 64^\circ \): actually, angle \( c \) is equal to \( 64^\circ \) because they are vertical angles? Wait, no, vertical angles are opposite each other. Wait, the angle \( 64^\circ \) and angle \( c \): if you look at the intersection, the angle \( c \) is opposite to the \( 64^\circ \) angle? Wait, no, the angle \( c \) is adjacent to \( b \). Wait, maybe \( c \) is equal to \( 64^\circ \) because they are vertical angles. Wait, let's check: the angle \( 64^\circ \) and angle \( c \): when two lines intersect, vertical angles are equal. So the angle opposite to \( 64^\circ \) is \( c \), so \( c = 64^\circ \). Alternatively, angle \( c \) and \( b \) are supplementary? No, \( b \) is \( 116^\circ \), so \( c = 180 - 116 = 64^\circ \). Yes, that makes sense. So \( c = 64^\circ \).

So:

\( b = 180 - 64 = 116^\circ \)

\( c = 64^\circ \) (because vertical angles are equal, or \( 180 - 116 = 64 \))

Answer:

\( b = \boxed{116}^\circ \), \( c = \boxed{64}^\circ \)