Question

find the measure of the missing angles.
answer attempt 2 out of 2
b = \boxed{\space}^\circ \quad c = \boxed{1}^\circ

Explanation:

Step1: Find angle \( b \)

Angles on a straight line sum to \( 180^\circ \). The given angle is \( 146^\circ \), so \( b + 146^\circ = 180^\circ \). Solving for \( b \), we get \( b = 180^\circ - 146^\circ = 34^\circ \). Wait, no, actually, \( b \) and the \( 146^\circ \) angle: Wait, looking at the diagram, the vertical line and the slanted line. Wait, \( b \) and the \( 146^\circ \) angle: Wait, maybe \( b \) is supplementary? Wait, no, actually, the vertical line is straight, so the angle adjacent to \( 146^\circ \) (which is \( c \)) and \( 146^\circ \) are supplementary? Wait, no, let's re-examine. The vertical line is a straight line, so the angle \( b \) and the \( 146^\circ \) angle: Wait, maybe \( b \) is equal to \( 180^\circ - 146^\circ \)? No, wait, the vertical line is a straight angle (180 degrees). The slanted line intersects it, creating angles. So the angle labeled \( 146^\circ \) and angle \( c \) are adjacent and form a linear pair? Wait, no, the vertical line is straight, so the angle between the vertical line and the slanted line: Let's see, the vertical line is up and down, the slanted line crosses it. So the angle \( 146^\circ \) and angle \( b \): Wait, maybe \( b \) is vertical to the angle that's supplementary? Wait, no, let's correct. The angle \( b \) and the \( 146^\circ \) angle: Wait, actually, the vertical line is a straight line, so the sum of \( b \) and \( 146^\circ \) is 180? No, wait, maybe \( b \) is equal to \( 180 - 146 = 34 \)? No, wait, no, the vertical line is straight, so the angle \( b \) and the angle adjacent to \( 146^\circ \) (which is \( c \)): Wait, maybe I mixed up. Let's start over.

The vertical line is a straight line, so the angle formed by the slanted line and the vertical line: the angle labeled \( 146^\circ \) and angle \( c \) are adjacent and form a linear pair? Wait, no, the vertical line is straight, so the sum of \( c \) and \( 146^\circ \) is \( 180^\circ \)? Wait, no, the vertical line is up and down, so the angle between the vertical line and the slanted line: the angle \( 146^\circ \) is on one side, and \( c \) is on the other. So \( c + 146^\circ = 180^\circ \), so \( c = 180 - 146 = 34^\circ \). Then, angle \( b \) and the \( 146^\circ \) angle: since \( b \) and the angle \( c \) are vertical angles? Wait, no, vertical angles are equal. Wait, the vertical line and the slanted line intersect, so vertical angles: angle \( b \) and the angle opposite to \( c \)? Wait, maybe I made a mistake. Let's look again. The diagram: vertical line (up and down), slanted line crossing it. The angle between the slanted line and the upper part of the vertical line is \( c \), and the angle between the slanted line and the lower part of the vertical line is \( b \). The angle between the slanted line and the right part (next to \( 146^\circ \)): Wait, the \( 146^\circ \) angle is between the slanted line and the vertical line (lower part? No, upper? Wait, the vertical line has an arrow up and down. The slanted line has an arrow up-left and down-right. So the angle between the slanted line (down-right) and the vertical line (down) is \( 146^\circ \)? No, that can't be. Wait, maybe the \( 146^\circ \) angle is adjacent to \( b \), so \( b + 146^\circ = 180^\circ \), so \( b = 180 - 146 = 34^\circ \). Then, angle \( c \) is equal to \( b \) because they are vertical angles? Wait, no, vertical angles are equal. Wait, no, angle \( c \) and the angle adjacent to \( 146^\circ \): Wait, maybe \( c \) is equal to \( 180 - 146 = 34^\circ \), and \( b \) is equal to \( 146^\circ \)? Wait, that makes more sense. Because the vertical line is straight, so the angle \( b \) and the \( 146^\circ \) angle are vertical angles? No, vertical angles are opposite each other. Wait, when two lines intersect, vertical angles are equal. So the slanted line and the vertical line intersect, creating two pairs of vertical angles. So the angle \( b \) and the angle opposite to \( 146^\circ \)? Wait, no, let's draw mentally: vertical line (V), slanted line (S) intersecting V. So the four angles: top left (c), top right (146°), bottom left (b), bottom right (let's say d). So top right (146°) and bottom left (b) are vertical angles? No, vertical angles are opposite. So top left (c) and bottom right (d) are vertical, top right (146°) and bottom left (b) are vertical? Wait, no, vertical angles are formed by two intersecting lines, so opposite angles. So if S intersects V, then the angle between S (upper left) and V (upper) is c, between S (upper right) and V (upper) is 146°, between S (lower left) and V (lower) is b, between S (lower right) and V (lower) is d. So c and d are vertical, 146° and b are vertical. Wait, that would mean \( b = 146^\circ \), and \( c = 180 - 146 = 34^\circ \). Ah, that's the mistake earlier. So vertical angles are equal, so \( b \) is vertical to \( 146^\circ \), so \( b = 146^\circ \). Then, \( c \) is adjacent to \( 146^\circ \) on the straight line (V), so \( c + 146^\circ = 180^\circ \), so \( c = 180 - 146 = 34^\circ \). Yes, that makes sense. So:

Step1: Find angle \( b \)

Vertical angles are equal. The angle \( b \) and the \( 146^\circ \) angle are vertical angles, so \( b = 146^\circ \).

Step2: Find angle \( c \)

Angles on a straight line sum to \( 180^\circ \). So \( c + 146^\circ = 180^\circ \), so \( c = 180^\circ - 146^\circ = 34^\circ \).

Answer:

\( b = 146^\circ \), \( c = 34^\circ \)