Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles \( h \) and \( 93^\circ \) are supplementary (form a linear pair), so \( h + 93^\circ = 180^\circ \).
\( h = 180^\circ - 93^\circ = 87^\circ \)

Step2: Find angle \( g \)

Angles \( g \) and \( 93^\circ \) are vertical angles? No, \( g \) and \( h \) are vertical? Wait, \( h \) and \( g \): Wait, the horizontal line and vertical line intersect, so \( h \) and the angle opposite to \( 93^\circ \)? Wait, no, \( h \) and \( 93^\circ \) are adjacent, forming a linear pair. Then \( g \) is vertical to \( 93^\circ \)? Wait, no, the vertical line and horizontal line intersect, so \( h + 93^\circ = 180^\circ \) (linear pair), and \( g \) is vertical to \( 93^\circ \)? Wait, no, \( g \) and \( h \) are adjacent? Wait, maybe I made a mistake. Wait, the vertical line and horizontal line: so the angle \( h \) and \( 93^\circ \) are adjacent, sum to \( 180^\circ \), so \( h = 87^\circ \). Then \( g \) is vertical to \( 93^\circ \)? No, \( g \) is adjacent to \( h \), forming a linear pair? Wait, no, the vertical line is straight, so \( h + 93^\circ = 180^\circ \), and \( g \) is equal to \( 93^\circ \)? Wait, no, vertical angles: when two lines intersect, vertical angles are equal. Wait, the horizontal line and vertical line intersect, so the angle \( h \) and the angle opposite to \( 93^\circ \) (let's say angle \( x \)) are vertical? No, \( h \) and \( 93^\circ \) are adjacent, so \( h = 180 - 93 = 87 \). Then \( g \) is vertical to \( 93^\circ \)? Wait, no, \( g \) is adjacent to \( h \), so \( g + h = 180^\circ \)? No, the vertical line is straight, so the sum of angles on a straight line is \( 180^\circ \). Wait, maybe \( g \) is equal to \( 93^\circ \)? No, that can't be. Wait, let's re-examine. The horizontal line and vertical line intersect, creating four angles. \( h \) and \( 93^\circ \) are adjacent, so they are supplementary. So \( h = 180 - 93 = 87^\circ \). Then \( g \) is vertical to \( 93^\circ \)? No, \( g \) is vertical to \( h \)? Wait, no, vertical angles: when two lines intersect, opposite angles are equal. So the angle opposite to \( h \) is \( 93^\circ \), and the angle opposite to \( 93^\circ \) is \( h \). Wait, no, that's not right. Wait, if \( h \) and \( 93^\circ \) are adjacent, then \( h + 93 = 180 \), so \( h = 87 \). Then the angle opposite to \( h \) is \( 93^\circ \), and the angle opposite to \( 93^\circ \) is \( h = 87^\circ \). Wait, maybe \( g \) is equal to \( 93^\circ \)? No, I think I messed up. Wait, the vertical line is straight, so the angle \( g \) and \( h \) are adjacent, forming a linear pair? No, the horizontal line is straight, so \( h + 93 = 180 \), so \( h = 87 \). Then the vertical line: the angle \( g \) is adjacent to \( h \), so \( g + h = 180 \)? No, the vertical line is straight, so the sum of angles on the vertical line? Wait, no, the horizontal line is horizontal, vertical line is vertical, so they intersect at a point, forming four angles: \( h \), \( 93^\circ \), \( g \), and the angle opposite to \( h \). So \( h + 93 = 180 \) (linear pair), so \( h = 87 \). Then \( g \) is equal to \( 93^\circ \) (vertical angle to \( 93^\circ \))? Wait, no, vertical angles are opposite. So \( h \) and the angle opposite to \( 93^\circ \) are vertical? No, let's label the intersection: horizontal line (left-right) and vertical line (up-down) intersect at a point. Let's call the intersection point O. The angle between vertical line (up) and horizontal line (right) is \( 93^\circ \), so angle \( \angle AOB = 93^\circ \), where A is up, B is right. Then angle \( \angle BOC = h \), where C is left. So \( \angle AOB + \angle BOC = 180^\circ \) (linear pair), so \( h = 180 - 93 = 87^\circ \). Then angle \( \angle COD = 93^\circ \) (vertical to \( \angle AOB \)), and angle \( \angle DOA = h = 87^\circ \) (vertical to \( \angle BOC \)). Wait, maybe \( g \) is \( 93^\circ \)? No, maybe \( g \) is the angle below the horizontal line, on the vertical line. Wait, the vertical line has two parts: up and down. The horizontal line intersects the vertical line, so above the horizontal line: \( h \) and \( 93^\circ \), below the horizontal line: \( g \) and the angle opposite to \( h \). So \( g \) is equal to \( 93^\circ \)? No, I think I need to correct. Let's move to angle \( m \) and \( k \).

Step3: Find angle \( m \)

Angles \( m \) and \( 122^\circ \) are supplementary (linear pair), so \( m + 122^\circ = 180^\circ \).
\( m = 180^\circ - 122^\circ = 58^\circ \)

Step4: Find angle \( k \)

Angle \( k \) is vertical to \( 122^\circ \), so \( k = 122^\circ \) (vertical angles are equal). Also, angle \( k \) and \( m \) are supplementary, so \( k + m = 180^\circ \), which checks out: \( 122 + 58 = 180 \).

Wait, going back to \( g \): The vertical line is straight, so the angle above the horizontal line (93°) and the angle below the horizontal line (let's say \( g \)): Wait, no, the horizontal line is straight, so the sum of angles on the horizontal line is 180°, but the vertical line is also straight. Wait, maybe \( g \) is equal to 93°? No, earlier we found \( h = 87° \), and \( g \) is adjacent to \( h \) on the vertical line? Wait, no, the vertical line is straight, so the angle \( h \) and \( g \) are adjacent, forming a linear pair? No, \( h \) is on the horizontal line, \( g \) is on the vertical line. Wait, I think I made a mistake in identifying \( g \). Let's re-express:

The horizontal line (left-right) and vertical line (up-down) intersect, creating four angles:
- Top-right: 93°
- Top-left: \( h \)
- Bottom-right: equal to \( h \) (vertical angle)
- Bottom-left: \( g \), equal to 93° (vertical angle to top-right)

So \( h + 93° = 180° \) (linear pair), so \( h = 87° \). Then \( g = 93° \) (vertical angle to top-right 93°).

So summarizing:

- \( h \): supplementary to 93°, so \( h = 180 - 93 = 87° \)
- \( g \): vertical to 93°, so \( g = 93° \) (or supplementary to \( h \): \( g = 180 - 87 = 93° \), same result)
- \( m \): supplementary to 122°, so \( m = 180 - 122 = 58° \)
- \( k \): vertical to 122°, so \( k = 122° \) (or supplementary to \( m \): \( k = 180 - 58 = 122° \), same result)

Answer:

• \( h = 87^\circ \)
• \( g = 93^\circ \)
• \( m = 58^\circ \)
• \( k = 122^\circ \)