Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles on a straight line sum to \( 180^\circ \). So \( h + 112^\circ = 180^\circ \).
\( h = 180^\circ - 112^\circ = 68^\circ \)

Step2: Find angle \( g \)

Vertical angles are equal, or angles on a straight line. Also, \( g \) and \( 112^\circ \) are vertical? Wait, no, \( h \) and the angle opposite to \( 112^\circ \)? Wait, actually, \( h \) and \( g \): Wait, the vertical line and the slant line. Wait, \( h \) and \( g \) are adjacent? Wait, no, let's re - examine. The straight vertical line, and the slant line. So \( h + g = 180^\circ \)? No, wait, \( h \) and the \( 112^\circ \) are adjacent on a straight line, so \( h = 68^\circ \). Then \( g \) and \( 112^\circ \) are vertical angles? Wait, no, vertical angles are opposite. Wait, the slant line intersects the vertical line. So the angle opposite to \( 112^\circ \) (let's say angle \( x \)) would be equal to \( 112^\circ \), and \( h \) is adjacent to \( 112^\circ \), so \( h = 68^\circ \), and \( g \) is vertical to \( h \)? No, wait, maybe \( g \) and \( 112^\circ \): Wait, no, let's look at the lower part. The horizontal and vertical lines: angle \( m \) and \( 92^\circ \) are adjacent on a straight line, so \( m + 92^\circ = 180^\circ \), so \( m = 88^\circ \). And angle \( k \) is vertical to \( 92^\circ \)? No, \( k \) is adjacent to \( 92^\circ \) on a straight line? Wait, no, the horizontal and vertical lines are perpendicular? Wait, no, the angle given is \( 92^\circ \), so they are not perpendicular. So for the lower intersection:
Angles on a straight line sum to \( 180^\circ \), so \( m + 92^\circ = 180^\circ \), so \( m = 180 - 92=88^\circ \). And angle \( k \): since \( m \) and \( k \) are adjacent on a straight line (vertical line), \( m + k = 180^\circ \)? No, wait, the horizontal line and vertical line intersect, so the angles around the point sum to \( 360^\circ \), and opposite angles are equal. Wait, the angle opposite to \( 92^\circ \) would be equal to \( 92^\circ \), and the angle opposite to \( m \) would be equal to \( m \). So \( m = 88^\circ \), and \( k \): if we consider the vertical line, the angle adjacent to \( 92^\circ \) (on the vertical line) is \( k \), so \( 92^\circ + k = 180^\circ \)? No, that's for a straight line. Wait, the horizontal line is a straight line, so \( m + 92^\circ = 180^\circ \), so \( m = 88^\circ \). Then, the vertical line: the angle \( k \) and the angle opposite to \( m \) (which is also \( 88^\circ \)): Wait, no, let's get back to the upper angles.

Upper intersection: slant line and vertical line. So angle \( h \) and \( 112^\circ \) are supplementary (sum to \( 180^\circ \)), so \( h = 68^\circ \). Then angle \( g \): since \( h \) and \( g \) are supplementary? No, \( h \) and \( g \) are adjacent on the vertical line? Wait, no, the slant line intersects the vertical line, so the angle \( g \) and \( 112^\circ \) are vertical angles? Wait, no, vertical angles are formed by two intersecting lines. So the two lines (slant and vertical) intersect, so the angle opposite to \( 112^\circ \) is equal to \( 112^\circ \), and the angle opposite to \( h \) is equal to \( h \). So \( h + 112^\circ = 180^\circ \) (supplementary), so \( h = 68^\circ \), and \( g \) is equal to \( 112^\circ \)? Wait, no, that can't be. Wait, maybe I made a mistake. Let's re - draw mentally: vertical line, slant line crossing it. So at the intersection, four angles: \( 112^\circ \), \( h \), \( g \), and the angle opposite to \( 112^\circ \). So \( 112^\circ + h = 180^\circ \) (linear pair), so \( h = 68^\circ \). Then \( g \) is equal to \( 112^\circ \) (vertical angle to \( 112^\circ \))? Wait, no, vertical angles are opposite. So if one angle is \( 112^\circ \), its vertical angle is also \( 112^\circ \), and \( h \) and its vertical angle are \( 68^\circ \). So \( g \) is equal to \( 112^\circ \)? Wait, no, maybe \( g \) is adjacent to \( h \). Wait, maybe the vertical line is straight, so \( h + g = 180^\circ \)? No, that would be if they are on a straight line, but they are on the intersection of two lines. Wait, I think I messed up. Let's do the lower angles first.

Lower intersection: horizontal and vertical lines. So angle \( m \) and \( 92^\circ \) are on a straight line (horizontal), so \( m + 92^\circ = 180^\circ \), so \( m = 180 - 92 = 88^\circ \). Angle \( k \): since \( m \) and \( k \) are on a straight line (vertical), \( m + k = 180^\circ \)? No, the vertical line is straight, so the angle adjacent to \( 92^\circ \) (on the vertical line) is \( k \), so \( 92^\circ + k = 180^\circ \)? No, that's for the horizontal line. Wait, the two lines (horizontal and vertical) intersect, so the angles around the point sum to \( 360^\circ \), and opposite angles are equal. So the angle opposite to \( 92^\circ \) is \( 92^\circ \), and the angle opposite to \( m \) is \( m \). So \( m = 88^\circ \), and \( k \): if we consider the angle between the vertical line and the horizontal line, the angle \( k \) and \( 92^\circ \) are supplementary? No, \( k \) is adjacent to \( 92^\circ \) on the vertical line? Wait, no, the horizontal line is a straight line, so the sum of angles on a straight line is \( 180^\circ \). So for the horizontal line, \( m + 92^\circ = 180^\circ \), so \( m = 88^\circ \). Then, for the vertical line, the angle \( k \) and the angle opposite to \( m \) (which is \( 88^\circ \)): Wait, no, the vertical line is also a straight line, so the sum of angles on the vertical line is \( 180^\circ \). So the angle adjacent to \( m \) (on the vertical line) is \( k \), so \( m + k = 180^\circ \)? No, \( m \) is on the horizontal line. I think I confused the lines. Let's start over.

For angle \( h \):

The angle of \( 112^\circ \) and angle \( h \) form a linear pair (they are adjacent and on a straight line). The sum of angles in a linear pair is \( 180^\circ \).
So, \( h+ 112^\circ=180^\circ\)
\(h = 180^\circ - 112^\circ=68^\circ\)

For angle \( g \):

Angle \( g \) and the \( 112^\circ \) angle are vertical angles? Wait, no. Wait, angle \( g \) and the \( 112^\circ \) angle: when two lines intersect, vertical angles are equal. Wait, the slant line and the vertical line intersect, so the angle opposite to \( 112^\circ \) is equal to \( 112^\circ \), and angle \( g \) is equal to that opposite angle? Wait, no, angle \( h \) and angle \( g \): if we look at the vertical line, angle \( h \) and angle \( g \) are adjacent and form a linear pair? No, the vertical line is straight, so the sum of angles on the vertical line (formed by the slant line) should be \( 180^\circ \). Wait, no, the slant line intersects the vertical line, creating two linear pairs. So \( h + 112^\circ = 180^\circ \) (linear pair 1), and \( g + h= 180^\circ \)? No, that would mean \( g = 112^\circ \). Yes, because if \( h + 112^\circ = 180^\circ \) and \( g + h= 180^\circ \), then \( g = 112^\circ \) (by transitive property: \( 180^\circ - h=112^\circ \) and \( 180^\circ - h = g \), so \( g = 112^\circ \))

For angle \( m \):

Angle \( m \) and the \( 92^\circ \) angle form a linear pair (they are adjacent and on a straight line - the horizontal line). The sum of angles in a linear pair is \( 180^\circ \).
So, \( m + 92^\circ=180^\circ\)
\(m=180^\circ - 92^\circ = 88^\circ\)

For angle \( k \):

Angle \( k \) and the \( 92^\circ \) angle are vertical angles? Wait, no. The horizontal and vertical lines intersect, so angle \( k \) and the \( 92^\circ \) angle: angle \( k \) and \( m \) form a linear pair? Wait, the vertical line is straight, so angle \( k \) and \( m \) are adjacent and form a linear pair? No, \( m \) is on the horizontal line. Wait, when two lines (horizontal and vertical) intersect, the angle opposite to \( 92^\circ \) is equal to \( 92^\circ \), and the angle opposite to \( m \) (which is \( 88^\circ \)) is equal to \( m \). Also, angle \( k \) and \( 92^\circ \) are supplementary? Wait, no, angle \( k \) and \( m \) are on the vertical line. Wait, the sum of angles on the vertical line (formed by the horizontal line) is \( 180^\circ \). So angle \( k \) and the angle opposite to \( 92^\circ \) (which is \( 92^\circ \)): No, let's use the linear pair for angle \( k \). The angle \( k \) and \( m \): Wait, no, \( m \) is \( 88^\circ \), and angle \( k \) and \( 92^\circ \) are vertical angles? Wait, no, vertical angles are opposite. So if the horizontal line and vertical line intersect, then angle \( k \) is opposite to the angle adjacent to \( 92^\circ \) on the horizontal line. Wait, I think the correct way is: angle \( k \) and \( 92^\circ \) are supplementary? No, angle \( k \) and \( m \) are supplementary? Wait, no, let's use the fact that in the lower intersection, the sum of angles around a point is \( 360^\circ \), but linear pairs are easier.

Since \( m = 88^\circ \), and angle \( k \) and \( m \) are adjacent on the vertical line (forming a linear pair), so \( k + m=180^\circ \)? No, that's not right. Wait, the horizontal line is straight, so \( m + 92^\circ = 180^\circ \) (linear pair). The vertical line is straight, so the angle \( k \) and the angle opposite to \( 92^\circ \) (which is \( 92^\circ \)): Wait, no, the angle opposite to \( m \) is \( 88^\circ \), and the angle opposite to \( 92^\circ \) is \( 92^\circ \). So angle \( k \) is equal to \( 92^\circ \)? No, that can't be. Wait, I think I made a mistake in identifying the lines. Let's assume that the lower two lines (horizontal and vertical) are intersecting, so angle \( m \) and \( 92^\circ \) are adjacent on the horizontal line (linear pair), so \( m = 88^\circ \). Then angle \( k \) is adjacent to \( 92^\circ \) on the vertical line, so \( k + 92^\circ = 180^\circ \), so \( k = 88^\circ \)? No, that's conflicting. Wait, no, when two lines intersect, vertical angles are equal, and linear pairs sum to \( 180^\circ \). So for the lower intersection:

- Let the horizontal line be \( L_1 \) and vertical line be \( L_2 \). They intersect at a point.
- Angle \( A = 92^\circ \) (between \( L_1 \) (right) and \( L_2 \) (up)).
- Angle \( m \) is between \( L_1 \) (left) and \( L_2 \) (up), so it's a linear pair with \( A \), so \( m + A=180^\circ \), so \( m = 88^\circ \).
- Angle \( k \) is between \( L_2 \) (down) and \( L_1 \) (left), so it's vertical to \( A \) (since vertical angles are opposite), so \( k = A = 92^\circ \)? No, vertical angles are opposite. So angle \( A \) (92°) and angle \( k \) are vertical angles? Wait, angle \( A \) is between \( L_1 \) (right) and \( L_2 \) (up), angle \( k \) is between \( L_1 \) (left) and \( L_2 \) (down)? No, that's not vertical. Vertical angles are opposite, so angle \( A \) (92°) and the angle between \( L_1 \) (left) and \( L_2 \) (down) are not vertical. Wait, I think I need to use the fact that the sum of angles around a point is \( 360^\circ \). So \( 92^\circ + m + k + \) (angle opposite to \( m \)) \( = 360^\circ \). But since vertical angles are equal, angle opposite to \( m \) is \( m \), and angle opposite to \( 92^\circ \) is \( 92^\circ \). So \( 92^\circ + m + k + 92^\circ + m=360^\circ \)? No, that's not right. Wait, no, when two lines intersect, there are two pairs of vertical angles. So for two intersecting lines, the four angles are \( \alpha, 180 - \alpha, \alpha, 180 - \alpha \). So in the lower intersection, one angle is \( 92^\circ \), so the opposite angle is \( 92^\circ \), and the other two angles are \( 180 - 92 = 88^\circ \), so \( m = 88^\circ \), and the angle opposite to \( m \) is \( 88^\circ \), and \( k \) is equal to \( 92^\circ \) (since it's opposite to the \( 92^\circ \) angle). Wait, that makes sense. So:

- \( m = 88^\circ \) (linear pair with \( 92^\circ \))
- \( k = 92^\circ \) (vertical angle with \( 92^\circ \))
- \( h = 68^\circ \) (linear pair with \( 112^\circ \))
- \( g = 112^\circ \) (vertical angle with \( 112^\circ \))

Answer:

\( h = 68^\circ \), \( g = 112^\circ \), \( m = 88^\circ \), \( k = 92^\circ \)