Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles on a straight line sum to \( 180^\circ \). So, \( h + 109^\circ = 180^\circ \).
\( h = 180^\circ - 109^\circ = 71^\circ \)

Step2: Find angle \( g \)

Vertical angles are equal, or angles on a straight line. Since \( h + 109^\circ = 180^\circ \) and \( g + h = 180^\circ \) (or vertical to \( 109^\circ \)? Wait, actually, \( g \) and \( 109^\circ \) are vertical? No, \( h \) and the angle opposite to \( 109^\circ \)? Wait, no, let's re - look. The straight line (vertical) and the slant line. So \( h + 109^\circ = 180^\circ \), and \( g \) and \( 109^\circ \) are vertical? Wait, no, \( h \) and \( g \) are adjacent to the slant line. Wait, actually, \( h + 109^\circ = 180^\circ \), so \( h = 71^\circ \), and \( g \) is vertical to \( 109^\circ \)? No, \( g \) and \( 109^\circ \): Wait, the vertical line and the slant line intersect, so \( h + 109^\circ = 180^\circ \), and \( g \) is equal to \( 109^\circ \)? No, wait, no. Wait, the two angles \( h \) and \( 109^\circ \) are supplementary (sum to \( 180^\circ \)), and \( g \) and \( h \) are supplementary? No, \( g \) and \( 109^\circ \) are vertical angles? Wait, no, let's draw mentally. The vertical line and the slant line cross, so the angle \( h \) and \( 109^\circ \) are adjacent on a straight line, so \( h = 180 - 109 = 71^\circ \). Then \( g \) is vertical to \( 109^\circ \)? No, \( g \) is adjacent to \( h \) on the vertical line? Wait, no, the vertical line is straight, so \( h + g = 180^\circ \)? No, the vertical line is a straight line, so the angles on one side of the slant line: \( h + 109^\circ = 180^\circ \), and on the other side, \( g \) and the angle equal to \( h \)? Wait, maybe I made a mistake. Wait, vertical angles: when two lines intersect, vertical angles are equal. So the slant line and vertical line intersect, so the angle opposite to \( 109^\circ \) is \( g \)? No, \( h \) and the angle opposite to \( 109^\circ \): Wait, no, let's label the intersection. The vertical line is \( L \), slant line is \( M \). They intersect at a point. So angle between \( L \) (upper part) and \( M \) (right part) is \( 109^\circ \), angle between \( L \) (upper part) and \( M \) (left part) is \( h \), so \( h + 109^\circ = 180^\circ \) (linear pair), so \( h = 71^\circ \). Then angle between \( L \) (lower part) and \( M \) (left part) is \( g \), which is vertical to \( 109^\circ \)? No, vertical angle to \( 109^\circ \) is \( g \)? Wait, no, vertical angles are opposite each other. So if \( 109^\circ \) is at the top - right, then the vertical angle is at the bottom - left, which is \( g \)? Wait, no, the vertical line is straight, so the lower part of the vertical line is a straight extension. So \( g \) is equal to \( 109^\circ \)? No, that can't be, because \( h + 109 = 180 \), so \( h = 71 \), and \( g \) should be equal to \( 109 \)? Wait, no, let's check the other intersection.

Now for the horizontal and vertical lines: they are perpendicular? No, the angle between vertical and horizontal is \( m \) and \( 84^\circ \). So \( m + 84^\circ = 180^\circ \) (linear pair), so \( m = 180 - 84 = 96^\circ \)? Wait, no, if they are intersecting lines, vertical angles: \( m \) and \( k \): Wait, horizontal and vertical lines intersect, so the angle \( m \) and \( 84^\circ \) are supplementary (linear pair), so \( m = 180 - 84 = 96^\circ \), and \( k \) is equal to \( 84^\circ \) (vertical angle), or \( k = 180 - m = 84^\circ \).

Wait, let's re - organize:

For the slant and vertical lines:

- Linear pair: \( h + 109^\circ = 180^\circ \) ⇒ \( h = 180 - 109 = 71^\circ \)
- Vertical angle to \( 109^\circ \): The angle opposite to \( 109^\circ \) (let's say \( g \)'s adjacent? Wait, no, \( g \) and \( 109^\circ \): Wait, the vertical line is straight, so the angle \( g \) and \( h \) are adjacent on the vertical line? No, the slant line cuts the vertical line, so two angles at the intersection: \( h \) and \( 109^\circ \) (supplementary), and the other two angles (vertical to \( h \) and vertical to \( 109^\circ \)). So the angle vertical to \( h \) is equal to \( 109^\circ \)? No, vertical angles are equal. So if \( h \) and \( 109^\circ \) are supplementary, then the angle vertical to \( h \) is also supplementary to the angle vertical to \( 109^\circ \). Wait, maybe I was wrong. Let's do it step by step.

1. Angle \( h \):
Since \( h \) and \( 109^\circ \) form a linear pair (they are adjacent and on a straight line), their sum is \( 180^\circ \).
\( h + 109^\circ = 180^\circ \)
\( h = 180^\circ - 109^\circ = 71^\circ \)

2. Angle \( g \):
\( g \) and \( 109^\circ \) are vertical angles (opposite each other when two lines intersect), so vertical angles are equal. Wait, no, if \( h \) and \( 109^\circ \) are supplementary, then \( g \) and \( h \) are supplementary? Wait, no, the vertical line is a straight line, so the sum of \( h \) and \( g \) should be \( 180^\circ \)? No, that's not right. Wait, the slant line and vertical line intersect, so there are two pairs of vertical angles. One pair is \( h \) and the angle opposite to \( 109^\circ \), and the other pair is \( 109^\circ \) and \( g \). Wait, yes! So \( g \) and \( 109^\circ \) are vertical angles? No, that would mean \( g = 109^\circ \), but then \( h + g = 71 + 109 = 180 \), which is correct (linear pair). Oh right! So \( g = 109^\circ \) (vertical angle to the \( 109^\circ \) angle? Wait, no, vertical angles are opposite. So if the angle between vertical (upper) and slant (right) is \( 109^\circ \), then the angle between vertical (lower) and slant (left) is \( g \), which is vertical to \( 109^\circ \)? Wait, no, the angle between vertical (lower) and slant (right) would be vertical to \( h \). I think I confused the labels. Anyway, using linear pair: \( h + 109 = 180 \), so \( h = 71 \), and \( g \) is equal to \( 109^\circ \) because they are vertical angles? Wait, no, \( h \) and \( g \) are adjacent on the vertical line, so \( h + g = 180 \), so \( g = 180 - h = 180 - 71 = 109^\circ \). Yes, that makes sense.

For the horizontal and vertical lines:

- Angle \( m \):
\( m \) and \( 84^\circ \) form a linear pair, so \( m + 84^\circ = 180^\circ \)
\( m = 180^\circ - 84^\circ = 96^\circ \)

- Angle \( k \):
\( k \) and \( 84^\circ \) are vertical angles, so \( k = 84^\circ \) (or \( k = 180 - m = 180 - 96 = 84^\circ \))

Answer:

\( h = 71^\circ \), \( g = 109^\circ \), \( m = 96^\circ \), \( k = 84^\circ \)