Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles on a straight line sum to \( 180^\circ \). So, \( h + 119^\circ = 180^\circ \).
\( h = 180^\circ - 119^\circ = 61^\circ \)

Step2: Find angle \( g \)

Vertical angles are equal, or angles on a straight line. \( g \) and \( 119^\circ \) are vertical? Wait, no, \( h \) and \( g \) with \( 119^\circ \): Wait, actually, \( g \) and \( 119^\circ \) are adjacent? Wait, no, \( h + 119^\circ = 180^\circ \), and \( g \) is vertical to the angle adjacent to \( h \)? Wait, no, let's re - look. The straight line, so \( h \) and \( 119^\circ \) are supplementary, so \( h = 61^\circ \), and \( g \) is equal to \( 119^\circ \) because they are vertical angles? Wait, no, wait the lines: the two intersecting lines, so vertical angles. Wait, the upper intersection: the angle \( 119^\circ \) and \( g \): Wait, no, \( h \) and \( 119^\circ \) are supplementary, so \( h = 61^\circ \), and \( g \) is equal to \( 119^\circ \) (vertical angles with the \( 119^\circ \) angle? Wait, no, let's correct. At the upper intersection, the two angles \( h \) and \( 119^\circ \) are supplementary (they form a linear pair), so \( h=180 - 119=61^\circ \). Then \( g \) is equal to \( 119^\circ \) because \( g \) and the \( 119^\circ \) angle are vertical angles? Wait, no, \( g \) and \( h \) are supplementary? Wait, no, the straight line, so \( h + g=180^\circ \)? No, wait the diagram: the vertical line and the slanted line intersect, creating two angles: \( h \) and \( 119^\circ \) which are adjacent and form a linear pair, so \( h + 119 = 180\), so \( h = 61^\circ \). Then \( g \) is vertical to the angle that is \( 119^\circ \)? Wait, no, \( g \) is adjacent to \( h \), so \( g=119^\circ \) (since \( h + g = 180\), no, \( h=61\), so \( g = 180 - 61=119^\circ \), yes, because they are supplementary.

Now for the lower intersection: the angle \( 121^\circ \) and \( m \) are supplementary (linear pair), so \( m=180 - 121 = 59^\circ \)? Wait, no, wait the other angle \( k \): \( k \) is equal to \( 121^\circ \) (vertical angles), and \( m \) is equal to \( 59^\circ \) (since \( m+121 = 180\)). Wait, but also, looking at the upper and lower slanted lines: are they parallel? Wait, the vertical line is a transversal. If we assume the slanted lines are parallel, then corresponding angles should be equal. Wait, but \( h = 61^\circ \) and \( m \) should be equal if the lines are parallel. But \( 180 - 121=59\), which is not 61. Wait, maybe I made a mistake. Wait, let's re - examine the problem.

Wait, the problem is about finding missing angles. Let's list the angles:

1. For angle \( h \):
- \( h \) and \( 119^\circ \) form a linear pair (they are adjacent and on a straight line), so \( h + 119^\circ=180^\circ \)
- \( h = 180^\circ- 119^\circ = 61^\circ \)

2. For angle \( g \):
- \( g \) and \( h \) form a linear pair (or \( g \) is vertical to the \( 119^\circ \) angle). Since \( h + g = 180^\circ \) (linear pair) or \( g = 119^\circ \) (vertical angles with the \( 119^\circ \) angle). Let's use linear pair: \( g=180^\circ - h=180 - 61 = 119^\circ \)

3. For angle \( m \):
- \( m \) and \( 121^\circ \) form a linear pair, so \( m + 121^\circ=180^\circ \)
- \( m=180^\circ - 121^\circ = 59^\circ \)? Wait, but this seems inconsistent. Wait, maybe the slanted lines are parallel, so \( h \) and \( m \) should be equal? But 61 and 59 are not equal. Wait, maybe I misread the angles. Wait, the upper angle is \( 119^\circ \), lower is \( 121^\circ \). Wait, \( 119+121 = 240\), no, that can't be. Wait, maybe the diagram has the vertical line, and two slanted lines (parallel) cut by the vertical transversal. Then the consecutive interior angles should be supplementary. Wait, \( 119 + 121=240\), which is not 180. So maybe the slanted lines are not parallel. So we just use linear pairs and vertical angles.

4. For angle \( k \):
- \( k \) and \( 121^\circ \) are vertical angles, so \( k = 121^\circ \)
- \( m \) and \( k \) are supplementary (linear pair), so \( m=180 - 121 = 59^\circ \)

Wait, let's summarize:

- Angle \( h \): Linear pair with \( 119^\circ \), so \( h = 180 - 119=61^\circ \)
- Angle \( g \): Linear pair with \( h \), so \( g = 180 - 61 = 119^\circ \) (or vertical angle with \( 119^\circ \))
- Angle \( m \): Linear pair with \( 121^\circ \), so \( m=180 - 121 = 59^\circ \)
- Angle \( k \): Vertical angle with \( 121^\circ \), so \( k = 121^\circ \)

Answer:

\( h = 61^\circ \), \( g = 119^\circ \), \( m = 59^\circ \), \( k = 121^\circ \)